```python
import numpy as np
import math
import scipy as sp
from scipy import linalg
import pandas as pd
import matplotlib.pyplot as plt
import sympy
```

```python
sp.info(sp.linalg.solve)
```

     solve(a, b, sym_pos=False, lower=False, overwrite_a=False, overwrite_b=False,
           debug=None, check_finite=True, assume_a='gen', transposed=False)

    Solves the linear equation set ``a * x = b`` for the unknown ``x``
    for square ``a`` matrix.

    If the data matrix is known to be a particular type then supplying the
    corresponding string to ``assume_a`` key chooses the dedicated solver.
    The available options are

    ===================  ========
     generic matrix       'gen'
     symmetric            'sym'
     hermitian            'her'
     positive definite    'pos'
    ===================  ========

    If omitted, ``'gen'`` is the default structure.

    The datatype of the arrays define which solver is called regardless
    of the values. In other words, even when the complex array entries have
    precisely zero imaginary parts, the complex solver will be called based
    on the data type of the array.

    Parameters
    ----------
    a : (N, N) array_like
        Square input data
    b : (N, NRHS) array_like
        Input data for the right hand side.
    sym_pos : bool, optional
        Assume `a` is symmetric and positive definite. This key is deprecated
        and assume_a = 'pos' keyword is recommended instead. The functionality
        is the same. It will be removed in the future.
    lower : bool, optional
        If True, only the data contained in the lower triangle of `a`. Default
        is to use upper triangle. (ignored for ``'gen'``)
    overwrite_a : bool, optional
        Allow overwriting data in `a` (may enhance performance).
        Default is False.
    overwrite_b : bool, optional
        Allow overwriting data in `b` (may enhance performance).
        Default is False.
    check_finite : bool, optional
        Whether to check that the input matrices contain only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
    assume_a : str, optional
        Valid entries are explained above.
    transposed: bool, optional
        If True, depending on the data type ``a^T x = b`` or ``a^H x = b`` is
        solved (only taken into account for ``'gen'``).

    Returns
    -------
    x : (N, NRHS) ndarray
        The solution array.

    Raises
    ------
    ValueError
        If size mismatches detected or input a is not square.
    LinAlgError
        If the matrix is singular.
    RuntimeWarning
        If an ill-conditioned input a is detected.

    Examples
    --------
    Given `a` and `b`, solve for `x`:

    >>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])
    >>> b = np.array([2, 4, -1])
    >>> from scipy import linalg
    >>> x = linalg.solve(a, b)
    >>> x
    array([ 2., -2.,  9.])
    >>> np.dot(a, x) == b
    array([ True,  True,  True], dtype=bool)

    Notes
    -----
    If the input b matrix is a 1D array with N elements, when supplied
    together with an NxN input a, it is assumed as a valid column vector
    despite the apparent size mismatch. This is compatible with the
    numpy.dot() behavior and the returned result is still 1D array.

    The generic, symmetric, hermitian and positive definite solutions are
    obtained via calling ?GESVX, ?SYSVX, ?HESVX, and ?POSVX routines of
    LAPACK respectively.

```python
sp.source(sp.linalg.lu)
```

    In file: /home/me/anaconda3/lib/python3.6/site-packages/scipy/linalg/decomp_lu.py

    def lu(a, permute_l=False, overwrite_a=False, check_finite=True):
        """
        Compute pivoted LU decomposition of a matrix.

        The decomposition is::

            A = P L U

        where P is a permutation matrix, L lower triangular with unit
        diagonal elements, and U upper triangular.

        Parameters
        ----------
        a : (M, N) array_like
            Array to decompose
        permute_l : bool, optional
            Perform the multiplication P*L  (Default: do not permute)
        overwrite_a : bool, optional
            Whether to overwrite data in a (may improve performance)
        check_finite : bool, optional
            Whether to check that the input matrix contains only finite numbers.
            Disabling may give a performance gain, but may result in problems
            (crashes, non-termination) if the inputs do contain infinities or NaNs.

        Returns
        -------
        **(If permute_l == False)**

        p : (M, M) ndarray
            Permutation matrix
        l : (M, K) ndarray
            Lower triangular or trapezoidal matrix with unit diagonal.
            K = min(M, N)
        u : (K, N) ndarray
            Upper triangular or trapezoidal matrix

        **(If permute_l == True)**

        pl : (M, K) ndarray
            Permuted L matrix.
            K = min(M, N)
        u : (K, N) ndarray
            Upper triangular or trapezoidal matrix

        Notes
        -----
        This is a LU factorization routine written for Scipy.

        """
        if check_finite:
            a1 = asarray_chkfinite(a)
        else:
            a1 = asarray(a)
        if len(a1.shape) != 2:
            raise ValueError('expected matrix')
        overwrite_a = overwrite_a or (_datacopied(a1, a))
        flu, = get_flinalg_funcs(('lu',), (a1,))
        p, l, u, info = flu(a1, permute_l=permute_l, overwrite_a=overwrite_a)
        if info < 0:
            raise ValueError('illegal value in %d-th argument of '
                                                'internal lu.getrf' % -info)
        if permute_l:
            return l, u
        return p, l, u

```python
dir(sp.linalg)
```

    ['LinAlgError',
     'Tester',
     '__all__',
     '__builtins__',
     '__cached__',
     '__doc__',
     '__file__',
     '__loader__',
     '__name__',
     '__package__',
     '__path__',
     '__spec__',
     '__version__',
     '_decomp_polar',
     '_decomp_qz',
     '_decomp_update',
     '_expm_frechet',
     '_fblas',
     '_flapack',
     '_flinalg',
     '_matfuncs_sqrtm',
     '_procrustes',
     '_solve_toeplitz',
     '_solvers',
     'absolute_import',
     'basic',
     'blas',
     'block_diag',
     'cho_factor',
     'cho_solve',
     'cho_solve_banded',
     'cholesky',
     'cholesky_banded',
     'circulant',
     'companion',
     'coshm',
     'cosm',
     'cython_blas',
     'cython_lapack',
     'decomp',
     'decomp_cholesky',
     'decomp_lu',
     'decomp_qr',
     'decomp_schur',
     'decomp_svd',
     'det',
     'dft',
     'diagsvd',
     'division',
     'eig',
     'eig_banded',
     'eigh',
     'eigvals',
     'eigvals_banded',
     'eigvalsh',
     'expm',
     'expm2',
     'expm3',
     'expm_cond',
     'expm_frechet',
     'find_best_blas_type',
     'flinalg',
     'fractional_matrix_power',
     'funm',
     'get_blas_funcs',
     'get_lapack_funcs',
     'hadamard',
     'hankel',
     'helmert',
     'hessenberg',
     'hilbert',
     'inv',
     'invhilbert',
     'invpascal',
     'kron',
     'lapack',
     'leslie',
     'linalg_version',
     'logm',
     'lstsq',
     'lu',
     'lu_factor',
     'lu_solve',
     'matfuncs',
     'matrix_balance',
     'misc',
     'norm',
     'ordqz',
     'orth',
     'orthogonal_procrustes',
     'pascal',
     'pinv',
     'pinv2',
     'pinvh',
     'polar',
     'print_function',
     'qr',
     'qr_delete',
     'qr_insert',
     'qr_multiply',
     'qr_update',
     'qz',
     'rq',
     'rsf2csf',
     'schur',
     'signm',
     'sinhm',
     'sinm',
     'solve',
     'solve_banded',
     'solve_circulant',
     'solve_continuous_are',
     'solve_discrete_are',
     'solve_discrete_lyapunov',
     'solve_lyapunov',
     'solve_sylvester',
     'solve_toeplitz',
     'solve_triangular',
     'solveh_banded',
     'special_matrices',
     'sqrtm',
     'svd',
     'svdvals',
     'tanhm',
     'tanm',
     'test',
     'toeplitz',
     'tri',
     'tril',
     'triu']

# Matrix Creation

```python
a = np.empty((4,5))
a
```

    array([[  6.93542489e-310,   2.13381972e-316,   6.93538267e-310,
              6.93542578e-310,   6.93538762e-310],
           [  6.93542582e-310,   6.93542582e-310,   6.93542582e-310,
              6.93538267e-310,   6.93540180e-310],
           [  6.93538751e-310,   4.00193173e-322,   2.12576764e-316,
              6.93542489e-310,   6.93542157e-310],
           [  6.93541501e-310,   6.93542583e-310,   6.93542143e-310,
              6.93541525e-310,   6.93538782e-310]])

```python
A = np.zeros((3,4))
A
```

    array([[ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.]])

```python
B = np.ones((4,2))
B
```

    array([[ 1.,  1.],
           [ 1.,  1.],
           [ 1.,  1.],
           [ 1.,  1.]])

```python
A = np.eye(3,3)
A
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.]])

```python
B = np.arange(9).reshape((3,3))
B
```

    array([[0, 1, 2],
           [3, 4, 5],
           [6, 7, 8]])

```python
C = np.fromfunction(lambda x, y: x + y, (3,3))
C
```

    array([[ 0.,  1.,  2.],
           [ 1.,  2.,  3.],
           [ 2.,  3.,  4.]])

```python
fn = lambda x: math.sin(x)
vfn = np.vectorize(fn)
D = vfn(np.arange(9)).reshape(3,3)
D
```

    array([[ 0.        ,  0.84147098,  0.90929743],
           [ 0.14112001, -0.7568025 , -0.95892427],
           [-0.2794155 ,  0.6569866 ,  0.98935825]])

```python
fn2 = lambda x: np.sin(x)
D = fn2(np.arange(9)).reshape(3,3)
D
```

    array([[ 0.        ,  0.84147098,  0.90929743],
           [ 0.14112001, -0.7568025 , -0.95892427],
           [-0.2794155 ,  0.6569866 ,  0.98935825]])

```python
map1 = map(math.sin, np.arange(9))
E = np.fromiter(map1, dtype = 'float').reshape(3,3)
E
```

    array([[ 0.        ,  0.84147098,  0.90929743],
           [ 0.14112001, -0.7568025 , -0.95892427],
           [-0.2794155 ,  0.6569866 ,  0.98935825]])

```python
F = np.array([math.sin(x) for x in range(9)]).reshape(3,3)
F
```

    array([[ 0.        ,  0.84147098,  0.90929743],
           [ 0.14112001, -0.7568025 , -0.95892427],
           [-0.2794155 ,  0.6569866 ,  0.98935825]])

## Time required in each 3x3 matrix creation

```python
%%timeit -n 5
fn = lambda x: math.sin(x)
vfn = np.vectorize(fn)
D = vfn(np.arange(9)).reshape(3,3)
```

    5 loops, best of 3: 28.7 µs per loop

```python
%%timeit -n 5
fn2 = lambda x: np.sin(x)
D = fn2(np.arange(9)).reshape(3,3)
```

    5 loops, best of 3: 5.22 µs per loop

```python
%%timeit -n 5
map1 = map(math.sin, np.arange(9))
E = np.fromiter(map1, dtype = 'float').reshape(3,3)
```

    5 loops, best of 3: 8.61 µs per loop

```python
%%timeit -n 5
F = np.array([math.sin(x) for x in range(9)]).reshape(3,3)
```

    5 loops, best of 3: 3.41 µs per loop

## Time required to create 100 x 100 matrix
Note: Numpy can has some overhead costs. Yet, applying numpy function to numpy array directly is fast for large matrix. (faster than vectorize non-numpy function / map non-numpy function / array of list comprehension)

```python
%%timeit -n 5
fn = lambda x: math.sin(x)
vfn = np.vectorize(fn)
D = vfn(np.arange(10000)).reshape(100,100)
```

    5 loops, best of 3: 2.82 ms per loop

```python
%%timeit -n 5
fn2 = lambda x: np.sin(x)
D = fn2(np.arange(10000)).reshape(100,100)
```

    5 loops, best of 3: 290 µs per loop

```python
%%timeit -n 5
map1 = map(math.sin, np.arange(10000))
E = np.fromiter(map1, dtype = 'float').reshape(100,100)
```

    5 loops, best of 3: 1.39 ms per loop

```python
%%timeit -n 5
F = np.array([math.sin(x) for x in range(10000)]).reshape(100,100)
```

    5 loops, best of 3: 1.82 ms per loop

# Matrix Concatenation

```python
A = np.eye(3,3)
B = np.zeros((3,3))
C = np.ones((3,3))
```

```python
A
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.]])

```python
B
```

    array([[ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.]])

```python
C
```

    array([[ 1.,  1.,  1.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.]])

```python
A + B # not for concatenation
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.]])

```python
np.concatenate([A,B,C])
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.]])

```python
np.vstack([A,B,C])
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.]])

```python
np.r_[A,B,C]
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 0.,  0.,  0.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.],
           [ 1.,  1.,  1.]])

```python
np.concatenate([A,B,C],axis = 1)
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  1.,  1.,  1.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  1.,  1.,  1.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  1.,  1.,  1.]])

```python
np.hstack([A,B,C])
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  1.,  1.,  1.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  1.,  1.,  1.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  1.,  1.,  1.]])

```python
np.c_[A,B,C]
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  1.,  1.,  1.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  1.,  1.,  1.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  1.,  1.,  1.]])

# Row Swapping

```python
A = np.eye(4,4)
A
```

    array([[ 1.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
A[0], A[3] = A[3], A[0]  #does not swap!
```

```python
A
```

    array([[ 0.,  0.,  0.,  1.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
A = np.eye(4,4)
```

```python
A[0,:], A[3,:] = A[3,:], A[0,:] #does not swap either!
```

```python
A
```

    array([[ 0.,  0.,  0.,  1.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
A = np.eye(4,4)
A[[0,3]]
```

    array([[ 1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
A[[0,3]] = A[[3,0]]  #Will swap!
A
```

    array([[ 0.,  0.,  0.,  1.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 1.,  0.,  0.,  0.]])

# Column Swapping

```python
A = np.arange(16).reshape((4,4))
A
```

    array([[ 0,  1,  2,  3],
           [ 4,  5,  6,  7],
           [ 8,  9, 10, 11],
           [12, 13, 14, 15]])

```python
A[:,[0,3]] = A[:,[3,0]]
```

```python
A
```

    array([[ 3,  1,  2,  0],
           [ 7,  5,  6,  4],
           [11,  9, 10,  8],
           [15, 13, 14, 12]])

# Matrix Expansion

```python
A = np.eye(3,3)
A
```

    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.]])

```python
A[:,:,np.newaxis]
```

    array([[[ 1.],
            [ 0.],
            [ 0.]],

           [[ 0.],
            [ 1.],
            [ 0.]],

           [[ 0.],
            [ 0.],
            [ 1.]]])

```python
A[:,np.newaxis,:]
```

    array([[[ 1.,  0.,  0.]],

           [[ 0.,  1.,  0.]],

           [[ 0.,  0.,  1.]]])

```python
A[np.newaxis,:,:]
```

    array([[[ 1.,  0.,  0.],
            [ 0.,  1.,  0.],
            [ 0.,  0.,  1.]]])

# Element by element operation

```python
%%timeit -n 5
A = np.arange(100)
B = np.arange(100)
C = A * B
```

    The slowest run took 4.87 times longer than the fastest. This could mean that an intermediate result is being cached.
    5 loops, best of 3: 4.87 µs per loop

```python
np.__version__
```

    '1.13.1'

```python
%%timeit -n 5
a = range(100)
b = range(100)
c = []
for i in range(len(a)):
    c.append(a[i]*b[i])
```

    5 loops, best of 3: 27.2 µs per loop

```python
A = np.arange(100)
B = np.arange(100)
C = A * B
C
```

    array([   0,    1,    4,    9,   16,   25,   36,   49,   64,   81,  100,
            121,  144,  169,  196,  225,  256,  289,  324,  361,  400,  441,
            484,  529,  576,  625,  676,  729,  784,  841,  900,  961, 1024,
           1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849,
           1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916,
           3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4096, 4225,
           4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776,
           5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569,
           7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409, 9604,
           9801])

```python
a = range(100)
b = range(100)
c = []
for i in range(len(a)):
    c.append(a[i]*b[i])
c
```

    [0,
     1,
     4,
     9,
     16,
     25,
     36,
     49,
     64,
     81,
     100,
     121,
     144,
     169,
     196,
     225,
     256,
     289,
     324,
     361,
     400,
     441,
     484,
     529,
     576,
     625,
     676,
     729,
     784,
     841,
     900,
     961,
     1024,
     1089,
     1156,
     1225,
     1296,
     1369,
     1444,
     1521,
     1600,
     1681,
     1764,
     1849,
     1936,
     2025,
     2116,
     2209,
     2304,
     2401,
     2500,
     2601,
     2704,
     2809,
     2916,
     3025,
     3136,
     3249,
     3364,
     3481,
     3600,
     3721,
     3844,
     3969,
     4096,
     4225,
     4356,
     4489,
     4624,
     4761,
     4900,
     5041,
     5184,
     5329,
     5476,
     5625,
     5776,
     5929,
     6084,
     6241,
     6400,
     6561,
     6724,
     6889,
     7056,
     7225,
     7396,
     7569,
     7744,
     7921,
     8100,
     8281,
     8464,
     8649,
     8836,
     9025,
     9216,
     9409,
     9604,
     9801]

# Matrix - Matrix Multiplication

Time for numpy

```python
%%timeit -n 5
A = np.array([[1,2],[3,4]])
np.dot(A,A)
```

    5 loops, best of 3: 3.63 µs per loop

```python
A = np.array([[1,2],[3,4]])
np.dot(A,A)
```

    array([[ 7, 10],
           [15, 22]])

```python
A.dot(A)
```

    array([[ 7, 10],
           [15, 22]])

Time with no direct multiplication function

```python
%%timeit -n 5
B = [[1,2],[3,4]]
C = []
for i in range(2):
    C.append([])
    for j in range(2):
        sum = 0
        for k in range(2):
            sum = sum + B[i][k] * B[k][j]
        C[i].append(sum)
```

    5 loops, best of 3: 3.6 µs per loop

```python
B = [[1,2],[3,4]]
C = []
for i in range(2):
    C.append([])
    for j in range(2):
        sum = 0
        for k in range(2):
            sum = sum + B[i][k] * B[k][j]
        C[i].append(sum)

#printing part
N = 2
print('[', end = '')
for i in range(N):
    if i != 0:
        print(' ', end = '')
    print('[', end = '')
    for j in range(N):
        print('{:2d}'.format(C[i][j]), end = '')
        if j != N-1:
            print(', ', end = '')
    print(']', end = '')
    if i != N-1:
        print(',')
print(']', end = '')
```

    [[ 7, 10],
     [15, 22]]

4 x 4 Matrix Multiplication

```python
np.random.seed(1)
A = np.ceil(10*np.random.random((4,4)))
np.random.seed(2)
B = np.ceil(10*np.random.random((4,4)))
print('A = ')
print(A, end = '\n\n')
print('B = ')
print(B)
```

    A =
    [[ 5.  8.  1.  4.]
     [ 2.  1.  2.  4.]
     [ 4.  6.  5.  7.]
     [ 3.  9.  1.  7.]]

    B =
    [[ 5.  1.  6.  5.]
     [ 5.  4.  3.  7.]
     [ 3.  3.  7.  6.]
     [ 2.  6.  2.  8.]]

```python
np.dot(A,B)
```

    array([[  76.,   64.,   69.,  119.],
           [  29.,   36.,   37.,   61.],
           [  79.,   85.,   91.,  148.],
           [  77.,   84.,   66.,  140.]])

```python
ans = np.zeros((4,4))
for i in range(4):
    for j in range(4):
        sum = 0
        for k in range(4):
            sum = sum + A[i,k] * B[k,j]
        ans[i,j] = sum
ans
```

    array([[  76.,   64.,   69.,  119.],
           [  29.,   36.,   37.,   61.],
           [  79.,   85.,   91.,  148.],
           [  77.,   84.,   66.,  140.]])

Pythonic way for matrix multiplication

```python
%%timeit -n 5
np.dot(A,B)
```

    The slowest run took 54.15 times longer than the fastest. This could mean that an intermediate result is being cached.
    5 loops, best of 3: 1.36 µs per loop

non-Pythonic way for matrix multiplication

```python
%%timeit -n 5
ans = np.zeros((4,4))
for i in range(4):
    for j in range(4):
        sum = 0
        for k in range(4):
            sum = sum + A[i,k] * B[k,j]
        ans[i,j] = sum
ans
```

    5 loops, best of 3: 32.8 µs per loop

# Gaussian Elimination

```python
np.random.seed(2)
a = np.ceil(10*np.random.random((4,4)))
a
```

    array([[ 5.,  1.,  6.,  5.],
           [ 5.,  4.,  3.,  7.],
           [ 3.,  3.,  7.,  6.],
           [ 2.,  6.,  2.,  8.]])

```python
b = np.floor(10*np.random.random((4,1)))
b
```

    array([[ 8.],
           [ 4.],
           [ 8.],
           [ 0.]])

```python
A = np.hstack((a,b))
A
```

    array([[ 5.,  1.,  6.,  5.,  8.],
           [ 5.,  4.,  3.,  7.,  4.],
           [ 3.,  3.,  7.,  6.,  8.],
           [ 2.,  6.,  2.,  8.,  0.]])

```python
A.shape[0]
```

    4

### Use:   $L_{ij}^{new} = \left[L_{ij} - \frac{L_{ik}}{L_{kk}} L_{kj}\right]^{old}$

```python
N = A.shape[0]
for k in range(N-1):
    for i in range(k+1, N):
        r = -A[i,k] / A[k,k]
        for j in range(k+1, N+1):
            A[i,j] = A[i,j] + r * A[k,j]
        #lines below are not used during back substitution
        for j in range(k+1):
            A[i,j] = 0
```

```python
A
```

    array([[ 5.        ,  1.        ,  6.        ,  5.        ,  8.        ],
           [ 0.        ,  3.        , -3.        ,  2.        , -4.        ],
           [ 0.        ,  0.        ,  5.8       ,  1.4       ,  6.4       ],
           [ 0.        ,  0.        ,  0.        ,  1.01149425, -1.47126437]])

# Back Substitution:   $x_i = \frac{a_{im} - \sum_{j=i+1}^n a_{ij}x_j}{a_{ii}}$

```python
#For ax = b
# A = np.hstack((a,b))
A[N-1,N] = A[N-1,-1] / A[N-1, -2]
for i in range(N-2, -1, -1): #2, 1, 0
    sum = 0
    for j in range(i+1, N): #i+1 to N-1
        sum = sum + A[i,j] * A[j,N]
    A[i,N] = (A[i,N] - sum)/A[i,i]
A[:,N]
```

    array([ 1.09090909,  1.09090909,  1.45454545, -1.45454545])

```python
ans = A[:,N]
ans
```

    array([ 1.09090909,  1.09090909,  1.45454545, -1.45454545])

```python
ans = ans[:,np.newaxis]
ans
```

    array([[ 1.09090909],
           [ 1.09090909],
           [ 1.45454545],
           [-1.45454545]])

## Check answer Ax = b, by calculating A* ans

```python
np.dot(a,ans)
```

    array([[ 8.],
           [ 4.],
           [ 8.],
           [ 0.]])

```python
b
```

    array([[ 8.],
           [ 4.],
           [ 8.],
           [ 0.]])

### Answer from Scipy

```python
sp.__version__
```

    '0.19.1'

```python
ans_sp = linalg.solve(a,b)
ans_sp
```

    array([[ 1.09090909],
           [ 1.09090909],
           [ 1.45454545],
           [-1.45454545]])

```python
ans - ans_sp
```

    array([[  2.22044605e-16],
           [ -6.66133815e-16],
           [ -2.22044605e-16],
           [  4.44089210e-16]])

```python
a.dot(ans) - b
```

    array([[  0.00000000e+00],
           [ -8.88178420e-16],
           [ -1.77635684e-15],
           [  0.00000000e+00]])

```python
a.dot(ans_sp) - b
```

    array([[ -1.77635684e-15],
           [ -3.55271368e-15],
           [ -1.77635684e-15],
           [ -1.77635684e-15]])

### Up-scaling to 10 x 10

```python
def user_gaussian_solve(a,b):
    A = np.hstack((a,b))
    N = A.shape[0]
    for k in range(N-1):
        for i in range(k+1, N):
            r = -A[i,k] / A[k,k]
            for j in range(k+1, N+1):
                A[i,j] = A[i,j] + r * A[k,j]
            #lines below are not used during back substitution
    A[N-1,N] = A[N-1,-1] / A[N-1, -2]
    for i in range(N-2, -1, -1): #2, 1, 0
        sum = 0
        for j in range(i+1, N): #i+1 to N-1
            sum = sum + A[i,j] * A[j,N]
        A[i,N] = (A[i,N] - sum)/A[i,i]
    return A[:,N][:,np.newaxis]
```

```python
x = user_gaussian_solve(a,b)
a.dot(x) - b
```

    array([[  0.00000000e+00],
           [ -8.88178420e-16],
           [ -1.77635684e-15],
           [  0.00000000e+00]])

```python
x
```

    array([[ 1.09090909],
           [ 1.09090909],
           [ 1.45454545],
           [-1.45454545]])

```python
np.random.seed(2)
a = np.ceil(10*np.random.random((10,10)))
a
```

    array([[  5.,   1.,   6.,   5.,   5.,   4.,   3.,   7.,   3.,   3.],
           [  7.,   6.,   2.,   6.,   2.,   8.,   9.,   5.,   9.,   1.],
           [  6.,   1.,   5.,   1.,   2.,   6.,   3.,   2.,   3.,   4.],
           [  5.,   3.,   7.,   5.,   6.,   4.,   8.,   6.,   2.,   8.],
           [ 10.,   6.,   9.,   4.,   6.,   5.,   5.,   8.,   6.,  10.],
           [  6.,   1.,   4.,   9.,   5.,   1.,   3.,   1.,  10.,  10.],
           [  9.,   7.,   8.,   2.,   3.,   6.,   4.,   1.,  10.,   5.],
           [  6.,   4.,   3.,   4.,   9.,   8.,   4.,   1.,   8.,   3.],
           [  6.,   1.,   7.,   4.,   5.,   5.,   4.,   6.,  10.,   2.],
           [  4.,   1.,   8.,   7.,   3.,   5.,   7.,   7.,   2.,   9.]])

```python
b = np.ceil(10*np.random.random((10,1)))
b
```

    array([[  8.],
           [  2.],
           [  9.],
           [  8.],
           [  8.],
           [  6.],
           [  3.],
           [ 10.],
           [  6.],
           [  4.]])

```python
x = user_gaussian_solve(a,b)
a.dot(x) - b
```

    array([[  1.77635684e-15],
           [  2.66453526e-15],
           [  1.77635684e-15],
           [  0.00000000e+00],
           [  0.00000000e+00],
           [ -8.88178420e-16],
           [ -4.44089210e-16],
           [ -3.37507799e-14],
           [  1.77635684e-15],
           [  8.88178420e-15]])

```python
x
```

    array([[ 2.07221037],
           [-1.39430277],
           [-0.12657769],
           [-0.01830467],
           [ 1.07867709],
           [-0.07065234],
           [ 0.4425574 ],
           [-0.48488272],
           [-0.7342585 ],
           [-0.31908441]])

```python
x2 = linalg.solve(a,b)
a.dot(x2) - b
```

    array([[  0.00000000e+00],
           [ -8.88178420e-16],
           [  0.00000000e+00],
           [  1.77635684e-15],
           [  3.55271368e-15],
           [ -8.88178420e-16],
           [  4.88498131e-15],
           [ -1.77635684e-15],
           [ -1.77635684e-15],
           [ -8.88178420e-16]])

```python
x2
```

    array([[ 2.07221037],
           [-1.39430277],
           [-0.12657769],
           [-0.01830467],
           [ 1.07867709],
           [-0.07065234],
           [ 0.4425574 ],
           [-0.48488272],
           [-0.7342585 ],
           [-0.31908441]])

```python
%%timeit -n 5
user_gaussian_solve(a,b)
```

    5 loops, best of 3: 208 µs per loop

```python
%%timeit -n 5
linalg.solve(a,b)
```

    5 loops, best of 3: 59.7 µs per loop

# Partial Pivoting: max( abs(diagonal)) on each step of elimination
Find the maximum value that is on the same column as the pivot element, then swap

```python
np.random.seed(1)
a2 = np.ceil(10*np.random.random((4,4)))
a2[3,:] = a2[3,:] * 10000
a2
```

    array([[  5.00000000e+00,   8.00000000e+00,   1.00000000e+00,
              4.00000000e+00],
           [  2.00000000e+00,   1.00000000e+00,   2.00000000e+00,
              4.00000000e+00],
           [  4.00000000e+00,   6.00000000e+00,   5.00000000e+00,
              7.00000000e+00],
           [  3.00000000e+04,   9.00000000e+04,   1.00000000e+04,
              7.00000000e+04]])

```python
np.random.seed(1)
b2 = np.ceil(10*np.random.random((4,1)))
b2
```

    array([[ 5.],
           [ 8.],
           [ 1.],
           [ 4.]])

```python
ans2 = user_gaussian_solve(a2,b2)
```

```python
a2.dot(ans2) - b2
```

    array([[  1.77635684e-15],
           [  0.00000000e+00],
           [  1.77635684e-15],
           [  1.45519152e-11]])

```python
ans2b = linalg.solve(a2,b2)
```

```python
a2.dot(ans2b) - b2
```

    array([[  8.88178420e-16],
           [  0.00000000e+00],
           [ -5.32907052e-15],
           [ -5.82076609e-11]])

```python
np.random.seed(1)
a2 = np.floor(10*np.random.random((10,10))) - 6
a2[0,0] = 0
```

```python
a2
```

    array([[ 0.,  1., -6., -3., -5., -6., -5., -3., -3., -1.],
           [-2.,  0., -4.,  2., -6.,  0., -2., -1., -5., -5.],
           [ 2.,  3., -3.,  0.,  2.,  2., -6., -6., -5.,  2.],
           [-6., -2.,  3., -1.,  0., -3.,  0.,  2., -6.,  1.],
           [ 3.,  1., -4.,  1., -5., -2.,  3., -4., -4., -5.],
           [-6.,  0., -4., -4., -2., -6., -1., -5., -1.,  0.],
           [-5., -2.,  0., -2., -6., -1.,  0., -1.,  3., -1.],
           [ 3., -5., -5.,  2., -3., -5.,  3., -3.,  1.,  1.],
           [ 2.,  0.,  1., -3., -4.,  2., -2.,  3.,  0.,  0.],
           [-5.,  3., -2., -1., -2., -4.,  3., -1., -6.,  0.]])

```python
b
```

    array([[  8.],
           [  2.],
           [  9.],
           [  8.],
           [  8.],
           [  6.],
           [  3.],
           [ 10.],
           [  6.],
           [  4.]])

```python
linalg.solve(a2,b)
```

    array([[ 0.77432279],
           [-1.39446307],
           [ 0.61638638],
           [-1.23315046],
           [-1.0049791 ],
           [ 0.19168532],
           [ 0.28772884],
           [-1.2307709 ],
           [-1.45290852],
           [ 1.09384768]])

```python
user_gaussian_solve(a2,b)
```

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: divide by zero encountered in double_scalars

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in double_scalars

    array([[ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan]])

### Get "<font color=#FF0000> Not A Number </font>" (nan) because of zeros in the diagonal terms

```python
def user_gaussian_solve_pp(a,b):
    A = np.hstack((a,b))
    N = A.shape[0]
    for k in range(N-1):
        maxidx = np.abs(A[k:,k]).argmax() + k #get index of the max arg
        # +k is needed, because, argmax restart at 0 for the new slice
        A[[k,maxidx]] = A[[maxidx, k]]
        for i in range(k+1, N):
            r = -A[i,k] / A[k,k]
            for j in range(k+1, N+1):
                A[i,j] = A[i,j] + r * A[k,j]

    A[N-1,N] = A[N-1,-1] / A[N-1, -2]
    for i in range(N-2, -1, -1): #2, 1, 0
        sum = 0
        for j in range(i+1, N): #i+1 to N-1
            sum = sum + A[i,j] * A[j,N]
        A[i,N] = (A[i,N] - sum)/A[i,i]
    return A[:,N][:,np.newaxis]
```

```python
ans = user_gaussian_solve_pp(a2,b)
ans
```

    array([[ 0.77432279],
           [-1.39446307],
           [ 0.61638638],
           [-1.23315046],
           [-1.0049791 ],
           [ 0.19168532],
           [ 0.28772884],
           [-1.2307709 ],
           [-1.45290852],
           [ 1.09384768]])

```python
err_usr = a2.dot(ans)-b
err_usr
```

    array([[ -2.66453526e-15],
           [  0.00000000e+00],
           [ -3.55271368e-15],
           [  0.00000000e+00],
           [  1.77635684e-15],
           [ -2.66453526e-15],
           [  0.00000000e+00],
           [ -1.77635684e-15],
           [  8.88178420e-16],
           [  5.32907052e-15]])

```python
ans_sp = linalg.solve(a2,b)
err_sp = a2.dot(ans_sp)-b
err_sp
```

    array([[  0.00000000e+00],
           [  8.88178420e-16],
           [  0.00000000e+00],
           [  0.00000000e+00],
           [  1.77635684e-15],
           [  0.00000000e+00],
           [  2.66453526e-15],
           [  0.00000000e+00],
           [  8.88178420e-16],
           [  2.66453526e-15]])

```python
ans_sp
```

    array([[ 0.77432279],
           [-1.39446307],
           [ 0.61638638],
           [-1.23315046],
           [-1.0049791 ],
           [ 0.19168532],
           [ 0.28772884],
           [-1.2307709 ],
           [-1.45290852],
           [ 1.09384768]])

```python
print('err scipy', np.linalg.norm(err_sp))
print('err user ', np.linalg.norm(err_usr))
```

    err scipy 4.35116785763e-15
    err user  7.89430247156e-15

# Checking the condition number of A (square matrix)
well conditioned: 1
<br>ill conditioned: very large number
## For Ax = b, if x change a little (contain a little error), what will happen to b
## $A(x+\delta x) = b + \delta b$
## $A \delta x = \delta b$
## $\frac{\|\delta x\|}{\|x\|} = \frac{\|A^{-1}\delta b\|}{\|x\|} <= \frac{\|A^{-1}\|\|\delta b\|}{\|x\|}$
## $\frac{\|A^{-1}\|\|\delta b\|}{\|x\|} \cdot \frac{\|b\|}{\|b\|} =
\frac{\|A^{-1}\|\| b\|}{\|x\|} \cdot \frac{\| \delta b\|}{\|b\|} = \|A^{-1}\|\cdot\|A\|\cdot \frac{\| \delta b\|}{\|b\|}$
## $\therefore \; \frac{\|\delta x\|}{\|x\|} <= \|A^{-1}\|\cdot\|A\|\cdot \frac{\| \delta b\|}{\|b\|}$
## $\|A^{-1}\|\cdot\|A\|$ is the condition number of matrix A
### The relative error in x (answer) is bounded by the relative error in b * condition number

```python
a2
```

    array([[ 0.,  1., -6., -3., -5., -6., -5., -3., -3., -1.],
           [-2.,  0., -4.,  2., -6.,  0., -2., -1., -5., -5.],
           [ 2.,  3., -3.,  0.,  2.,  2., -6., -6., -5.,  2.],
           [-6., -2.,  3., -1.,  0., -3.,  0.,  2., -6.,  1.],
           [ 3.,  1., -4.,  1., -5., -2.,  3., -4., -4., -5.],
           [-6.,  0., -4., -4., -2., -6., -1., -5., -1.,  0.],
           [-5., -2.,  0., -2., -6., -1.,  0., -1.,  3., -1.],
           [ 3., -5., -5.,  2., -3., -5.,  3., -3.,  1.,  1.],
           [ 2.,  0.,  1., -3., -4.,  2., -2.,  3.,  0.,  0.],
           [-5.,  3., -2., -1., -2., -4.,  3., -1., -6.,  0.]])

```python
np.linalg.cond(a2)
```

    18.062377254421843

```python
a
```

    array([[  5.,   1.,   6.,   5.,   5.,   4.,   3.,   7.,   3.,   3.],
           [  7.,   6.,   2.,   6.,   2.,   8.,   9.,   5.,   9.,   1.],
           [  6.,   1.,   5.,   1.,   2.,   6.,   3.,   2.,   3.,   4.],
           [  5.,   3.,   7.,   5.,   6.,   4.,   8.,   6.,   2.,   8.],
           [ 10.,   6.,   9.,   4.,   6.,   5.,   5.,   8.,   6.,  10.],
           [  6.,   1.,   4.,   9.,   5.,   1.,   3.,   1.,  10.,  10.],
           [  9.,   7.,   8.,   2.,   3.,   6.,   4.,   1.,  10.,   5.],
           [  6.,   4.,   3.,   4.,   9.,   8.,   4.,   1.,   8.,   3.],
           [  6.,   1.,   7.,   4.,   5.,   5.,   4.,   6.,  10.,   2.],
           [  4.,   1.,   8.,   7.,   3.,   5.,   7.,   7.,   2.,   9.]])

```python
np.linalg.cond(a)
```

    40.145162515000834

```python
c_num_test = a + 50 * np.eye((a.shape[0]))
c_num_test
```

    array([[ 55.,   1.,   6.,   5.,   5.,   4.,   3.,   7.,   3.,   3.],
           [  7.,  56.,   2.,   6.,   2.,   8.,   9.,   5.,   9.,   1.],
           [  6.,   1.,  55.,   1.,   2.,   6.,   3.,   2.,   3.,   4.],
           [  5.,   3.,   7.,  55.,   6.,   4.,   8.,   6.,   2.,   8.],
           [ 10.,   6.,   9.,   4.,  56.,   5.,   5.,   8.,   6.,  10.],
           [  6.,   1.,   4.,   9.,   5.,  51.,   3.,   1.,  10.,  10.],
           [  9.,   7.,   8.,   2.,   3.,   6.,  54.,   1.,  10.,   5.],
           [  6.,   4.,   3.,   4.,   9.,   8.,   4.,  51.,   8.,   3.],
           [  6.,   1.,   7.,   4.,   5.,   5.,   4.,   6.,  60.,   2.],
           [  4.,   1.,   8.,   7.,   3.,   5.,   7.,   7.,   2.,  59.]])

```python
np.linalg.cond(c_num_test,p='fro') # p = 'fro' is just to select Frobenius norm
```

    11.250797180551318

## large diagonal term make it easier to get the answer, better accuracy (lower condition number)

```python
np.linalg.norm(sp.linalg.inv(c_num_test)) * np.linalg.norm(c_num_test)
```

    11.250797180551317

<font size = '4', style = "line-height:2">
Frobenius norm is defined as
<br>$\|A\|_{\rm F}=\sqrt{\sum_{i=1}^m\sum_{j=1}^n |a_{ij}|^2}=\sqrt{\operatorname{trace}(A^{\dagger}A)}$
<br> dagger is conjugate transpose (or just transpose for real number)
<br> trace is summation of the diagonal term

```python
np.linalg.norm(c_num_test)
```

    183.01639270841287

```python
np.linalg.norm(sp.linalg.inv(c_num_test))
```

    0.061474259294774868

```python
for row in c_num_test:
    print(row)
```

    [ 55.   1.   6.   5.   5.   4.   3.   7.   3.   3.]
    [  7.  56.   2.   6.   2.   8.   9.   5.   9.   1.]
    [  6.   1.  55.   1.   2.   6.   3.   2.   3.   4.]
    [  5.   3.   7.  55.   6.   4.   8.   6.   2.   8.]
    [ 10.   6.   9.   4.  56.   5.   5.   8.   6.  10.]
    [  6.   1.   4.   9.   5.  51.   3.   1.  10.  10.]
    [  9.   7.   8.   2.   3.   6.  54.   1.  10.   5.]
    [  6.   4.   3.   4.   9.   8.   4.  51.   8.   3.]
    [  6.   1.   7.   4.   5.   5.   4.   6.  60.   2.]
    [  4.   1.   8.   7.   3.   5.   7.   7.   2.  59.]

```python
#this is non-pythonic way.
#use just to show the direct calculation
sum_ = 0
for row in c_num_test:
    for i in row:
        sum_ = sum_ + i**2
sum_**0.5
```

    183.01639270841287

```python
#this is non-pythonic way.
#use just to show the direct calculation
sum_ = 0
for i in range(c_num_test.shape[0]):
    dia = c_num_test.dot(c_num_test.T)[i,i]
    sum_ += dia
sum_**0.5
```

    183.01639270841287

## Condition number before and after partial pivoting is the same

```python
def partial_pivoting(a):
    A = a.copy()
    N = A.shape[0]
    p = np.eye(N,N)
    for k in range(N-1):
        maxidx = np.abs(A[k:,k]).argmax() + k #get index of the max arg
        # +k is needed, because, argmax restart at 0 for the new slice
        if (k != maxidx):
            p[[k, maxidx]] = p[[maxidx, k]]
            A[[k, maxidx]] = A[[maxidx, k]]
    return A, p

```

```python
np.random.seed(1)
a2 = np.floor(10*np.random.random((10,10))) - 6
a2[0,0] = 0
a2
```

    array([[ 0.,  1., -6., -3., -5., -6., -5., -3., -3., -1.],
           [-2.,  0., -4.,  2., -6.,  0., -2., -1., -5., -5.],
           [ 2.,  3., -3.,  0.,  2.,  2., -6., -6., -5.,  2.],
           [-6., -2.,  3., -1.,  0., -3.,  0.,  2., -6.,  1.],
           [ 3.,  1., -4.,  1., -5., -2.,  3., -4., -4., -5.],
           [-6.,  0., -4., -4., -2., -6., -1., -5., -1.,  0.],
           [-5., -2.,  0., -2., -6., -1.,  0., -1.,  3., -1.],
           [ 3., -5., -5.,  2., -3., -5.,  3., -3.,  1.,  1.],
           [ 2.,  0.,  1., -3., -4.,  2., -2.,  3.,  0.,  0.],
           [-5.,  3., -2., -1., -2., -4.,  3., -1., -6.,  0.]])

```python
a2_pp, p = partial_pivoting(a2)
print(a2_pp)
```

    [[-6. -2.  3. -1.  0. -3.  0.  2. -6.  1.]
     [ 3. -5. -5.  2. -3. -5.  3. -3.  1.  1.]
     [ 0.  1. -6. -3. -5. -6. -5. -3. -3. -1.]
     [-6.  0. -4. -4. -2. -6. -1. -5. -1.  0.]
     [-5. -2.  0. -2. -6. -1.  0. -1.  3. -1.]
     [-5.  3. -2. -1. -2. -4.  3. -1. -6.  0.]
     [ 2.  3. -3.  0.  2.  2. -6. -6. -5.  2.]
     [ 3.  1. -4.  1. -5. -2.  3. -4. -4. -5.]
     [-2.  0. -4.  2. -6.  0. -2. -1. -5. -5.]
     [ 2.  0.  1. -3. -4.  2. -2.  3.  0.  0.]]

```python
np.linalg.cond(a2,p='fro')
```

    33.563995874635459

```python
np.linalg.cond(a2_pp,p='fro')
```

    33.563995874635467

```python
a2
```

    array([[ 0.,  1., -6., -3., -5., -6., -5., -3., -3., -1.],
           [-2.,  0., -4.,  2., -6.,  0., -2., -1., -5., -5.],
           [ 2.,  3., -3.,  0.,  2.,  2., -6., -6., -5.,  2.],
           [-6., -2.,  3., -1.,  0., -3.,  0.,  2., -6.,  1.],
           [ 3.,  1., -4.,  1., -5., -2.,  3., -4., -4., -5.],
           [-6.,  0., -4., -4., -2., -6., -1., -5., -1.,  0.],
           [-5., -2.,  0., -2., -6., -1.,  0., -1.,  3., -1.],
           [ 3., -5., -5.,  2., -3., -5.,  3., -3.,  1.,  1.],
           [ 2.,  0.,  1., -3., -4.,  2., -2.,  3.,  0.,  0.],
           [-5.,  3., -2., -1., -2., -4.,  3., -1., -6.,  0.]])

```python
a2_pp
```

    array([[-6., -2.,  3., -1.,  0., -3.,  0.,  2., -6.,  1.],
           [ 3., -5., -5.,  2., -3., -5.,  3., -3.,  1.,  1.],
           [ 0.,  1., -6., -3., -5., -6., -5., -3., -3., -1.],
           [-6.,  0., -4., -4., -2., -6., -1., -5., -1.,  0.],
           [-5., -2.,  0., -2., -6., -1.,  0., -1.,  3., -1.],
           [-5.,  3., -2., -1., -2., -4.,  3., -1., -6.,  0.],
           [ 2.,  3., -3.,  0.,  2.,  2., -6., -6., -5.,  2.],
           [ 3.,  1., -4.,  1., -5., -2.,  3., -4., -4., -5.],
           [-2.,  0., -4.,  2., -6.,  0., -2., -1., -5., -5.],
           [ 2.,  0.,  1., -3., -4.,  2., -2.,  3.,  0.,  0.]])

<font size = '4', style = "line-height:2">
In the above case, the condition number does not change after row switching
<br> Can condition number change after just row switching?
<br> No.
<br>Scaling / adding with other row can change the condition number.
<br> https://math.stackexchange.com/questions/149516/effects-of-elementary-row-operation-on-condition-number
</font>

```python
p
```

    array([[ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.]])

```python
np.linalg.cond(p)
```

    1.0

```python
p2, _ = partial_pivoting(p)
```

```python
p2
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.]])

```python
np.linalg.cond(p2)
```

    1.0

```python
p[[2]] = p[[2]] * 5
p
```

    array([[ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 5.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.]])

```python
np.linalg.cond(p)
```

    5.0

```python
p[[2]] = p[[2]] + p[[4]] * -5
p
```

    array([[ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 5.,  0.,  0.,  0.,  0.,  0., -5.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.]])

```python
np.linalg.cond(p)
```

    10.100999900019994

# LU Decomposition

```python
np.random.seed(1)
B = np.ceil(10*np.random.random((4,4))) + 10 * np.eye(4,4)
B
```

    array([[ 15.,   8.,   1.,   4.],
           [  2.,  11.,   2.,   4.],
           [  4.,   6.,  15.,   7.],
           [  3.,   9.,   1.,  17.]])

```python
p, l, u = sp.linalg.lu(B ,permute_l = False)
print(p, end = '\n\n')
print(l, end = '\n\n')
print(u, end = '\n\n')
```

    [[ 1.  0.  0.  0.]
     [ 0.  1.  0.  0.]
     [ 0.  0.  1.  0.]
     [ 0.  0.  0.  1.]]

    [[ 1.          0.          0.          0.        ]
     [ 0.13333333  1.          0.          0.        ]
     [ 0.26666667  0.38926174  1.          0.        ]
     [ 0.2         0.74496644 -0.04216579  1.        ]]

    [[ 15.           8.           1.           4.        ]
     [  0.           9.93333333   1.86666667   3.46666667]
     [  0.           0.          14.00671141   4.58389262]
     [  0.           0.           0.          13.81073311]]

```python
i = 2
j = 2
B[0:i,j].dot(B[i,0:i])
```

    16.0

```python
B[0:i,j]
```

    array([ 1.,  2.])

```python
B[i,0:i]
```

    array([ 4.,  6.])

```python
N = B.shape[0]
A = B.copy()
for i in range(1, N):
    #loop for L
    for j in range(i): #A[a:b] is [a,b) not [a,b]
        Sum = A[0:j,j].dot(A[i,0:j])
        A[i,j] = (A[i,j] - Sum)/A[j,j]
    #loop for U
    for j in range(i,N):
        Sum = A[0:i,j].dot(A[i,0:i])
        A[i,j] = A[i,j] - Sum

```

```python
U_ans = np.triu(A)
U_ans
```

    array([[ 15.        ,   8.        ,   1.        ,   4.        ],
           [  0.        ,   9.93333333,   1.86666667,   3.46666667],
           [  0.        ,   0.        ,  14.00671141,   4.58389262],
           [  0.        ,   0.        ,   0.        ,  13.81073311]])

```python
u
```

    array([[ 15.        ,   8.        ,   1.        ,   4.        ],
           [  0.        ,   9.93333333,   1.86666667,   3.46666667],
           [  0.        ,   0.        ,  14.00671141,   4.58389262],
           [  0.        ,   0.        ,   0.        ,  13.81073311]])

```python
L_ans = np.tril(A)
L_ans[np.diag_indices(4)] = 1
L_ans
```

    array([[ 1.        ,  0.        ,  0.        ,  0.        ],
           [ 0.13333333,  1.        ,  0.        ,  0.        ],
           [ 0.26666667,  0.38926174,  1.        ,  0.        ],
           [ 0.2       ,  0.74496644, -0.04216579,  1.        ]])

```python
l
```

    array([[ 1.        ,  0.        ,  0.        ,  0.        ],
           [ 0.13333333,  1.        ,  0.        ,  0.        ],
           [ 0.26666667,  0.38926174,  1.        ,  0.        ],
           [ 0.2       ,  0.74496644, -0.04216579,  1.        ]])

```python
L_ans.dot(U_ans)
```

    array([[ 15.,   8.,   1.,   4.],
           [  2.,  11.,   2.,   4.],
           [  4.,   6.,  15.,   7.],
           [  3.,   9.,   1.,  17.]])

```python
B
```

    array([[ 15.,   8.,   1.,   4.],
           [  2.,  11.,   2.,   4.],
           [  4.,   6.,  15.,   7.],
           [  3.,   9.,   1.,  17.]])

```python
L_ans.dot(U_ans) - B
```

    array([[ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.]])

# A = PLU (LU with partial pivoting)
Just row permutation (use permutation matrix p)
## $p = p^T$ and $pp^T = I$
### $Ax = d$, do left-multiply the permutation matrix on both sides
### $PAx = Pd$, do LU-decomposition with "PA" (so that there is no division by zero)
### $PAx = LUx = Pd$
### $Ly = c$, Solve for y. y and c are column vectors where $y = Ux$, and $c = Pd$
### $Ux = y$, Use backward substitution to solve for x
### $PA = LU$ and $A = P^TLU$ because $P^T = P^{-1}$

```python
np.random.seed(1)
B = np.ceil(10*np.random.random((4,4)))
B
```

    array([[ 5.,  8.,  1.,  4.],
           [ 2.,  1.,  2.,  4.],
           [ 4.,  6.,  5.,  7.],
           [ 3.,  9.,  1.,  7.]])

```python
p, l, u = sp.linalg.lu(B ,permute_l = False)
print(p, end = '\n\n')
print(l, end = '\n\n')
print(u, end = '\n\n')
```

    [[ 1.  0.  0.  0.]
     [ 0.  0.  0.  1.]
     [ 0.  0.  1.  0.]
     [ 0.  1.  0.  0.]]

    [[ 1.          0.          0.          0.        ]
     [ 0.6         1.          0.          0.        ]
     [ 0.8        -0.0952381   1.          0.        ]
     [ 0.4        -0.52380952  0.42696629  1.        ]]

    [[ 5.          8.          1.          4.        ]
     [ 0.          4.2         0.4         4.6       ]
     [ 0.          0.          4.23809524  4.23809524]
     [ 0.          0.          0.          3.        ]]

```python
np.dot(np.dot(p,l),u)
```

    array([[ 5.,  8.,  1.,  4.],
           [ 2.,  1.,  2.,  4.],
           [ 4.,  6.,  5.,  7.],
           [ 3.,  9.,  1.,  7.]])

```python
np.dot(l,u)
```

    array([[ 5.,  8.,  1.,  4.],
           [ 3.,  9.,  1.,  7.],
           [ 4.,  6.,  5.,  7.],
           [ 2.,  1.,  2.,  4.]])

```python
np.dot(p,p)
```

    array([[ 1.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
B
```

    array([[ 5.,  8.,  1.,  4.],
           [ 2.,  1.,  2.,  4.],
           [ 4.,  6.,  5.,  7.],
           [ 3.,  9.,  1.,  7.]])

```python
max(range(4), key = lambda i:B[1,i])
```

    3

```python
N = B.shape[0]
A = B.copy()
p_ans = np.eye(N,N)
for i in range(N):
    maxidx = np.abs(A[i:,i]).argmax() + i
    #may use max(range(i,N), key = lambda x: A[x,i])
    if (i != maxidx):
        p_ans[[i, maxidx]] = p_ans[[maxidx, i]]
        A[[i, maxidx]] = A[[maxidx, i]]
```

```python
p_ans
```

    array([[ 1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  1.,  0.,  0.]])

```python
p_ans.dot(p_ans)
```

    array([[ 1.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
N = B.shape[0]
A = p_ans.dot(B.copy())
for i in range(1, N):
    #loop for L
    for j in range(i): #A[a:b] is [a,b) not [a,b]
        Sum = A[0:j,j].dot(A[i,0:j])
        A[i,j] = (A[i,j] - Sum)/A[j,j]
    #loop for U
    for j in range(i,N):
        Sum = A[0:i,j].dot(A[i,0:i])
        A[i,j] = A[i,j] - Sum

```

```python
U_ans = np.triu(A)
U_ans
```

    array([[ 5.        ,  8.        ,  1.        ,  4.        ],
           [ 0.        ,  4.2       ,  0.4       ,  4.6       ],
           [ 0.        ,  0.        ,  4.23809524,  4.23809524],
           [ 0.        ,  0.        ,  0.        ,  3.        ]])

```python
L_ans = np.tril(A)
L_ans[np.diag_indices(4)] = 1
L_ans
```

    array([[ 1.        ,  0.        ,  0.        ,  0.        ],
           [ 0.6       ,  1.        ,  0.        ,  0.        ],
           [ 0.8       , -0.0952381 ,  1.        ,  0.        ],
           [ 0.4       , -0.52380952,  0.42696629,  1.        ]])

```python
(p_ans.T).dot(L_ans.dot(U_ans))
```

    array([[ 5.,  8.,  1.,  4.],
           [ 2.,  1.,  2.,  4.],
           [ 4.,  6.,  5.,  7.],
           [ 3.,  9.,  1.,  7.]])

```python
B
```

    array([[ 5.,  8.,  1.,  4.],
           [ 2.,  1.,  2.,  4.],
           [ 4.,  6.,  5.,  7.],
           [ 3.,  9.,  1.,  7.]])

# Forward substitution $x_m = \frac{b_m - \sum_{i=1}^{m-1} \ell_{m,i}x_i}{\ell_{m,m}}$

## After we know that code works, wrap it up into functions

```python
#LU with partial pivoting
#PA = LU
#take A, return PLU

def p_LU(B):
    N = B.shape[0]
    A = B.copy()
    p_ans = np.eye(N,N)
    #pivoting
    for i in range(N):
        maxidx = np.abs(A[i:,i]).argmax() + i
        #may use max(range(i,N), key = lambda x: A[x,i])
        if (i != maxidx):
            p_ans[[i, maxidx]] = p_ans[[maxidx, i]]
            A[[i, maxidx]] = A[[maxidx, i]]
    for i in range(1, N):
        #loop for L
        for j in range(i): #A[a:b] is [a,b) not [a,b]
#             Sum = A[0:j,j].dot(A[i,0:j])
            Sum = 0
            for k in range(j):
                Sum = Sum + A[k,j] * A[i,k]
            A[i,j] = (A[i,j] - Sum)/A[j,j]
        #loop for U
        for j in range(i,N):
#             Sum = A[0:i,j].dot(A[i,0:i])
            Sum = 0
            for k in range(i):
                Sum = Sum + A[k,j] * A[i,k]
            A[i,j] = A[i,j] - Sum
    U_ans = np.triu(A)
    L_ans = np.tril(A)
    L_ans[np.diag_indices(N)] = 1 #need to change from 4 to N
    return p_ans, L_ans, U_ans
```

```python
def solve_LU(B,d):
    p, l, u = p_LU(B)
    c = p.dot(d)
    #PAx = LUx = Ly = Pd = c
    #Ly = c
    #solve for y (forward sub)
    N = d.shape[0]
    y = np.empty(N)
    for i in range(N):
        Sum = 0
        for k in range(i):
            Sum = Sum + l[i,k] * y[k]
        y[i] = (c[i] - Sum)/l[i,i]
    #solve for x (backward sub)
    #Ux = y
    x = np.empty(N)
    for i in range(N-1,-1,-1): #3 to 0
        Sum = 0
        for k in range(i+1, N):
            Sum = Sum + u[i,k] * x[k]
        x[i] = (y[i] - Sum)/u[i,i]
    return x
```

```python
np.random.seed(1)
B = np.ceil(10*np.random.random((4,4)))
np.random.seed(10)
D = np.ceil(10*np.random.random(4))
```

```python
x = solve_LU(B,D)
```

```python
B.dot(x)
```

    array([ 8.,  1.,  7.,  8.])

```python
D
```

    array([ 8.,  1.,  7.,  8.])

```python
x
```

    array([ 0.06741573,  1.04494382,  0.51685393, -0.30337079])

```python
np.random.seed(1)
a2 = np.ceil(10*np.random.random((5,5)))
np.random.seed(2)
b = np.ceil(10*np.random.random((5,1)))
```

```python
a2
```

    array([[  5.,   8.,   1.,   4.,   2.],
           [  1.,   2.,   4.,   4.,   6.],
           [  5.,   7.,   3.,   9.,   1.],
           [  7.,   5.,   6.,   2.,   2.],
           [  9.,  10.,   4.,   7.,   9.]])

```python
b
```

    array([[ 5.],
           [ 1.],
           [ 6.],
           [ 5.],
           [ 5.]])

```python
p, l, u = p_LU(a2)
```

```python
x = solve_LU(a2,b)
x
```

    array([-0.03360414,  0.64881832,  0.43426883,  0.07884047, -0.38607829])

```python
a2.dot(x) -b.T
```

    array([[  0.00000000e+00,   0.00000000e+00,   8.88178420e-16,
              0.00000000e+00,   0.00000000e+00]])

```python
a2.dot(x)
```

    array([ 5.,  1.,  6.,  5.,  5.])

```python
p
```

    array([[ 0.,  0.,  0.,  0.,  1.],
           [ 1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  1.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.]])

```python
l
```

    array([[ 1.        ,  0.        ,  0.        ,  0.        ,  0.        ],
           [ 0.55555556,  1.        ,  0.        ,  0.        ,  0.        ],
           [ 0.77777778, -1.13636364,  1.        ,  0.        ,  0.        ],
           [ 0.55555556,  0.59090909,  1.        ,  1.        ,  0.        ],
           [ 0.11111111,  0.36363636,  2.66666667,  1.4384058 ,  1.        ]])

```python
u
```

    array([[  9.        ,  10.        ,   4.        ,   7.        ,   9.        ],
           [  0.        ,   2.44444444,  -1.22222222,   0.11111111,  -3.        ],
           [  0.        ,   0.        ,   1.5       ,  -3.31818182,
             -8.40909091],
           [  0.        ,   0.        ,   0.        ,   8.36363636,
              6.18181818],
           [  0.        ,   0.        ,   0.        ,   0.        ,
             19.62318841]])

```python
(p.T).dot(l.dot(u))
```

    array([[  5.,   8.,   1.,   4.,   2.],
           [  1.,   2.,   4.,   4.,   6.],
           [  5.,   7.,   3.,   9.,   1.],
           [  7.,   5.,   6.,   2.,   2.],
           [  9.,  10.,   4.,   7.,   9.]])

```python
a2
```

    array([[  5.,   8.,   1.,   4.,   2.],
           [  1.,   2.,   4.,   4.,   6.],
           [  5.,   7.,   3.,   9.,   1.],
           [  7.,   5.,   6.,   2.,   2.],
           [  9.,  10.,   4.,   7.,   9.]])

```python
x2 = user_gaussian_solve_pp(a2,b)
```

```python
x2
```

    array([[-0.03360414],
           [ 0.64881832],
           [ 0.43426883],
           [ 0.07884047],
           [-0.38607829]])

```python
a2.dot(x2)
```

    array([[ 5.],
           [ 1.],
           [ 6.],
           [ 5.],
           [ 5.]])

```python
p, l, u = p_LU(a2)
```

```python
p.dot(p)
```

    array([[ 0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.],
           [ 1.,  0.,  0.,  0.,  0.]])

```python
p2, l2, u2 = sp.linalg.lu(a2)
```

```python
p2
```

    array([[ 0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.],
           [ 0.,  1.,  0.,  0.,  0.],
           [ 1.,  0.,  0.,  0.,  0.]])

```python
sp.linalg.inv(p2).dot(p2)
```

    array([[ 1.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  1.]])

```python
sp.linalg.inv(p2)
```

    array([[ 0.,  0., -0.,  0.,  1.],
           [ 0.,  0., -0.,  1.,  0.],
           [ 0.,  1., -0.,  0.,  0.],
           [ 1.,  0., -0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.]])

```python
(p2.T).dot(p2)
```

    array([[ 1.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  1.]])

```python
(l.dot(u)).astype(int)
```

    array([[ 9, 10,  4,  7,  9],
           [ 5,  8,  1,  4,  2],
           [ 7,  5,  6,  2,  2],
           [ 5,  7,  3,  9,  1],
           [ 1,  2,  3,  4,  6]])

```python
np.random.seed(1)
a2 = np.ceil(10*np.random.random((10,10))) - 6
np.random.seed(2)
b = np.ceil(10*np.random.random((10,1)))
```

```python
x = solve_LU(a2,b)
x
```

    array([ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
            0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528])

```python
a2.dot(x)
```

    array([ 5.,  1.,  6.,  5.,  5.,  4.,  3.,  7.,  3.,  3.])

```python
b.T
```

    array([[ 5.,  1.,  6.,  5.,  5.,  4.,  3.,  7.,  3.,  3.]])

```python
p, l, u = p_LU(a2)
```

```python
p2, l2, u2 = sp.linalg.lu(a2)
```

```python
p
```

    array([[ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.]])

```python
p2
```

    array([[ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.],
           [ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.]])

The linalg.lu and p_lu use different pivoting method

```python
p2, l2, u2 = sp.linalg.lu(a2)
```

```python
LU = sp.linalg.lu_factor(a2)
LU
```

    (array([[ -5.00000000e+00,  -1.00000000e+00,   4.00000000e+00,
               0.00000000e+00,   1.00000000e+00,  -2.00000000e+00,
               1.00000000e+00,   3.00000000e+00,  -5.00000000e+00,
               2.00000000e+00],
            [ -8.00000000e-01,  -4.80000000e+00,  -8.00000000e-01,
               3.00000000e+00,  -1.20000000e+00,  -5.60000000e+00,
               4.80000000e+00,   4.00000000e-01,  -2.00000000e+00,
               3.60000000e+00],
            [  1.00000000e+00,  -4.16666667e-01,  -7.33333333e+00,
              -1.75000000e+00,  -2.50000000e+00,  -5.33333333e+00,
               1.00000000e+00,  -6.83333333e+00,   4.16666667e+00,
               5.00000000e-01],
            [  2.00000000e-01,  -2.50000000e-01,   5.45454545e-01,
               4.70454545e+00,  -4.13636364e+00,   2.90909091e+00,
              -5.45454545e-01,   3.22727273e+00,  -5.77272727e+00,
              -3.77272727e+00],
            [ -6.00000000e-01,  -8.33333333e-02,  -5.90909091e-01,
              -5.91787440e-01,  -6.42512077e+00,  -9.66183575e-02,
               2.68115942e-01,   3.70531401e+00,  -3.12077295e+00,
               5.62801932e-01],
            [  8.00000000e-01,  -1.00000000e+00,   6.81818182e-01,
               8.91304348e-01,  -3.72180451e-01,  -5.99248120e+00,
               7.90413534e+00,   1.16165414e+00,  -1.85714286e+00,
               6.23120301e+00],
            [  2.00000000e-01,  -4.58333333e-01,   8.40909091e-01,
               1.79951691e-01,   2.96240602e-01,   5.30112923e-01,
              -7.01226474e+00,   1.03531995e+00,  -2.47264743e+00,
              -1.96151192e+00],
            [ -8.00000000e-01,  -2.50000000e-01,  -3.02788098e-17,
               5.84541063e-01,   1.68421053e-01,   9.48557089e-01,
               1.74528407e-01,  -4.29311432e+00,  -1.40684676e+00,
              -4.95778592e+00],
            [  8.00000000e-01,   4.16666667e-02,   2.95454545e-01,
              -1.29227053e-01,   8.63157895e-01,  -6.45545797e-01,
              -6.42461899e-01,   4.10865556e-01,   6.59057235e+00,
               1.92826107e+00],
            [ -6.00000000e-01,  -7.08333333e-01,   2.27272727e-02,
               6.72705314e-01,  -8.69924812e-01,   6.81932246e-01,
               8.28924943e-01,   7.82507580e-01,  -4.43952969e-01,
               1.18784156e+01]]),
     array([3, 7, 5, 7, 8, 9, 7, 8, 8, 9], dtype=int32))

```python
LU[1]
```

    array([3, 7, 5, 7, 8, 9, 7, 8, 8, 9], dtype=int32)

```python
len(LU)
```

    2

```python
ans_scipy = sp.linalg.lu_solve(LU,b)
ans_scipy
```

    array([[ 0.48316746],
           [-0.9890326 ],
           [ 0.67385345],
           [-0.73804389],
           [-1.47130591],
           [ 0.14987401],
           [ 0.21031571],
           [-1.27113437],
           [-1.25877634],
           [ 1.40582528]])

```python
x
```

    array([ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
            0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528])

```python
np.set_printoptions(precision=8)
```

```python
x
```

    array([ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
            0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528])

```python
a2.dot(x)
```

    array([ 5.,  1.,  6.,  5.,  5.,  4.,  3.,  7.,  3.,  3.])

```python

b.T
```

    array([[ 5.,  1.,  6.,  5.,  5.,  4.,  3.,  7.,  3.,  3.]])

```python
a2.dot(ans_scipy).T
```

    array([[ 5.,  1.,  6.,  5.,  5.,  4.,  3.,  7.,  3.,  3.]])

```python
ans_scipy.T
```

    array([[ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
             0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528]])

```python
x
```

    array([ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
            0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528])

# Speed Test

```python
a2
```

    array([[-1.,  2., -5., -2., -4., -5., -4., -2., -2.,  0.],
           [-1.,  1., -3.,  3., -5.,  1., -1.,  0., -4., -4.],
           [ 3.,  4., -2.,  1.,  3.,  3., -5., -5., -4.,  3.],
           [-5., -1.,  4.,  0.,  1., -2.,  1.,  3., -5.,  2.],
           [ 4.,  2., -3.,  2., -4., -1.,  4., -3., -3., -4.],
           [-5.,  1., -3., -3., -1., -5.,  0., -4.,  0.,  1.],
           [-4., -1.,  1., -1., -5.,  0.,  1.,  0.,  4.,  0.],
           [ 4., -4., -4.,  3., -2., -4.,  4., -2.,  2.,  2.],
           [ 3.,  1.,  2., -2., -3.,  3., -1.,  4.,  1.,  1.],
           [-4.,  4., -1.,  0., -1., -3.,  4.,  0., -5.,  1.]])

```python
b
```

    array([[ 5.],
           [ 1.],
           [ 6.],
           [ 5.],
           [ 5.],
           [ 4.],
           [ 3.],
           [ 7.],
           [ 3.],
           [ 3.]])

```python
%%timeit -n 5
LU = sp.linalg.lu_factor(a2)
ans = sp.linalg.lu_solve(LU,b)
```

    5 loops, best of 3: 46.7 µs per loop

```python
%%timeit -n 5
x = solve_LU(a2,b)
```

    5 loops, best of 3: 427 µs per loop

```python
LU = sp.linalg.lu_factor(a2)
ans = sp.linalg.lu_solve(LU,b)
ans.T
```

    array([[ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
             0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528]])

```python
x = solve_LU(a2,b)
x
```

    array([ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
            0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528])

```python
sp.linalg.lu
```

    <function scipy.linalg.decomp_lu.lu>

```python
sp_solve_ans = sp.linalg.solve(a2,b)
```

```python
sp_solve_ans.T
```

    array([[ 0.48316746, -0.9890326 ,  0.67385345, -0.73804389, -1.47130591,
             0.14987401,  0.21031571, -1.27113437, -1.25877634,  1.40582528]])

```python
%%timeit -n 5
sp_solve_ans = sp.linalg.solve(a2,b)
```

    5 loops, best of 3: 61.1 µs per loop

# Determinant: $det(A)=(-1)^s\Pi_{i=0}^{n-1}L_{ii}\Pi_{i=0}^{n-1}U_{ii}$
# $det(A)=(-1)^s\Pi_{i=0}^{n-1}U_{ii}$

```python
def det_LU(B):
    N = B.shape[0]
    A = B.copy()
    p_ans = np.eye(N,N)
    s = 0
    #pivoting
    for i in range(N):
        maxidx = np.abs(A[i:,i]).argmax() + i
        #may use max(range(i,N), key = lambda x: A[x,i])
        if (i != maxidx):
            p_ans[[i, maxidx]] = p_ans[[maxidx, i]]
            A[[i, maxidx]] = A[[maxidx, i]]
            s = s+1
    for i in range(1, N):
        #loop for L
        for j in range(i): #A[a:b] is [a,b) not [a,b]
#             Sum = A[0:j,j].dot(A[i,0:j])
            Sum = 0
            for k in range(j):
                Sum = Sum + A[k,j] * A[i,k]
            A[i,j] = (A[i,j] - Sum)/A[j,j]
        #loop for U
        for j in range(i,N):
#             Sum = A[0:i,j].dot(A[i,0:i])
            Sum = 0
            for k in range(i):
                Sum = Sum + A[k,j] * A[i,k]
            A[i,j] = A[i,j] - Sum
    U_ans = np.triu(A)
    L_ans = np.tril(A)
    L_ans[np.diag_indices(N)] = 1 #need to change from 4 to N

    U_ii = np.product(U_ans[range(N),range(N)])
    det_ans = (-1)**s * U_ii
    return det_ans
```

```python
#debug (do during making function)
np.random.seed(3)
a = np.ceil(10*np.random.random((5,5)))
a[range(5),range(5)] = a[range(5),range(5)] * 10
print(a, end = '\n\n')
a[range(5),range(5)]
```

    [[ 60.   8.   3.   6.   9.]
     [  9.  20.   3.   1.   5.]
     [  1.   5.  70.   3.   7.]
     [  6.   1.   6.  30.   5.]
     [  3.   7.   5.   2.  60.]]

    array([ 60.,  20.,  70.,  30.,  60.])

```python
np.product(_)
```

    151200000.0

```python
prod = 1
for i in a[range(5),range(5)]:
    prod = prod * i
prod
```

    151200000.0

```python
a
```

    array([[ 60.,   8.,   3.,   6.,   9.],
           [  9.,  20.,   3.,   1.,   5.],
           [  1.,   5.,  70.,   3.,   7.],
           [  6.,   1.,   6.,  30.,   5.],
           [  3.,   7.,   5.,   2.,  60.]])

```python
det_LU(a)
```

    131629550.0

```python
sp.linalg.det(a)
```

    131629549.99999996

```python
a2
```

    array([[-1.,  2., -5., -2., -4., -5., -4., -2., -2.,  0.],
           [-1.,  1., -3.,  3., -5.,  1., -1.,  0., -4., -4.],
           [ 3.,  4., -2.,  1.,  3.,  3., -5., -5., -4.,  3.],
           [-5., -1.,  4.,  0.,  1., -2.,  1.,  3., -5.,  2.],
           [ 4.,  2., -3.,  2., -4., -1.,  4., -3., -3., -4.],
           [-5.,  1., -3., -3., -1., -5.,  0., -4.,  0.,  1.],
           [-4., -1.,  1., -1., -5.,  0.,  1.,  0.,  4.,  0.],
           [ 4., -4., -4.,  3., -2., -4.,  4., -2.,  2.,  2.],
           [ 3.,  1.,  2., -2., -3.,  3., -1.,  4.,  1.,  1.],
           [-4.,  4., -1.,  0., -1., -3.,  4.,  0., -5.,  1.]])

```python
sp.linalg.det(a2)
```

    -75132998.00000001

```python
det_LU(a2)
```

    -75132997.99999994

### det_LU gives the same result as the one from scipy.linalg.det !

```python
%%timeit -n 5
sp.linalg.det(a2)
```

    5 loops, best of 3: 19.5 µs per loop

```python
%%timeit -n 5
det_LU(a2)
```

    5 loops, best of 3: 386 µs per loop

```python
# det from LU is a lot slower
```

```python
p, l, u = sp.linalg.lu(a2)
```

```python
sp.linalg.det(p)
```

    1.0

# Determinant if LU does not return s

```python
p, l, u = p_LU(a2)
```

```python
p = p.astype(int)
p
```

    array([[0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
           [0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
           [1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
           [0, 0, 0, 0, 0, 0, 0, 1, 0, 0]])

```python
p = np.eye(10,10)
p[[2,5]] = p[[5,2]]
```

```python
def det_p(p):
    p = p.astype(int)
    N = p.shape[0]
    if N == 1:
        return 1
    for i in range(N):
        for j in range(N):
            if p[i,j] == 1:
                new_p = np.delete(p,i,0)
                new_p = np.delete(new_p,j,1)
                return (-1)**(i+j) * det_p(new_p)
```

```python
p
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.]])

```python
p2 = np.delete(p,2,0)
p2 = np.delete(p2,5,1)
p2
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.]])

```python
p
```

    array([[ 1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  1.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  1.]])

## Testing function det_p

```python
p = np.eye(10,10)
p[[2,5]] = p[[5,2]]
det_p(p)
```

    -1

```python
p = np.eye(10,10)
det_p(p)
```

    1

```python
p = np.eye(10,10)
p[[2,5]] = p[[5,2]]
p[[0,1]] = p[[1,0]]
det_p(p)
```

    1

```python
p = np.eye(10,10)
p[[2,5]] = p[[5,2]]
p[[0,1]] = p[[1,0]]
p[[7,4]] = p[[4,7]]
det_p(p)
```

    -1

```python
p = np.eye(10,10)
p[[2,5]] = p[[5,2]]
p[[0,1]] = p[[1,0]]
p[[7,4]] = p[[4,7]]
p[[8,1]] = p[[1,8]]
det_p(p)
```

    1

```python
p = np.eye(10,10)
p[[2,5]] = p[[5,2]]
p[[0,1]] = p[[1,0]]
p[[7,4]] = p[[4,7]]
p[[8,1]] = p[[1,8]]
p[[8,1]] = p[[1,8]]
det_p(p)
```

    -1

```python
def det_LU_no_S(p,u):
    N = u.shape[0]
    return det_p(p) * np.product(u[range(N),range(N)])
```

```python
a2
```

    array([[-1.,  2., -5., -2., -4., -5., -4., -2., -2.,  0.],
           [-1.,  1., -3.,  3., -5.,  1., -1.,  0., -4., -4.],
           [ 3.,  4., -2.,  1.,  3.,  3., -5., -5., -4.,  3.],
           [-5., -1.,  4.,  0.,  1., -2.,  1.,  3., -5.,  2.],
           [ 4.,  2., -3.,  2., -4., -1.,  4., -3., -3., -4.],
           [-5.,  1., -3., -3., -1., -5.,  0., -4.,  0.,  1.],
           [-4., -1.,  1., -1., -5.,  0.,  1.,  0.,  4.,  0.],
           [ 4., -4., -4.,  3., -2., -4.,  4., -2.,  2.,  2.],
           [ 3.,  1.,  2., -2., -3.,  3., -1.,  4.,  1.,  1.],
           [-4.,  4., -1.,  0., -1., -3.,  4.,  0., -5.,  1.]])

```python
p,l,u = p_LU(a2)
```

```python
det_LU_no_S(p,u)
```

    -75132997.99999994

```python
sp.linalg.det(a2)
```

    -75132998.00000001

```python
det_p(p)
```

    1

```python
np.random.seed(6)
a3 = np.ceil(10*np.random.random((15,15)))
a3[range(15),range(15)] = a3[range(15),range(15)] - 10
print(a3, end = '\n\n')
```

    [[ -1.   4.   9.   1.   2.   6.   6.   5.   4.   7.   5.   8.   6.   6.
        7.]
     [ 10.  -1.   5.   9.   9.   1.   8.   9.   8.   8.   6.   2.  10.   5.
        3.]
     [  8.  10.  -7.   7.   6.   8.  10.   4.   3.   5.   8.   8.   5.  10.
        5.]
     [  4.   8.   1.  -6.   8.   8.   3.   2.   5.   4.   8.   2.   4.   9.
        7.]
     [  9.  10.   9.   2.  -3.   5.   5.   8.   9.   7.   2.   8.   1.   2.
       10.]
     [  5.   2.   7.   2.   7.  -7.   1.  10.   1.   6.   4.   7.   9.   7.
        6.]
     [  2.   8.   3.   5.   1.   4.  -1.   6.   6.   4.   9.   3.   2.   6.
        3.]
     [  8.   2.   8.   7.   1.   6.   7.   0.   8.   8.   5.   1.   9.   1.
        6.]
     [  6.   2.   1.   6.   5.   4.   1.   7.  -8.   1.   3.   4.   2.   1.
        8.]
     [  5.   3.   6.   3.   9.   5.   7.  10.   4.  -7.  10.   6.  10.  10.
        8.]
     [  1.   7.   3.   2.   4.   3.   8.  10.   1.   3.  -7.   2.  10.   3.
        4.]
     [  1.   3.   9.  10.   7.   3.   1.   9.   2.   3.   6.  -7.   9.   6.
        5.]
     [ 10.  10.   3.   1.   7.   3.   1.   1.   5.   2.   3.   8.  -5.   2.
        3.]
     [ 10.   1.   7.   7.   7.   6.   8.   9.   7.   2.   8.   3.   9.  -4.
        1.]
     [ 10.   5.  10.   1.   2.   8.   1.   5.   9.   5.   2.   9.   2.   3.
       -7.]]

```python
p,l,u = p_LU(a3)
```

```python
det_p(p)
```

    -1

```python
np.float128(det_LU_no_S(p,u))
```

    12001154699853958.0

```python
np.float128(sp.linalg.det(a3))
```

    12001154699853984.0

```python
np.float128(det_LU(a3))
```

    12001154699853958.0

# Matrix Inversion: Gauss-Jordan
### Use:   $L_{ij}^{new} = \left[L_{ij} - \frac{L_{ik}}{L_{kk}} L_{kj}\right]^{old}$

```python
np.random.seed(3)
a = np.ceil(10 * np.random.random([4,4]))
a
```

    array([[ 6.,  8.,  3.,  6.],
           [ 9.,  9.,  2.,  3.],
           [ 1.,  5.,  1.,  5.],
           [ 7.,  3.,  7.,  6.]])

```python
I = np.eye(4,4)
I
```

    array([[ 1.,  0.,  0.,  0.],
           [ 0.,  1.,  0.,  0.],
           [ 0.,  0.,  1.,  0.],
           [ 0.,  0.,  0.,  1.]])

```python
aI = np.hstack([a,I])
aI
```

    array([[ 6.,  8.,  3.,  6.,  1.,  0.,  0.,  0.],
           [ 9.,  9.,  2.,  3.,  0.,  1.,  0.,  0.],
           [ 1.,  5.,  1.,  5.,  0.,  0.,  1.,  0.],
           [ 7.,  3.,  7.,  6.,  0.,  0.,  0.,  1.]])

```python
N = I.shape[0]
N
```

    4

```python
A = aI.copy()
for k in range(N-1):
    for i in range(k+1, N):
        r = -A[i,k] / A[k,k]
        print('r = ', r)
        for j in range(0, 2*N):
            A[i,j] = A[i,j] + r * A[k,j]
        #lines below are not used during back substitution
        np.set_printoptions(precision=2)
        print(A, end = '\n\n')
```

    r =  -1.5
    [[ 6.   8.   3.   6.   1.   0.   0.   0. ]
     [ 0.  -3.  -2.5 -6.  -1.5  1.   0.   0. ]
     [ 1.   5.   1.   5.   0.   0.   1.   0. ]
     [ 7.   3.   7.   6.   0.   0.   0.   1. ]]

    r =  -0.166666666667
    [[ 6.    8.    3.    6.    1.    0.    0.    0.  ]
     [ 0.   -3.   -2.5  -6.   -1.5   1.    0.    0.  ]
     [ 0.    3.67  0.5   4.   -0.17  0.    1.    0.  ]
     [ 7.    3.    7.    6.    0.    0.    0.    1.  ]]

    r =  -1.16666666667
    [[ 6.    8.    3.    6.    1.    0.    0.    0.  ]
     [ 0.   -3.   -2.5  -6.   -1.5   1.    0.    0.  ]
     [ 0.    3.67  0.5   4.   -0.17  0.    1.    0.  ]
     [ 0.   -6.33  3.5  -1.   -1.17  0.    0.    1.  ]]

    r =  1.22222222222
    [[ 6.    8.    3.    6.    1.    0.    0.    0.  ]
     [ 0.   -3.   -2.5  -6.   -1.5   1.    0.    0.  ]
     [ 0.    0.   -2.56 -3.33 -2.    1.22  1.    0.  ]
     [ 0.   -6.33  3.5  -1.   -1.17  0.    0.    1.  ]]

    r =  -2.11111111111
    [[  6.     8.     3.     6.     1.     0.     0.     0.  ]
     [  0.    -3.    -2.5   -6.    -1.5    1.     0.     0.  ]
     [  0.     0.    -2.56  -3.33  -2.     1.22   1.     0.  ]
     [  0.     0.     8.78  11.67   2.    -2.11   0.     1.  ]]

    r =  3.4347826087
    [[ 6.    8.    3.    6.    1.    0.    0.    0.  ]
     [ 0.   -3.   -2.5  -6.   -1.5   1.    0.    0.  ]
     [ 0.    0.   -2.56 -3.33 -2.    1.22  1.    0.  ]
     [ 0.    0.    0.    0.22 -4.87  2.09  3.43  1.  ]]

```python
for k in range(N-1,-1,-1):
    for i in range(k-1, -1, -1):
        r = -A[i,k] / A[k,k]
        print('r = ', r)
        for j in range(0, 2*N):
            A[i,j] = A[i,j] + r * A[k,j]
        #lines below are not used during back substitution
        np.set_printoptions(precision=2)
        print(A, end = '\n\n')
```

    r =  15.3333333333
    [[  6.     8.     3.     6.     1.     0.     0.     0.  ]
     [  0.    -3.    -2.5   -6.    -1.5    1.     0.     0.  ]
     [  0.     0.    -2.56   0.   -76.67  33.22  53.67  15.33]
     [  0.     0.     0.     0.22  -4.87   2.09   3.43   1.  ]]

    r =  27.6
    [[   6.      8.      3.      6.      1.      0.      0.      0.  ]
     [   0.     -3.     -2.5     0.   -135.9    58.6    94.8    27.6 ]
     [   0.      0.     -2.56    0.    -76.67   33.22   53.67   15.33]
     [   0.      0.      0.      0.22   -4.87    2.09    3.43    1.  ]]

    r =  -27.6
    [[   6.      8.      3.      0.    135.4   -57.6   -94.8   -27.6 ]
     [   0.     -3.     -2.5     0.   -135.9    58.6    94.8    27.6 ]
     [   0.      0.     -2.56    0.    -76.67   33.22   53.67   15.33]
     [   0.      0.      0.      0.22   -4.87    2.09    3.43    1.  ]]

    r =  -0.978260869565
    [[   6.      8.      3.      0.    135.4   -57.6   -94.8   -27.6 ]
     [   0.     -3.      0.      0.    -60.9    26.1    42.3    12.6 ]
     [   0.      0.     -2.56    0.    -76.67   33.22   53.67   15.33]
     [   0.      0.      0.      0.22   -4.87    2.09    3.43    1.  ]]

    r =  1.17391304348
    [[  6.     8.     0.     0.    45.4  -18.6  -31.8   -9.6 ]
     [  0.    -3.     0.     0.   -60.9   26.1   42.3   12.6 ]
     [  0.     0.    -2.56   0.   -76.67  33.22  53.67  15.33]
     [  0.     0.     0.     0.22  -4.87   2.09   3.43   1.  ]]

    r =  2.66666666667
    [[   6.      0.      0.      0.   -117.     51.     81.     24.  ]
     [   0.     -3.      0.      0.    -60.9    26.1    42.3    12.6 ]
     [   0.      0.     -2.56    0.    -76.67   33.22   53.67   15.33]
     [   0.      0.      0.      0.22   -4.87    2.09    3.43    1.  ]]

```python
for i in range(N):
    for j in range(N,2*N):
        A[i,j] = A[i,j]/A[i,i]
    A[i,i] = 1
    print(A, end = '\n\n')
```

    [[  1.     0.     0.     0.   -19.5    8.5   13.5    4.  ]
     [  0.    -3.     0.     0.   -60.9   26.1   42.3   12.6 ]
     [  0.     0.    -2.56   0.   -76.67  33.22  53.67  15.33]
     [  0.     0.     0.     0.22  -4.87   2.09   3.43   1.  ]]

    [[  1.     0.     0.     0.   -19.5    8.5   13.5    4.  ]
     [  0.     1.     0.     0.    20.3   -8.7  -14.1   -4.2 ]
     [  0.     0.    -2.56   0.   -76.67  33.22  53.67  15.33]
     [  0.     0.     0.     0.22  -4.87   2.09   3.43   1.  ]]

    [[  1.     0.     0.     0.   -19.5    8.5   13.5    4.  ]
     [  0.     1.     0.     0.    20.3   -8.7  -14.1   -4.2 ]
     [  0.     0.     1.     0.    30.   -13.   -21.    -6.  ]
     [  0.     0.     0.     0.22  -4.87   2.09   3.43   1.  ]]

    [[  1.    0.    0.    0.  -19.5   8.5  13.5   4. ]
     [  0.    1.    0.    0.   20.3  -8.7 -14.1  -4.2]
     [  0.    0.    1.    0.   30.  -13.  -21.   -6. ]
     [  0.    0.    0.    1.  -22.4   9.6  15.8   4.6]]

```python
a_inv = A[:,N:]
a_inv
```

    array([[-19.5,   8.5,  13.5,   4. ],
           [ 20.3,  -8.7, -14.1,  -4.2],
           [ 30. , -13. , -21. ,  -6. ],
           [-22.4,   9.6,  15.8,   4.6]])

```python
a.dot(a_inv)
```

    array([[  1.00e+00,   1.24e-14,   2.13e-14,  -7.11e-15],
           [  0.00e+00,   1.00e+00,   1.42e-14,  -1.42e-14],
           [ -1.78e-14,   1.78e-14,   1.00e+00,   1.78e-15],
           [ -1.42e-14,   0.00e+00,   2.13e-14,   1.00e+00]])

```python
np.set_printoptions(precision=2, suppress=True)
a.dot(a_inv)
```

    array([[ 1.,  0.,  0., -0.],
           [ 0.,  1.,  0., -0.],
           [-0.,  0.,  1.,  0.],
           [-0.,  0.,  0.,  1.]])

```python
## Making function out of the tested code
def GJ_inv(a):
    N = a.shape[0]
    I = np.eye(N,N)
    A = np.hstack([a,I])
    for k in range(N-1):
        for i in range(k+1, N):
            r = -A[i,k] / A[k,k]
            for j in range(k+1, N+k+1): #optimize: 0, 2N to k+1, N+k+1
                A[i,j] = A[i,j] + r * A[k,j]
    for k in range(N-1,-1,-1):
        for i in range(k-1, -1, -1):
            r = -A[i,k] / A[k,k]
            for j in range(k+1, 2*N): #optimize by change 0 to k+1
                A[i,j] = A[i,j] + r * A[k,j]
    for i in range(N):
        for j in range(N,2*N):
            A[i,j] = A[i,j]/A[i,i]
    return A[:,N:]
```

```python
np.random.seed(3)
a = np.ceil(10 * np.random.random([4,4]))
fn_cal = GJ_inv(a)
fn_cal
```

    array([[-19.5,   8.5,  13.5,   4. ],
           [ 20.3,  -8.7, -14.1,  -4.2],
           [ 30. , -13. , -21. ,  -6. ],
           [-22.4,   9.6,  15.8,   4.6]])

```python
np.set_printoptions(precision=2, suppress=True)
a.dot(fn_cal)
```

    array([[ 1.,  0.,  0., -0.],
           [ 0.,  1.,  0., -0.],
           [-0.,  0.,  1.,  0.],
           [-0.,  0.,  0.,  1.]])

```python
np.set_printoptions(precision=2, suppress=False)
a.dot(fn_cal)
```

    array([[  1.00e+00,   1.24e-14,   2.13e-14,  -7.11e-15],
           [  0.00e+00,   1.00e+00,   1.42e-14,  -1.42e-14],
           [ -1.78e-14,   1.78e-14,   1.00e+00,   1.78e-15],
           [ -1.42e-14,   0.00e+00,   2.13e-14,   1.00e+00]])

```python
def pivoting(B):
    A = B.copy()
    N = A.shape[0]
    p_ans = np.eye(N,N)
    #pivoting
    for i in range(N):
        maxidx = np.abs(A[i:,i]).argmax() + i
        if (i != maxidx):
            p_ans[[i, maxidx]] = p_ans[[maxidx, i]]
            A[[i, maxidx]] = A[[maxidx, i]]

    return A, p_ans

def GJ_inv_pp(a):
    N = a.shape[0]
    I = np.eye(N,N)
    b, p = pivoting(a)
    A = np.hstack([b.copy(),I])
    for k in range(N-1):
        for i in range(k+1, N):
            r = -A[i,k] / A[k,k]
            for j in range(k+1, N+k+1): #optimize: 0, 2N to k+1, N+k+1
                A[i,j] = A[i,j] + r * A[k,j]
    for k in range(N-1,-1,-1):
        for i in range(k-1, -1, -1):
            r = -A[i,k] / A[k,k]
            for j in range(k+1, 2*N): #optimize by change 0 to k+1
                A[i,j] = A[i,j] + r * A[k,j]
    for i in range(N):
        for j in range(N,2*N):
            A[i,j] = A[i,j]/A[i,i]
    return A[:,N:].dot(p)
```

```python
a3.shape
```

    (15, 15)

```python
a3[range(15),range(15)] = 0
a3.astype(int)
```

    array([[ 0,  4,  9,  1,  2,  6,  6,  5,  4,  7,  5,  8,  6,  6,  7],
           [10,  0,  5,  9,  9,  1,  8,  9,  8,  8,  6,  2, 10,  5,  3],
           [ 8, 10,  0,  7,  6,  8, 10,  4,  3,  5,  8,  8,  5, 10,  5],
           [ 4,  8,  1,  0,  8,  8,  3,  2,  5,  4,  8,  2,  4,  9,  7],
           [ 9, 10,  9,  2,  0,  5,  5,  8,  9,  7,  2,  8,  1,  2, 10],
           [ 5,  2,  7,  2,  7,  0,  1, 10,  1,  6,  4,  7,  9,  7,  6],
           [ 2,  8,  3,  5,  1,  4,  0,  6,  6,  4,  9,  3,  2,  6,  3],
           [ 8,  2,  8,  7,  1,  6,  7,  0,  8,  8,  5,  1,  9,  1,  6],
           [ 6,  2,  1,  6,  5,  4,  1,  7,  0,  1,  3,  4,  2,  1,  8],
           [ 5,  3,  6,  3,  9,  5,  7, 10,  4,  0, 10,  6, 10, 10,  8],
           [ 1,  7,  3,  2,  4,  3,  8, 10,  1,  3,  0,  2, 10,  3,  4],
           [ 1,  3,  9, 10,  7,  3,  1,  9,  2,  3,  6,  0,  9,  6,  5],
           [10, 10,  3,  1,  7,  3,  1,  1,  5,  2,  3,  8,  0,  2,  3],
           [10,  1,  7,  7,  7,  6,  8,  9,  7,  2,  8,  3,  9,  0,  1],
           [10,  5, 10,  1,  2,  8,  1,  5,  9,  5,  2,  9,  2,  3,  0]])

```python
a3_inv = GJ_inv_pp(a3)
a3_inv
```

    array([[-0.14, -0.07,  0.07,  0.03,  0.08,  0.12, -0.06,  0.02, -0.06,
            -0.02, -0.06,  0.01, -0.06,  0.07, -0.  ],
           [-0.03, -0.05,  0.02,  0.  ,  0.03,  0.02,  0.02,  0.  , -0.05,
            -0.03,  0.03,  0.03,  0.05,  0.02, -0.03],
           [-0.01, -0.07,  0.05,  0.02,  0.11,  0.05, -0.1 , -0.03, -0.12,
            -0.04, -0.08,  0.11, -0.03,  0.09, -0.03],
           [ 0.02,  0.05,  0.01, -0.08, -0.05, -0.09,  0.03,  0.02,  0.07,
             0.02,  0.01,  0.04,  0.04, -0.08,  0.03],
           [ 0.08,  0.06, -0.04,  0.05, -0.03, -0.05, -0.05, -0.06,  0.02,
            -0.03, -0.  ,  0.04,  0.07,  0.01, -0.02],
           [ 0.02, -0.03,  0.  ,  0.06, -0.04, -0.03, -0.02,  0.  ,  0.07,
            -0.04,  0.03, -0.  , -0.06,  0.02,  0.07],
           [ 0.02,  0.01,  0.07,  0.  ,  0.08, -0.02, -0.08, -0.05, -0.07,
            -0.01, -0.03,  0.02, -0.03,  0.07, -0.06],
           [-0.02,  0.  , -0.  ,  0.02,  0.04,  0.03,  0.02, -0.07,  0.01,
            -0.03,  0.01, -0.01, -0.07,  0.05,  0.01],
           [ 0.06,  0.15, -0.13, -0.06, -0.09, -0.18,  0.1 ,  0.04,  0.08,
             0.12,  0.08, -0.07,  0.08, -0.17,  0.08],
           [ 0.05,  0.01,  0.03,  0.07,  0.02,  0.09, -0.01, -0.03, -0.02,
            -0.15, -0.03, -0.01, -0.04,  0.06, -0.04],
           [ 0.01, -0.07,  0.02,  0.03,  0.03,  0.09,  0.04, -0.01, -0.06,
            -0.04, -0.08, -0.02, -0.02,  0.14, -0.09],
           [ 0.11,  0.05, -0.04, -0.11, -0.12, -0.06,  0.07,  0.02,  0.1 ,
             0.06,  0.04, -0.08,  0.1 , -0.09,  0.04],
           [-0.02, -0.01, -0.06, -0.06, -0.12,  0.  ,  0.08,  0.12,  0.05,
             0.07,  0.11, -0.07,  0.05, -0.08,  0.04],
           [-0.08,  0.02,  0.06,  0.01,  0.04,  0.01, -0.04, -0.01, -0.06,
             0.06, -0.03,  0.04, -0.07, -0.08,  0.05],
           [ 0.01,  0.01, -0.04,  0.  ,  0.02, -0.03, -0.  ,  0.04,  0.06,
             0.05,  0.  , -0.02,  0.01, -0.06, -0.03]])

```python
a3.dot(a3_inv)
```

    array([[  1.00e+00,  -1.61e-15,  -4.27e-15,   6.41e-16,   4.16e-15,
              1.31e-15,  -2.65e-15,   5.75e-15,   1.98e-15,   6.76e-15,
             -3.14e-16,   6.28e-16,  -1.99e-15,  -6.66e-16,  -1.60e-15],
           [  2.19e-16,   1.00e+00,   4.86e-17,   1.93e-17,   1.11e-16,
              1.77e-16,  -6.16e-17,  -1.32e-16,  -6.94e-17,  -2.36e-16,
              1.75e-17,  -1.08e-16,   1.32e-16,  -1.67e-16,  -1.60e-16],
           [ -1.08e-16,  -2.56e-16,   1.00e+00,   1.37e-16,   6.59e-16,
              2.26e-16,  -4.25e-17,  -1.73e-16,  -2.64e-16,  -1.53e-16,
             -1.02e-16,  -3.12e-17,  -2.01e-16,   3.33e-16,  -2.01e-16],
           [  1.77e-16,   1.34e-16,   3.96e-16,   1.00e+00,   1.60e-16,
              1.21e-16,   1.15e-16,  -1.87e-16,  -1.39e-17,  -6.94e-17,
             -1.17e-16,  -1.49e-16,  -1.18e-16,  -2.78e-16,   3.47e-17],
           [ -1.39e-15,  -8.31e-16,   2.93e-15,   2.08e-16,   1.00e+00,
              4.23e-15,  -2.28e-15,  -1.57e-15,  -1.30e-15,  -7.49e-16,
             -8.90e-16,   2.71e-16,  -1.30e-15,   3.89e-15,  -1.51e-15],
           [ -4.23e-16,   8.52e-16,   1.79e-15,  -9.52e-16,   1.71e-15,
              1.00e+00,  -1.39e-15,  -1.60e-15,  -9.16e-16,  -1.94e-16,
             -1.16e-15,  -1.32e-16,  -3.33e-16,   9.44e-16,  -3.75e-16],
           [ -1.98e-16,   1.78e-15,  -1.60e-16,  -2.10e-15,  -2.21e-15,
             -1.54e-15,   1.00e+00,   4.86e-17,   1.12e-15,   9.02e-16,
             -6.38e-16,  -6.90e-16,   3.68e-16,  -1.67e-15,   1.03e-15],
           [ -4.93e-16,  -3.27e-15,   1.80e-16,   3.33e-15,   3.54e-15,
              2.74e-15,  -2.60e-17,   1.00e+00,  -2.69e-15,  -1.75e-15,
              8.85e-16,   1.91e-15,  -7.36e-16,   3.28e-15,  -2.76e-15],
           [  2.25e-15,   2.78e-15,   3.11e-15,   7.42e-16,  -6.66e-16,
             -3.08e-15,  -1.46e-15,  -5.61e-15,   1.00e+00,  -4.55e-15,
             -3.09e-15,  -1.94e-16,   8.74e-16,  -1.61e-15,   2.08e-15],
           [ -1.39e-16,  -2.29e-16,   5.00e-16,   1.34e-16,   0.00e+00,
              2.22e-16,   3.26e-16,  -4.44e-16,  -6.66e-16,   1.00e+00,
             -1.36e-16,  -8.33e-17,  -3.61e-16,  -1.11e-16,   1.11e-16],
           [ -9.85e-16,  -1.84e-16,  -1.94e-16,  -1.16e-15,   5.55e-17,
              1.01e-15,   1.08e-16,   2.78e-17,   5.55e-17,   5.55e-16,
              1.00e+00,  -8.47e-16,   3.19e-16,   3.89e-16,   3.19e-16],
           [ -3.30e-16,  -8.11e-16,   4.86e-17,   4.28e-16,   7.63e-17,
             -7.98e-17,   2.91e-16,  -2.57e-16,  -8.19e-16,  -2.64e-16,
             -3.10e-16,   1.00e+00,   2.08e-17,   3.33e-16,   2.15e-16],
           [ -2.92e-15,  -2.37e-15,   2.85e-15,  -2.29e-15,   4.85e-15,
              3.52e-15,  -2.52e-15,  -1.88e-15,  -9.58e-16,   2.15e-15,
              6.87e-16,   1.14e-16,   1.00e+00,   3.77e-15,  -1.33e-15],
           [  2.12e-16,   4.60e-17,   1.69e-15,   4.57e-16,  -1.66e-15,
             -1.12e-15,  -1.92e-16,  -2.53e-15,  -2.21e-15,  -2.87e-15,
             -1.64e-15,   7.18e-16,   1.24e-15,   1.00e+00,   1.33e-15],
           [  0.00e+00,  -4.72e-16,   6.25e-16,   0.00e+00,   2.22e-16,
              1.39e-16,   0.00e+00,  -3.33e-16,  -4.44e-16,  -2.22e-16,
              0.00e+00,   1.11e-16,  -1.80e-16,   1.67e-16,   1.00e+00]])

```python
np.set_printoptions(suppress=True, precision=2)
result = a3.dot(a3_inv)
result
```

    array([[ 1., -0., -0.,  0.,  0.,  0., -0.,  0.,  0.,  0., -0.,  0., -0.,
            -0., -0.],
           [ 0.,  1.,  0.,  0.,  0.,  0., -0., -0., -0., -0.,  0., -0.,  0.,
            -0., -0.],
           [-0., -0.,  1.,  0.,  0.,  0., -0., -0., -0., -0., -0., -0., -0.,
             0., -0.],
           [ 0.,  0.,  0.,  1.,  0.,  0.,  0., -0., -0., -0., -0., -0., -0.,
            -0.,  0.],
           [-0., -0.,  0.,  0.,  1.,  0., -0., -0., -0., -0., -0.,  0., -0.,
             0., -0.],
           [-0.,  0.,  0., -0.,  0.,  1., -0., -0., -0., -0., -0., -0., -0.,
             0., -0.],
           [-0.,  0., -0., -0., -0., -0.,  1.,  0.,  0.,  0., -0., -0.,  0.,
            -0.,  0.],
           [-0., -0.,  0.,  0.,  0.,  0., -0.,  1., -0., -0.,  0.,  0., -0.,
             0., -0.],
           [ 0.,  0.,  0.,  0., -0., -0., -0., -0.,  1., -0., -0., -0.,  0.,
            -0.,  0.],
           [-0., -0.,  0.,  0.,  0.,  0.,  0., -0., -0.,  1., -0., -0., -0.,
            -0.,  0.],
           [-0., -0., -0., -0.,  0.,  0.,  0.,  0.,  0.,  0.,  1., -0.,  0.,
             0.,  0.],
           [-0., -0.,  0.,  0.,  0., -0.,  0., -0., -0., -0., -0.,  1.,  0.,
             0.,  0.],
           [-0., -0.,  0., -0.,  0.,  0., -0., -0., -0.,  0.,  0.,  0.,  1.,
             0., -0.],
           [ 0.,  0.,  0.,  0., -0., -0., -0., -0., -0., -0., -0.,  0.,  0.,
             1.,  0.],
           [ 0., -0.,  0.,  0.,  0.,  0.,  0., -0., -0., -0.,  0.,  0., -0.,
             0.,  1.]])

```python
result.round().astype(int)
```

    array([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
           [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]])

## without pivoting, error occur

```python
GJ_inv(a3)
```

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:8: RuntimeWarning: divide by zero encountered in double_scalars

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:8: RuntimeWarning: invalid value encountered in double_scalars

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:15: RuntimeWarning: invalid value encountered in double_scalars
      from ipykernel import kernelapp as app

    array([[ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan],
           [ nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,  nan,
             nan,  nan,  nan,  nan]])

```python
a3_inv_scipy = sp.linalg.inv(a3)
```

```python
diff = a3_inv - a3_inv_scipy
diff
```

    array([[-0.,  0.,  0.,  0., -0.,  0.,  0., -0., -0., -0., -0.,  0.,  0.,
             0.,  0.],
           [-0., -0.,  0., -0.,  0.,  0., -0., -0., -0.,  0.,  0., -0.,  0.,
             0., -0.],
           [-0., -0.,  0.,  0.,  0.,  0., -0.,  0., -0., -0.,  0.,  0., -0.,
             0., -0.],
           [ 0., -0., -0.,  0.,  0., -0.,  0.,  0.,  0.,  0.,  0., -0., -0.,
            -0., -0.],
           [-0., -0., -0., -0.,  0.,  0., -0.,  0.,  0.,  0.,  0.,  0., -0.,
            -0.,  0.],
           [ 0.,  0., -0.,  0., -0., -0.,  0., -0.,  0., -0., -0.,  0.,  0.,
            -0.,  0.],
           [-0., -0., -0.,  0.,  0.,  0., -0.,  0., -0.,  0.,  0.,  0., -0.,
             0., -0.],
           [ 0.,  0.,  0., -0., -0., -0.,  0., -0.,  0., -0., -0., -0.,  0.,
            -0.,  0.],
           [-0., -0., -0., -0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -0., -0.,
            -0., -0.],
           [ 0.,  0., -0., -0., -0., -0., -0.,  0.,  0.,  0., -0.,  0.,  0.,
            -0.,  0.],
           [-0.,  0.,  0., -0., -0., -0., -0., -0., -0., -0., -0.,  0.,  0.,
            -0.,  0.],
           [ 0.,  0., -0., -0.,  0., -0., -0.,  0.,  0.,  0.,  0., -0., -0.,
            -0., -0.],
           [-0., -0., -0., -0.,  0., -0.,  0., -0.,  0.,  0.,  0., -0.,  0.,
            -0., -0.],
           [ 0., -0.,  0.,  0., -0.,  0.,  0.,  0., -0., -0.,  0., -0.,  0.,
             0., -0.],
           [ 0., -0.,  0.,  0.,  0.,  0., -0., -0.,  0., -0., -0.,  0., -0.,
             0., -0.]])

```python
np.linalg.norm(diff)
```

    3.8769658820079069e-15

```python
Sum = 0
for i in diff:
    for j in i:
        Sum = j**2 + Sum
Sum**0.5
```

    3.8769658820079069e-15

# Cholesky Factorization

```python
np.random.seed(4)
a = np.ceil(10*np.random.random([4,4])).astype(int)
b = np.tril(a)
b
```

    array([[10,  0,  0,  0],
           [ 7,  3,  0,  0],
           [ 3,  5,  8,  0],
           [ 9, 10,  2,  6]])

```python
A = b.dot(b.T)
A
```

    array([[100,  70,  30,  90],
           [ 70,  58,  36,  93],
           [ 30,  36,  98,  93],
           [ 90,  93,  93, 221]])

```python
sp.linalg.cholesky(A)
```

    array([[ 10.,   7.,   3.,   9.],
           [  0.,   3.,   5.,  10.],
           [  0.,   0.,   8.,   2.],
           [  0.,   0.,   0.,   6.]])

```python
sp.linalg.cholesky(A, lower=True)
```

    array([[ 10.,   0.,   0.,   0.],
           [  7.,   3.,   0.,   0.],
           [  3.,   5.,   8.,   0.],
           [  9.,  10.,   2.,   6.]])

```python
def cholesky_user(A):
    N = A.shape[0]
    L = np.zeros([N,N])
    for i in range(N):
        for j in range(i+1):
            if i == j:
                Sum = 0
                for k in range(j):
                    Sum = Sum + L[j,k]**2
                try:
                    L[i,j] = (A[i,i] - Sum)**0.5
                except Exception as e:
                    print(e)
                    return None
            else:
                Sum = 0
                for k in range(j):
                    Sum = Sum + L[i,k]*L[j,k]
                L[i,j] = (A[i,j] - Sum) / L[j,j]
    return L
```

```python
cholesky_user(A)
```

    array([[ 10.,   0.,   0.,   0.],
           [  7.,   3.,   0.,   0.],
           [  3.,   5.,   8.,   0.],
           [  9.,  10.,   2.,   6.]])

```python
sp.linalg.cholesky(A, lower=True)
```

    array([[ 10.,   0.,   0.,   0.],
           [  7.,   3.,   0.,   0.],
           [  3.,   5.,   8.,   0.],
           [  9.,  10.,   2.,   6.]])

```python
_ - __
```

    array([[ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.]])

```python
%%timeit -n 5
cholesky_user(A)
```

    5 loops, best of 3: 39.2 µs per loop

```python
%%timeit -n 5
sp.linalg.cholesky(A, lower=True)
```

    5 loops, best of 3: 14.8 µs per loop

## with zero on one of the diagonal terms

```python
np.random.seed(4)
a = np.ceil(10*np.random.random([4,4])).astype(int)
b = np.tril(a)
b[2,2] = 0
b
```

    array([[10,  0,  0,  0],
           [ 7,  3,  0,  0],
           [ 3,  5,  0,  0],
           [ 9, 10,  2,  6]])

```python
A = b.dot(b.T)
A
```

    array([[100,  70,  30,  90],
           [ 70,  58,  36,  93],
           [ 30,  36,  34,  77],
           [ 90,  93,  77, 221]])

```python
try:
    sp.linalg.cholesky(A)
except Exception as e:
    print(e)
```

    3-th leading minor not positive definite

```python
cholesky_user(A)
```

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:19: RuntimeWarning: invalid value encountered in double_scalars

    array([[ 10.,   0.,   0.,   0.],
           [  7.,   3.,   0.,   0.],
           [  3.,   5.,   0.,   0.],
           [  9.,  10.,  nan,  nan]])

## if not symmetric

```python
np.random.seed(4)
A = np.ceil(10*np.random.random([4,4])).astype(int)
A
```

    array([[10,  6, 10,  8],
           [ 7,  3, 10,  1],
           [ 3,  5,  8,  2],
           [ 9, 10,  2,  6]])

```python
try:
    sp.linalg.cholesky(A)
except Exception as e:
    print(e)
```

    2-th leading minor not positive definite

```python
cholesky_user(A)
```

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:11: RuntimeWarning: invalid value encountered in double_scalars
      # This is added back by InteractiveShellApp.init_path()

    array([[ 3.16,  0.  ,  0.  ,  0.  ],
           [ 2.21,   nan,  0.  ,  0.  ],
           [ 0.95,   nan,   nan,  0.  ],
           [ 2.85,   nan,   nan,   nan]])

# Compare Cholesky with LU

```python
np.random.seed(42)
a = np.ceil(10*np.random.random([400,400])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
```

```python
%%timeit -n 5
p1,l1,u1 = sp.linalg.lu(A)
```

    5 loops, best of 3: 2.84 ms per loop

```python
%%timeit -n 5
ans1 = sp.linalg.cholesky(A, lower = True)
```

    5 loops, best of 3: 836 µs per loop

```python
%%timeit -n 5
ans2 = np.linalg.cholesky(A)
```

    5 loops, best of 3: 1.59 ms per loop

```python
p,l,u = sp.linalg.lu(A)
```

```python
p
```

    array([[ 0.,  0.,  0., ...,  0.,  0.,  1.],
           [ 0.,  0.,  0., ...,  0.,  0.,  0.],
           [ 0.,  0.,  0., ...,  0.,  0.,  0.],
           ...,
           [ 0.,  0.,  0., ...,  0.,  0.,  0.],
           [ 0.,  0.,  0., ...,  0.,  0.,  0.],
           [ 0.,  0.,  0., ...,  0.,  0.,  0.]])

```python
l
```

    array([[ 1.  ,  0.  ,  0.  , ...,  0.  ,  0.  ,  0.  ],
           [ 0.1 ,  1.  ,  0.  , ...,  0.  ,  0.  ,  0.  ],
           [ 0.2 , -0.  ,  1.  , ...,  0.  ,  0.  ,  0.  ],
           ...,
           [ 0.4 , -0.  ,  0.41, ...,  1.  ,  0.  ,  0.  ],
           [ 0.4 , -0.11,  0.37, ...,  0.58,  1.  ,  0.  ],
           [ 0.4 , -0.21, -0.12, ..., -0.03, -0.25,  1.  ]])

```python
u
```

    array([[   40. ,    70. ,   126. , ...,   718. ,   846. ,   559. ],
           [    0. ,    95. ,    57.4, ...,  4376.2,  4369.4,  3672.1],
           [    0. ,     0. ,    52.8, ...,   167.4,   178.8,    81.2],
           ...,
           [    0. ,     0. ,     0. , ...,    -0. ,    -0. ,    -0. ],
           [    0. ,     0. ,     0. , ...,     0. ,     0. ,     0. ],
           [    0. ,     0. ,     0. , ...,     0. ,     0. ,     0. ]])

```python
np.linalg.cholesky(A)
```

    array([[  4.,   0.,   0., ...,   0.,   0.,   0.],
           [  2.,  10.,   0., ...,   0.,   0.,   0.],
           [  8.,   2.,   6., ...,   0.,   0.,   0.],
           ...,
           [  3.,   2.,   5., ...,   5.,   0.,   0.],
           [  6.,   3.,  10., ...,   4.,   7.,   0.],
           [  9.,   1.,   4., ...,   3.,   8.,   5.]])

```python
sp.linalg.cholesky(A, lower = True)
```

    array([[  4.,   0.,   0., ...,   0.,   0.,   0.],
           [  2.,  10.,   0., ...,   0.,   0.,   0.],
           [  8.,   2.,   6., ...,   0.,   0.,   0.],
           ...,
           [  3.,   2.,   5., ...,   5.,   0.,   0.],
           [  6.,   3.,  10., ...,   4.,   7.,   0.],
           [  9.,   1.,   4., ...,   3.,   8.,   5.]])

# Determinant after Cholesky Decomposition

```python
np.random.seed(42)
a = np.ceil(10*np.random.random([5,5])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
A
```

    array([[ 16,   8,   4,   8,  28],
           [  8,   5,  12,   8,  16],
           [  4,  12, 182,  96,  54],
           [  8,   8,  96,  81,  60],
           [ 28,  16,  54,  60, 103]])

```python
sp.linalg.det(A)
```

    809999.9999999998

```python
L = cholesky_user(A)
```

```python
det_ans = np.product(L[range(5),range(5)]) **2 #**2 to account for L and L^T
det_ans
```

    810000.0

# Iterative method: Jacobi
# $\mathbf{x}^{(k+1)} = D^{-1} (\mathbf{b} - R \mathbf{x}^{(k)})$

```python
def jacobi(A,b):
    """This function use Jacobi method to solve Ax = b
    A is decomposed into D + R where D is the diagonal term and R is the rest
    D^-1 is 1/a_ii for all diagonal terms
    Function input: A = nxn numpy array, b = nx1 numpy array
    This function return x
    This function is not the most efficient way of Jacobi method
    It is just to show the algorithm and how it works"""
    N = A.shape[0]
    D_inv = np.zeros((N,N))
    D_inv[range(N),range(N)] = 1/A[range(N),range(N)]
    R = A.copy()
    R[range(N),range(N)] = 0
    x = np.ones((N,1))
    for i in range(2000):
        x_old = x
        x = D_inv.dot(b-R.dot(x))
        diff = np.abs(x_old-x)
        norm_av = sp.linalg.norm(diff) / N**2
        if norm_av < 1e-14:
            break
        if i%5 == 0:
            print(x,end = '\n\n')

    print('#iteration = ', i, 'diff_av = ', norm_av)
    return x
```

```python
np.random.seed(100)
A = np.ceil(100*np.random.random((5,5)))
A[range(5),range(5)] = A[range(5),range(5)] + 200
A
```

    array([[ 255.,   28.,   43.,   85.,    1.],
           [  13.,  268.,   83.,   14.,   58.],
           [  90.,   21.,  219.,   11.,   22.],
           [  98.,   82.,   18.,  282.,   28.],
           [  44.,   95.,   82.,   34.,  218.]])

```python
np.random.seed(1)
b = np.ceil(100*np.random.random((5,1)))
b
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

```python
jacobi_ans = jacobi(A,b)
```

    [[-0.45]
     [-0.35]
     [-0.65]
     [-0.69]
     [-1.1 ]]

    [[ 0.27]
     [ 0.43]
     [ 0.04]
     [ 0.13]
     [ 0.14]]

    [[ 0.13]
     [ 0.28]
     [-0.1 ]
     [-0.05]
     [-0.1 ]]

    [[ 0.16]
     [ 0.31]
     [-0.07]
     [-0.01]
     [-0.05]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    [[ 0.15]
     [ 0.3 ]
     [-0.08]
     [-0.02]
     [-0.06]]

    #iteration =  96 diff_av =  8.05882946335e-15

```python
np.set_printoptions(precision=8)
jacobi_ans
```

    array([[ 0.15168993],
           [ 0.30401938],
           [-0.07977452],
           [-0.02002359],
           [-0.06116461]])

```python
A.dot(jacobi_ans)
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

```python
b
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

Jacobi method converges and give the correct answer

```python
np.random.seed(42)
a = np.ceil(10*np.random.random([50,50])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
A[range(50),range(50)] = A[range(50),range(50)] + 200000
A
```

    array([[200016,     40,      4, ...,     28,     16,     12],
           [    40, 200164,     66, ...,    150,     48,    110],
           [     4,     66, 200066, ...,    105,     31,     97],
           ...,
           [    28,    150,    105, ..., 201804,   1352,   1635],
           [    16,     48,     31, ...,   1352, 201658,   1525],
           [    12,    110,     97, ...,   1635,   1525, 202131]])

```python
np.random.seed(1)
b = np.ceil(100*np.random.random((50,1)))
b
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.],
           [ 10.],
           [ 19.],
           [ 35.],
           [ 40.],
           [ 54.],
           [ 42.],
           [ 69.],
           [ 21.],
           [ 88.],
           [  3.],
           [ 68.],
           [ 42.],
           [ 56.],
           [ 15.],
           [ 20.],
           [ 81.],
           [ 97.],
           [ 32.],
           [ 70.],
           [ 88.],
           [ 90.],
           [  9.],
           [  4.],
           [ 17.],
           [ 88.],
           [ 10.],
           [ 43.],
           [ 96.],
           [ 54.],
           [ 70.],
           [ 32.],
           [ 69.],
           [ 84.],
           [  2.],
           [ 76.],
           [ 99.],
           [ 75.],
           [ 29.],
           [ 79.],
           [ 11.],
           [ 45.],
           [ 91.],
           [ 30.],
           [ 29.],
           [ 14.]])

```python
try:
    ans = jacobi(A,b)
except Exception as e:
    print(e)
```

    [[-0.00488961]
     [-0.02266641]
     [-0.01536993]
     [-0.02573724]
     [-0.03386385]
     [-0.03582099]
     [-0.05650403]
     [-0.04919678]
     [-0.06817672]
     [-0.05730834]
     [-0.09476197]
     [-0.08457448]
     [-0.05980896]
     [-0.08441507]
     [-0.08665247]
     [-0.1007044 ]
     [-0.12825209]
     [-0.08796209]
     [-0.10641925]
     [-0.10555284]
     [-0.14321314]
     [-0.1484513 ]
     [-0.14534297]
     [-0.15250588]
     [-0.13321586]
     [-0.11809788]
     [-0.1561532 ]
     [-0.17393875]
     [-0.14555345]
     [-0.1791533 ]
     [-0.15114521]
     [-0.17394804]
     [-0.15181981]
     [-0.17476239]
     [-0.15668771]
     [-0.17725182]
     [-0.17085036]
     [-0.20487094]
     [-0.17415022]
     [-0.17865119]
     [-0.16724991]
     [-0.18681123]
     [-0.20059019]
     [-0.17835191]
     [-0.18949224]
     [-0.19752603]
     [-0.1859551 ]
     [-0.19036788]
     [-0.16870642]
     [-0.19760452]]

    [[ 0.00020929]
     [ 0.00036167]
     [ 0.00000274]
     [ 0.00015129]
     [ 0.00007027]
     [ 0.0000449 ]
     [ 0.00008733]
     [ 0.00016804]
     [ 0.00019041]
     [ 0.00026192]
     [ 0.00019699]
     [ 0.00033318]
     [ 0.00009684]
     [ 0.0004282 ]
     [ 0.00000324]
     [ 0.00032595]
     [ 0.00019243]
     [ 0.00026781]
     [ 0.00006072]
     [ 0.00008564]
     [ 0.00038468]
     [ 0.00046419]
     [ 0.00013962]
     [ 0.00032866]
     [ 0.00042177]
     [ 0.00043401]
     [ 0.00002379]
     [-0.0000037 ]
     [ 0.00006535]
     [ 0.00041532]
     [ 0.00002969]
     [ 0.00019083]
     [ 0.00045935]
     [ 0.00024614]
     [ 0.00032852]
     [ 0.00013633]
     [ 0.00032183]
     [ 0.00039151]
     [-0.00001318]
     [ 0.00035565]
     [ 0.00047229]
     [ 0.00034926]
     [ 0.0001179 ]
     [ 0.00037057]
     [ 0.00002918]
     [ 0.00019828]
     [ 0.00042952]
     [ 0.00012443]
     [ 0.00012272]
     [ 0.00004338]]

    [[ 0.0002089 ]
     [ 0.00035989]
     [ 0.00000154]
     [ 0.00014928]
     [ 0.00006754]
     [ 0.00004204]
     [ 0.0000827 ]
     [ 0.00016408]
     [ 0.00018488]
     [ 0.0002573 ]
     [ 0.00018909]
     [ 0.00032618]
     [ 0.00009174]
     [ 0.00042104]
     [-0.00000416]
     [ 0.00031742]
     [ 0.00018141]
     [ 0.00026026]
     [ 0.00005141]
     [ 0.00007658]
     [ 0.00037225]
     [ 0.00045145]
     [ 0.0001269 ]
     [ 0.00031539]
     [ 0.0004098 ]
     [ 0.00042343]
     [ 0.00000999]
     [-0.0000191 ]
     [ 0.00005211]
     [ 0.00039917]
     [ 0.00001604]
     [ 0.0001754 ]
     [ 0.00044532]
     [ 0.00022996]
     [ 0.00031425]
     [ 0.00011996]
     [ 0.00030613]
     [ 0.000373  ]
     [-0.00002933]
     [ 0.00033931]
     [ 0.00045666]
     [ 0.00033207]
     [ 0.00009951]
     [ 0.00035397]
     [ 0.00001186]
     [ 0.00018035]
     [ 0.00041237]
     [ 0.0001069 ]
     [ 0.00010686]
     [ 0.00002511]]

    #iteration =  15 diff_av =  2.16372742506e-15

Jacobi method work fine for the above case

```python
np.set_printoptions(suppress=False)
sp.linalg.norm(A.dot(ans) - b)/sp.linalg.norm(b) * 100
```

    4.259427776293408e-08

## Check convergence condition
### Sufficient but not necessary condition $|a_{ii}| \geq \sum_{j\neq i} |a_{ij}| \quad\text{for all } i$
### Standard convergence condition $\text{max abs( Eigenvalues( } D^{-1}R\text{ ))}<1$

```python
for i in range(A.shape[0]):
    off_diag = A[i,:].sum() - A[i,i]
    print('diagonal - sum(non-diagonal) = ', A[i,i] - off_diag)
```

    diagonal - sum(non-diagonal) =  198996
    diagonal - sum(non-diagonal) =  195554
    diagonal - sum(non-diagonal) =  196990
    diagonal - sum(non-diagonal) =  194956
    diagonal - sum(non-diagonal) =  193390
    diagonal - sum(non-diagonal) =  192982
    diagonal - sum(non-diagonal) =  188984
    diagonal - sum(non-diagonal) =  190370
    diagonal - sum(non-diagonal) =  186730
    diagonal - sum(non-diagonal) =  188753
    diagonal - sum(non-diagonal) =  181575
    diagonal - sum(non-diagonal) =  183461
    diagonal - sum(non-diagonal) =  188272
    diagonal - sum(non-diagonal) =  183441
    diagonal - sum(non-diagonal) =  183104
    diagonal - sum(non-diagonal) =  180328
    diagonal - sum(non-diagonal) =  175100
    diagonal - sum(non-diagonal) =  182762
    diagonal - sum(non-diagonal) =  179265
    diagonal - sum(non-diagonal) =  179390
    diagonal - sum(non-diagonal) =  172170
    diagonal - sum(non-diagonal) =  171043
    diagonal - sum(non-diagonal) =  171825
    diagonal - sum(non-diagonal) =  170367
    diagonal - sum(non-diagonal) =  174042
    diagonal - sum(non-diagonal) =  176891
    diagonal - sum(non-diagonal) =  169708
    diagonal - sum(non-diagonal) =  166278
    diagonal - sum(non-diagonal) =  171831
    diagonal - sum(non-diagonal) =  165265
    diagonal - sum(non-diagonal) =  170693
    diagonal - sum(non-diagonal) =  166214
    diagonal - sum(non-diagonal) =  170490
    diagonal - sum(non-diagonal) =  166219
    diagonal - sum(non-diagonal) =  169596
    diagonal - sum(non-diagonal) =  165746
    diagonal - sum(non-diagonal) =  166969
    diagonal - sum(non-diagonal) =  160315
    diagonal - sum(non-diagonal) =  166377
    diagonal - sum(non-diagonal) =  165411
    diagonal - sum(non-diagonal) =  167666
    diagonal - sum(non-diagonal) =  163959
    diagonal - sum(non-diagonal) =  161427
    diagonal - sum(non-diagonal) =  165677
    diagonal - sum(non-diagonal) =  163637
    diagonal - sum(non-diagonal) =  161749
    diagonal - sum(non-diagonal) =  164145
    diagonal - sum(non-diagonal) =  163357
    diagonal - sum(non-diagonal) =  167608
    diagonal - sum(non-diagonal) =  162175

```python
N = A.shape[0]
D_inv = np.zeros((N,N))
D_inv[range(N),range(N)] = 1/A[range(N),range(N)]
R = A.copy()
R[range(N),range(N)] = 0
np.abs(sp.linalg.eigvals(D_inv.dot(R))).max()
```

    0.15525468366230136

## compare value one by one (& export data)

```python

```

```python
col1 = A.dot(ans)
```

```python
col2 = b
```

```python
dt = pd.DataFrame(data = np.hstack([col1,col2,col1-col2]),
                  columns=['A.dot(ans)','b', 'A.dot(ans) - b'])
dt.to_csv('Jacobi_result.csv')
dt
```

<div>
<style>
    .dataframe thead tr:only-child th {
        text-align: right;
    }

    .dataframe thead th {
        text-align: left;
    }

    .dataframe tbody tr th {
        vertical-align: top;
    }
</style>
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>A.dot(ans)</th>
      <th>b</th>
      <th>A.dot(ans) - b</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>42.0</td>
      <td>42.0</td>
      <td>7.316743e-10</td>
    </tr>
    <tr>
      <th>1</th>
      <td>73.0</td>
      <td>73.0</td>
      <td>3.352710e-09</td>
    </tr>
    <tr>
      <th>2</th>
      <td>1.0</td>
      <td>1.0</td>
      <td>2.248211e-09</td>
    </tr>
    <tr>
      <th>3</th>
      <td>31.0</td>
      <td>31.0</td>
      <td>3.789875e-09</td>
    </tr>
    <tr>
      <th>4</th>
      <td>15.0</td>
      <td>15.0</td>
      <td>5.141253e-09</td>
    </tr>
    <tr>
      <th>5</th>
      <td>10.0</td>
      <td>10.0</td>
      <td>5.381203e-09</td>
    </tr>
    <tr>
      <th>6</th>
      <td>19.0</td>
      <td>19.0</td>
      <td>8.715883e-09</td>
    </tr>
    <tr>
      <th>7</th>
      <td>35.0</td>
      <td>35.0</td>
      <td>7.442544e-09</td>
    </tr>
    <tr>
      <th>8</th>
      <td>40.0</td>
      <td>40.0</td>
      <td>1.043262e-08</td>
    </tr>
    <tr>
      <th>9</th>
      <td>54.0</td>
      <td>54.0</td>
      <td>8.685554e-09</td>
    </tr>
    <tr>
      <th>10</th>
      <td>42.0</td>
      <td>42.0</td>
      <td>1.489620e-08</td>
    </tr>
    <tr>
      <th>11</th>
      <td>69.0</td>
      <td>69.0</td>
      <td>1.319873e-08</td>
    </tr>
    <tr>
      <th>12</th>
      <td>21.0</td>
      <td>21.0</td>
      <td>9.607835e-09</td>
    </tr>
    <tr>
      <th>13</th>
      <td>88.0</td>
      <td>88.0</td>
      <td>1.349062e-08</td>
    </tr>
    <tr>
      <th>14</th>
      <td>3.0</td>
      <td>3.0</td>
      <td>1.394296e-08</td>
    </tr>
    <tr>
      <th>15</th>
      <td>68.0</td>
      <td>68.0</td>
      <td>1.606581e-08</td>
    </tr>
    <tr>
      <th>16</th>
      <td>42.0</td>
      <td>42.0</td>
      <td>2.080829e-08</td>
    </tr>
    <tr>
      <th>17</th>
      <td>56.0</td>
      <td>56.0</td>
      <td>1.422748e-08</td>
    </tr>
    <tr>
      <th>18</th>
      <td>15.0</td>
      <td>15.0</td>
      <td>1.755305e-08</td>
    </tr>
    <tr>
      <th>19</th>
      <td>20.0</td>
      <td>20.0</td>
      <td>1.709546e-08</td>
    </tr>
    <tr>
      <th>20</th>
      <td>81.0</td>
      <td>81.0</td>
      <td>2.348943e-08</td>
    </tr>
    <tr>
      <th>21</th>
      <td>97.0</td>
      <td>97.0</td>
      <td>2.404808e-08</td>
    </tr>
    <tr>
      <th>22</th>
      <td>32.0</td>
      <td>32.0</td>
      <td>2.403634e-08</td>
    </tr>
    <tr>
      <th>23</th>
      <td>70.0</td>
      <td>70.0</td>
      <td>2.508342e-08</td>
    </tr>
    <tr>
      <th>24</th>
      <td>88.0</td>
      <td>88.0</td>
      <td>2.258433e-08</td>
    </tr>
    <tr>
      <th>25</th>
      <td>90.0</td>
      <td>90.0</td>
      <td>1.995809e-08</td>
    </tr>
    <tr>
      <th>26</th>
      <td>9.0</td>
      <td>9.0</td>
      <td>2.609357e-08</td>
    </tr>
    <tr>
      <th>27</th>
      <td>4.0</td>
      <td>4.0</td>
      <td>2.913193e-08</td>
    </tr>
    <tr>
      <th>28</th>
      <td>17.0</td>
      <td>17.0</td>
      <td>2.503499e-08</td>
    </tr>
    <tr>
      <th>29</th>
      <td>88.0</td>
      <td>88.0</td>
      <td>3.057720e-08</td>
    </tr>
    <tr>
      <th>30</th>
      <td>10.0</td>
      <td>10.0</td>
      <td>2.580600e-08</td>
    </tr>
    <tr>
      <th>31</th>
      <td>43.0</td>
      <td>43.0</td>
      <td>2.919178e-08</td>
    </tr>
    <tr>
      <th>32</th>
      <td>96.0</td>
      <td>96.0</td>
      <td>2.652394e-08</td>
    </tr>
    <tr>
      <th>33</th>
      <td>54.0</td>
      <td>54.0</td>
      <td>3.063180e-08</td>
    </tr>
    <tr>
      <th>34</th>
      <td>70.0</td>
      <td>70.0</td>
      <td>2.697900e-08</td>
    </tr>
    <tr>
      <th>35</th>
      <td>32.0</td>
      <td>32.0</td>
      <td>3.100639e-08</td>
    </tr>
    <tr>
      <th>36</th>
      <td>69.0</td>
      <td>69.0</td>
      <td>2.972990e-08</td>
    </tr>
    <tr>
      <th>37</th>
      <td>84.0</td>
      <td>84.0</td>
      <td>3.509113e-08</td>
    </tr>
    <tr>
      <th>38</th>
      <td>2.0</td>
      <td>2.0</td>
      <td>3.057989e-08</td>
    </tr>
    <tr>
      <th>39</th>
      <td>76.0</td>
      <td>76.0</td>
      <td>3.094534e-08</td>
    </tr>
    <tr>
      <th>40</th>
      <td>99.0</td>
      <td>99.0</td>
      <td>2.958207e-08</td>
    </tr>
    <tr>
      <th>41</th>
      <td>75.0</td>
      <td>75.0</td>
      <td>3.260220e-08</td>
    </tr>
    <tr>
      <th>42</th>
      <td>29.0</td>
      <td>29.0</td>
      <td>3.491621e-08</td>
    </tr>
    <tr>
      <th>43</th>
      <td>79.0</td>
      <td>79.0</td>
      <td>3.148396e-08</td>
    </tr>
    <tr>
      <th>44</th>
      <td>11.0</td>
      <td>11.0</td>
      <td>3.286018e-08</td>
    </tr>
    <tr>
      <th>45</th>
      <td>45.0</td>
      <td>45.0</td>
      <td>3.398931e-08</td>
    </tr>
    <tr>
      <th>46</th>
      <td>91.0</td>
      <td>91.0</td>
      <td>3.252404e-08</td>
    </tr>
    <tr>
      <th>47</th>
      <td>30.0</td>
      <td>30.0</td>
      <td>3.325217e-08</td>
    </tr>
    <tr>
      <th>48</th>
      <td>29.0</td>
      <td>29.0</td>
      <td>3.005018e-08</td>
    </tr>
    <tr>
      <th>49</th>
      <td>14.0</td>
      <td>14.0</td>
      <td>3.471172e-08</td>
    </tr>
  </tbody>
</table>
</div>

## Export using numpy.savetxt

```python
np.savetxt(fname = 'jacobi_via_np.csv', X = np.hstack([col1,col2,col1-col2]), delimiter=',')
```

# Case where Jacobi does not work
# When it seems that $|a_{ii}| \geq \sum_{j\neq i} |a_{ij}| \quad\text{for all } i$, but it is not
# Standard convergence condition
# $\text{max abs( Eigenvalues( } D^{-1}R\text{ ))}<1$

```python
np.random.seed(42)
a = np.ceil(10*np.random.random([50,50])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
A[range(50),range(50)] = A[range(50),range(50)] + 20000
np.random.seed(1)

b = np.ceil(100*np.random.random((50,1)))
try:
    ans = jacobi(A,b)
except Exception as e:
    print(e)
```

    [[-0.04886091]
     [-0.22500496]
     [-0.15324429]
     [-0.25578508]
     [-0.33586009]
     [-0.35561948]
     [-0.55698258]
     [-0.4863504 ]
     [-0.66870565]
     [-0.56583683]
     [-0.9216152 ]
     [-0.8276872 ]
     [-0.59089339]
     [-0.82743276]
     [-0.84828361]
     [-0.9807739 ]
     [-1.23234014]
     [-0.86220049]
     [-1.03489894]
     [-1.02866582]
     [-1.36824597]
     [-1.42240763]
     [-1.38623536]
     [-1.45307244]
     [-1.28096879]
     [-1.1459794 ]
     [-1.48681532]
     [-1.64418878]
     [-1.3859483 ]
     [-1.68309859]
     [-1.44065788]
     [-1.64621244]
     [-1.44573864]
     [-1.63891087]
     [-1.48768287]
     [-1.66170381]
     [-1.6040919 ]
     [-1.9021494 ]
     [-1.63459747]
     [-1.67558887]
     [-1.57015704]
     [-1.73518442]
     [-1.8440985 ]
     [-1.65531837]
     [-1.74639401]
     [-1.84212961]
     [-1.72468165]
     [-1.76192442]
     [-1.57082833]
     [-1.8047987 ]]

    [[  0.29072651]
     [  1.31684399]
     [  0.88505599]
     [  1.48812413]
     [  2.01145546]
     [  2.10827454]
     [  3.38644254]
     [  2.9027519 ]
     [  4.03315171]
     [  3.38374627]
     [  5.69878011]
     [  5.08625606]
     [  3.73957444]
     [  5.20688056]
     [  5.36774956]
     [  6.1539882 ]
     [  7.8438494 ]
     [  5.48662393]
     [  6.7038683 ]
     [  6.5449084 ]
     [  8.79772881]
     [  9.03740272]
     [  8.97590838]
     [  9.35946632]
     [  8.50590908]
     [  7.59292668]
     [  9.72071312]
     [ 10.76119679]
     [  9.31436252]
     [ 11.21378397]
     [  9.61413957]
     [ 10.79531926]
     [  9.86572195]
     [ 11.1984442 ]
     [ 10.00205229]
     [ 11.32403094]
     [ 10.87769789]
     [ 12.69537919]
     [ 11.17873976]
     [ 11.31326287]
     [ 10.81811989]
     [ 11.78598878]
     [ 12.48165736]
     [ 11.36686077]
     [ 11.78016624]
     [ 12.34939317]
     [ 11.73802582]
     [ 11.96593493]
     [ 10.87714578]
     [ 12.31503771]]

    [[ -1.92215915]
     [ -8.75141136]
     [ -5.90029799]
     [ -9.90939244]
     [-13.4043386 ]
     [-14.05184608]
     [-22.56940858]
     [-19.33891133]
     [-26.87312748]
     [-22.5383346 ]
     [-37.97643109]
     [-33.88271772]
     [-24.92270977]
     [-34.67942787]
     [-35.78418516]
     [-41.00127694]
     [-52.27679943]
     [-36.5565059 ]
     [-44.68678708]
     [-43.62545803]
     [-58.62152384]
     [-60.21323149]
     [-59.82791995]
     [-62.37050251]
     [-56.67245017]
     [-50.58501324]
     [-64.80154204]
     [-71.74010895]
     [-62.08902795]
     [-74.72511119]
     [-64.09025177]
     [-71.95298482]
     [-65.73439808]
     [-74.63534283]
     [-66.65358651]
     [-75.48068185]
     [-72.49132993]
     [-84.60419726]
     [-74.52346966]
     [-75.39253904]
     [-72.08242076]
     [-78.5445577 ]
     [-83.19963902]
     [-75.74854409]
     [-78.53016369]
     [-82.3119179 ]
     [-78.21851707]
     [-79.76097255]
     [-72.50223539]
     [-82.09440402]]

    [[  12.82772325]
     [  58.3580568 ]
     [  39.32714949]
     [  66.06020199]
     [  89.34889074]
     [  93.66265527]
     [ 150.43806204]
     [ 128.91181838]
     [ 179.13118097]
     [ 150.24404071]
     [ 253.13858133]
     [ 225.86309534]
     [ 166.1243639 ]
     [ 231.18083728]
     [ 238.51205332]
     [ 273.30986983]
     [ 348.45449351]
     [ 243.67995054]
     [ 297.85515671]
     [ 290.78270685]
     [ 390.75826669]
     [ 401.37354254]
     [ 398.78069632]
     [ 415.74216319]
     [ 377.77078249]
     [ 337.19725942]
     [ 431.92297919]
     [ 478.1683959 ]
     [ 413.84685988]
     [ 498.09648879]
     [ 427.18284639]
     [ 479.60187857]
     [ 438.17459579]
     [ 497.4856736 ]
     [ 444.29091742]
     [ 503.11199637]
     [ 483.20095464]
     [ 563.94148725]
     [ 496.72052621]
     [ 502.54084998]
     [ 480.48718752]
     [ 523.54939858]
     [ 554.559485  ]
     [ 504.9150406 ]
     [ 523.42903696]
     [ 548.64853656]
     [ 521.38253466]
     [ 531.640364  ]
     [ 483.25906642]
     [ 547.18723673]]

    [[  -85.48691341]
     [ -388.95689795]
     [ -262.13424152]
     [ -440.31150118]
     [ -595.54785174]
     [ -624.3032088 ]
     [-1002.73497016]
     [ -859.24633971]
     [-1193.98069778]
     [-1001.42862941]
     [-1687.27462968]
     [-1505.4602156 ]
     [-1107.29078082]
     [-1540.89806201]
     [-1589.79642332]
     [-1821.71606676]
     [-2322.60079976]
     [-1624.22276517]
     [-1985.34180907]
     [-1938.19894411]
     [-2604.56126378]
     [-2675.31105337]
     [-2658.05313127]
     [-2771.09497434]
     [-2517.98983488]
     [-2247.54695771]
     [-2878.97059243]
     [-3187.22047836]
     [-2758.48106869]
     [-3320.01852665]
     [-2847.37452512]
     [-3196.76071991]
     [-2920.60673507]
     [-3315.95963567]
     [-2961.38526967]
     [-3353.46986546]
     [-3220.73942709]
     [-3758.90879689]
     [-3310.87904443]
     [-3349.64655116]
     [-3202.63914456]
     [-3489.67910971]
     [-3696.39349539]
     [-3365.4703494 ]
     [-3488.90126414]
     [-3656.98823659]
     [-3475.22963741]
     [-3543.6260429 ]
     [-3221.14134642]
     [-3647.25948385]]

    [[   569.82460153]
     [  2592.5995682 ]
     [  1747.24226185]
     [  2934.88490088]
     [  3969.59867308]
     [  4161.26396651]
     [  6683.68479792]
     [  5727.27466642]
     [  7958.43094722]
     [  6674.99054687]
     [ 11246.45694997]
     [ 10034.59262169]
     [  7380.59704523]
     [ 10270.80920287]
     [ 10596.70640377]
     [ 12142.57925054]
     [ 15481.19068212]
     [ 10826.19357131]
     [ 13233.1985159 ]
     [ 12918.97178938]
     [ 17360.59800358]
     [ 17832.1831715 ]
     [ 17717.12674549]
     [ 18470.61575555]
     [ 16783.56425233]
     [ 14980.9421686 ]
     [ 19189.63237365]
     [ 21244.25399602]
     [ 18386.51922188]
     [ 22129.44474723]
     [ 18979.0312347 ]
     [ 21307.85858914]
     [ 19467.18980306]
     [ 22102.37793483]
     [ 19738.98669284]
     [ 22352.39233578]
     [ 21467.6982791 ]
     [ 25054.84330329]
     [ 22068.49403441]
     [ 22326.92458226]
     [ 21347.06350781]
     [ 23260.30383173]
     [ 24638.13056063]
     [ 22432.39860963]
     [ 23255.09472023]
     [ 24375.48280445]
     [ 23163.99790864]
     [ 23619.86754867]
     [ 21470.36267512]
     [ 24310.62479626]]

    [[  -3798.12300223]
     [ -17280.8226513 ]
     [ -11646.16084918]
     [ -19562.32551036]
     [ -26459.16744674]
     [ -27736.70547137]
     [ -44549.77904603]
     [ -38174.86651335]
     [ -53046.53173342]
     [ -44491.81483012]
     [ -74962.713151  ]
     [ -66885.07838714]
     [ -49195.01385924]
     [ -68459.56141829]
     [ -70631.84613628]
     [ -80935.76422169]
     [-103189.13109242]
     [ -72161.46186045]
     [ -88205.25514951]
     [ -86110.7901238 ]
     [-115716.2175134 ]
     [-118859.5401017 ]
     [-118092.66156003]
     [-123114.99256405]
     [-111870.02263162]
     [ -99854.72923261]
     [-127907.58665214]
     [-141602.57221241]
     [-122554.47028356]
     [-147502.73718294]
     [-126503.83388061]
     [-142026.51102123]
     [-129757.59873191]
     [-147322.33710612]
     [-131569.25840401]
     [-148988.80439851]
     [-143091.90407787]
     [-167001.84511285]
     [-147096.50504835]
     [-148819.03393851]
     [-142287.80697425]
     [-155040.42982688]
     [-164224.28621605]
     [-149522.0647743 ]
     [-155005.73316203]
     [-162473.6140276 ]
     [-154398.50095196]
     [-157437.10159819]
     [-143109.67806033]
     [-162041.31754326]]

    [[   25316.22312791]
     [  115184.5239548 ]
     [   77626.92788925]
     [  130391.75596994]
     [  176362.31978073]
     [  184877.68630024]
     [  296944.42718379]
     [  254452.75865839]
     [  353579.13328431]
     [  296558.08257201]
     [  499660.39261904]
     [  445819.31487094]
     [  327907.01844993]
     [  456313.96362288]
     [  470793.1911666 ]
     [  539473.48856775]
     [  687802.23074233]
     [  480988.79174849]
     [  587927.91283547]
     [  573967.35803105]
     [  771300.94027602]
     [  792252.612681  ]
     [  787140.99581871]
     [  820617.11725275]
     [  745664.31596746]
     [  665576.95369001]
     [  852561.90108029]
     [  943845.16996026]
     [  816880.96361587]
     [  983172.46946965]
     [  843205.25496651]
     [  946670.92841095]
     [  864893.10194655]
     [  981970.00890542]
     [  876968.62247727]
     [  993077.75880465]
     [  953772.26570694]
     [ 1113142.83847859]
     [  980464.71664844]
     [  991946.17827739]
     [  948412.60666561]
     [ 1033414.59434261]
     [ 1094629.13232793]
     [  996632.19799102]
     [ 1033183.30096127]
     [ 1082960.11658603]
     [ 1029135.85443113]
     [ 1049389.47857444]
     [  953890.72276006]
     [ 1080078.65354554]]

    [[ -168744.06043786]
     [ -767756.91701429]
     [ -517418.58121934]
     [ -869120.01200301]
     [-1175534.61613739]
     [-1232293.38712634]
     [-1979268.89393291]
     [-1696042.7003902 ]
     [-2356764.81556265]
     [-1976693.71927491]
     [-3330462.47363827]
     [-2971587.3305107 ]
     [-2185648.56669125]
     [-3041538.90321089]
     [-3138049.53533505]
     [-3595834.75380113]
     [-4584512.91209448]
     [-3206007.80401929]
     [-3918805.42477354]
     [-3825752.01172305]
     [-5141069.55909516]
     [-5280721.92498842]
     [-5246650.71150534]
     [-5469784.18602715]
     [-4970189.86025429]
     [-4436371.37014399]
     [-5682710.64003895]
     [-6291154.91213996]
     [-5444881.05155543]
     [-6553289.11695914]
     [-5620344.36132282]
     [-6309989.85555992]
     [-5764903.63673876]
     [-6545274.1859689 ]
     [-5845392.45256796]
     [-6619312.37063597]
     [-6357323.46936852]
     [-7419600.40749783]
     [-6535240.75178817]
     [-6611769.85831294]
     [-6321598.90622015]
     [-6888175.60605494]
     [-7296198.20339196]
     [-6643004.29764367]
     [-6886633.9554877 ]
     [-7218418.93018767]
     [-6859655.86319256]
     [-6994655.45089649]
     [-6358113.05377639]
     [-7199212.68738777]]

    [[  1124755.49092736]
     [  5117447.08655614]
     [  3448828.83907259]
     [  5793077.93855856]
     [  7835469.84067849]
     [  8213792.71561699]
     [ 13192722.28105748]
     [ 11304891.62198075]
     [ 15708903.32353805]
     [ 13175557.59047133]
     [ 22199038.5563776 ]
     [ 19806973.44148535]
     [ 14568336.13317007]
     [ 20273232.31558732]
     [ 20916519.29196163]
     [ 23967864.82436724]
     [ 30557851.8552133 ]
     [ 21369491.90607155]
     [ 26120610.3856452 ]
     [ 25500367.30717937]
     [ 34267553.61675256]
     [ 35198399.79733703]
     [ 34971299.72661053]
     [ 36458585.25769842]
     [ 33128563.14968915]
     [ 29570421.50219142]
     [ 37877836.34392033]
     [ 41933392.57571444]
     [ 36292594.58515716]
     [ 43680635.62953838]
     [ 37462136.89282602]
     [ 42058936.00605088]
     [ 38425689.87141533]
     [ 43627212.47736503]
     [ 38962184.21851858]
     [ 44120710.45279412]
     [ 42374435.95889802]
     [ 49454992.15612932]
     [ 43560335.08158994]
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     [  1.02085302e+296]
     [  9.27951201e+295]
     [  1.05070765e+296]]

    [[ -1.64155339e+295]
     [ -7.46879007e+295]
     [ -5.03348216e+295]
     [ -8.45485694e+295]
     [ -1.14356785e+296]
     [ -1.19878315e+296]
     [ -1.92544586e+296]
     [ -1.64992155e+296]
     [ -2.29267639e+296]
     [ -1.92294072e+296]
     [ -3.23989591e+296]
     [ -2.89077980e+296]
     [ -2.12621337e+296]
     [ -2.95882915e+296]
     [ -3.05271533e+296]
     [ -3.49805182e+296]
     [ -4.45984447e+296]
     [ -3.11882559e+296]
     [ -3.81223982e+296]
     [ -3.72171685e+296]
     [ -5.00126645e+296]
     [ -5.13712120e+296]
     [ -5.10397649e+296]
     [ -5.32104221e+296]
     [ -4.83503355e+296]
     [ -4.31573140e+296]
     [ -5.52817847e+296]
     [ -6.12007708e+296]
     [ -5.29681628e+296]
     [ -6.37508297e+296]
     [ -5.46750815e+296]
     [ -6.13839984e+296]
     [ -5.60813637e+296]
     [ -6.36728600e+296]
     [ -5.68643642e+296]
     [ -6.43931083e+296]
     [ -6.18444630e+296]
     [ -7.21783632e+296]
     [ -6.35752540e+296]
     [ -6.43197343e+296]
     [ -6.14969321e+296]
     [ -6.70086276e+296]
     [ -7.09778984e+296]
     [ -6.46235850e+296]
     [ -6.69936303e+296]
     [ -7.02212565e+296]
     [ -6.67311857e+296]
     [ -6.80444706e+296]
     [ -6.18521441e+296]
     [ -7.00344169e+296]]

    [[  1.09416957e+296]
     [  4.97828635e+296]
     [  3.35504349e+296]
     [  5.63554451e+296]
     [  7.62239686e+296]
     [  7.99043185e+296]
     [  1.28339674e+297]
     [  1.09974733e+297]
     [  1.52817250e+297]
     [  1.28172695e+297]
     [  2.15953714e+297]
     [  1.92683547e+297]
     [  1.41721737e+297]
     [  1.97219344e+297]
     [  2.03477282e+297]
     [  2.33160973e+297]
     [  2.97268803e+297]
     [  2.07883830e+297]
     [  2.54103024e+297]
     [  2.48069259e+297]
     [  3.33357027e+297]
     [  3.42412360e+297]
     [  3.40203115e+297]
     [  3.54671527e+297]
     [  3.22276852e+297]
     [  2.87663015e+297]
     [  3.68478096e+297]
     [  4.07930815e+297]
     [  3.53056759e+297]
     [  4.24928111e+297]
     [  3.64434144e+297]
     [  4.09152110e+297]
     [  3.73807651e+297]
     [  4.24408407e+297]
     [  3.79026704e+297]
     [  4.29209188e+297]
     [  4.12221314e+297]
     [  4.81101433e+297]
     [  4.23757819e+297]
     [  4.28720117e+297]
     [  4.09904863e+297]
     [  4.46642807e+297]
     [  4.73099792e+297]
     [  4.30745419e+297]
     [  4.46542843e+297]
     [  4.68056431e+297]
     [  4.44793531e+297]
     [  4.53547169e+297]
     [  4.12272512e+297]
     [  4.66811061e+297]]

    [[ -7.29313504e+296]
     [ -3.31825299e+297]
     [ -2.23628822e+297]
     [ -3.75634528e+297]
     [ -5.08067222e+297]
     [ -5.32598419e+297]
     [ -8.55441970e+297]
     [ -7.33031330e+297]
     [ -1.01859608e+298]
     [ -8.54328978e+297]
     [ -1.43942917e+298]
     [ -1.28432298e+298]
     [ -9.44639468e+297]
     [ -1.31455612e+298]
     [ -1.35626811e+298]
     [ -1.55412333e+298]
     [ -1.98143101e+298]
     [ -1.38563974e+298]
     [ -1.69371157e+298]
     [ -1.65349380e+298]
     [ -2.22197534e+298]
     [ -2.28233323e+298]
     [ -2.26760762e+298]
     [ -2.36404613e+298]
     [ -2.14812097e+298]
     [ -1.91740409e+298]
     [ -2.45607315e+298]
     [ -2.71904336e+298]
     [ -2.35328297e+298]
     [ -2.83233803e+298]
     [ -2.42911838e+298]
     [ -2.72718385e+298]
     [ -2.49159704e+298]
     [ -2.82887397e+298]
     [ -2.52638439e+298]
     [ -2.86087334e+298]
     [ -2.74764148e+298]
     [ -3.20675862e+298]
     [ -2.82453750e+298]
     [ -2.85761346e+298]
     [ -2.73220129e+298]
     [ -2.97707630e+298]
     [ -3.15342407e+298]
     [ -2.87111302e+298]
     [ -2.97640999e+298]
     [ -3.11980779e+298]
     [ -2.96475004e+298]
     [ -3.02309699e+298]
     [ -2.74798274e+298]
     [ -3.11150683e+298]]

    [[  4.86120433e+297]
     [  2.21176568e+298]
     [  1.49058723e+298]
     [  2.50377400e+298]
     [  3.38649780e+298]
     [  3.55000932e+298]
     [  5.70190759e+298]
     [  4.88598532e+298]
     [  6.78940352e+298]
     [  5.69448900e+298]
     [  9.59444638e+298]
     [  8.56059347e+298]
     [  6.29644926e+298]
     [  8.76211102e+298]
     [  9.04014029e+298]
     [  1.03589348e+299]
     [  1.32071337e+299]
     [  9.23591550e+298]
     [  1.12893536e+299]
     [  1.10212839e+299]
     [  1.48104705e+299]
     [  1.52127831e+299]
     [  1.51146303e+299]
     [  1.57574365e+299]
     [  1.43181977e+299]
     [  1.27803654e+299]
     [  1.63708383e+299]
     [  1.81236537e+299]
     [  1.56856952e+299]
     [  1.88788138e+299]
     [  1.61911725e+299]
     [  1.81779137e+299]
     [  1.66076211e+299]
     [  1.88557243e+299]
     [  1.68394945e+299]
     [  1.90690146e+299]
     [  1.83142731e+299]
     [  2.13744964e+299]
     [  1.88268198e+299]
     [  1.90472860e+299]
     [  1.82113572e+299]
     [  1.98435599e+299]
     [  2.10189975e+299]
     [  1.91372667e+299]
     [  1.98391187e+299]
     [  2.07949298e+299]
     [  1.97613998e+299]
     [  2.01503086e+299]
     [  1.83165477e+299]
     [  2.07396002e+299]]

    [[ -3.24021253e+298]
     [ -1.47424185e+299]
     [ -9.93543798e+298]
     [ -1.66887860e+299]
     [ -2.25725393e+299]
     [ -2.36624176e+299]
     [ -3.80057927e+299]
     [ -3.25673018e+299]
     [ -4.52544449e+299]
     [ -3.79563444e+299]
     [ -6.39513241e+299]
     [ -5.70602269e+299]
     [ -4.19686818e+299]
     [ -5.84034325e+299]
     [ -6.02566233e+299]
     [ -6.90469854e+299]
     [ -8.80315191e+299]
     [ -6.15615536e+299]
     [ -7.52486472e+299]
     [ -7.34618416e+299]
     [ -9.87184837e+299]
     [ -1.01400079e+300]
     [ -1.00745846e+300]
     [ -1.05030441e+300]
     [ -9.54372629e+299]
     [ -8.51869149e+299]
     [ -1.09119041e+300]
     [ -1.20802348e+300]
     [ -1.04552252e+300]
     [ -1.25835832e+300]
     [ -1.07921487e+300]
     [ -1.21164015e+300]
     [ -1.10697305e+300]
     [ -1.25681930e+300]
     [ -1.12242846e+300]
     [ -1.27103606e+300]
     [ -1.22072912e+300]
     [ -1.42470685e+300]
     [ -1.25489268e+300]
     [ -1.26958775e+300]
     [ -1.21386932e+300]
     [ -1.32266301e+300]
     [ -1.40101124e+300]
     [ -1.27558537e+300]
     [ -1.32236698e+300]
     [ -1.38607611e+300]
     [ -1.31718666e+300]
     [ -1.34310920e+300]
     [ -1.22088074e+300]
     [ -1.38238814e+300]]

    [[  2.15974818e+299]
     [  9.82648865e+299]
     [  6.62241873e+299]
     [  1.11238306e+300]
     [  1.50456183e+300]
     [  1.57720714e+300]
     [  2.53325796e+300]
     [  2.17075795e+300]
     [  3.01641341e+300]
     [  2.52996201e+300]
     [  4.26264496e+300]
     [  3.80332217e+300]
     [  2.79740244e+300]
     [  3.89285290e+300]
     [  4.01637645e+300]
     [  4.60229383e+300]
     [  5.86769886e+300]
     [  4.10335595e+300]
     [  5.01566264e+300]
     [  4.89656396e+300]
     [  6.58003338e+300]
     [  6.75877386e+300]
     [  6.71516626e+300]
     [  7.00075386e+300]
     [  6.36132517e+300]
     [  5.67809313e+300]
     [  7.27327753e+300]
     [  8.05202280e+300]
     [  6.96888044e+300]
     [  8.38752728e+300]
     [  7.19345519e+300]
     [  8.07612957e+300]
     [  7.37847601e+300]
     [  8.37726900e+300]
     [  7.48149330e+300]
     [  8.47203016e+300]
     [  8.13671167e+300]
     [  9.49631547e+300]
     [  8.36442725e+300]
     [  8.46237653e+300]
     [  8.09098795e+300]
     [  8.81614709e+300]
     [  9.33837350e+300]
     [  8.50235336e+300]
     [  8.81417393e+300]
     [  9.23882413e+300]
     [  8.77964479e+300]
     [  8.95243019e+300]
     [  8.13772225e+300]
     [  9.21424214e+300]]

    [[ -1.43956984e+300]
     [ -6.54979907e+300]
     [ -4.41414158e+300]
     [ -7.41453617e+300]
     [ -1.00285850e+301]
     [ -1.05127989e+301]
     [ -1.68853100e+301]
     [ -1.44690834e+301]
     [ -2.01057595e+301]
     [ -1.68633409e+301]
     [ -2.84124564e+301]
     [ -2.53508623e+301]
     [ -1.86459524e+301]
     [ -2.59476251e+301]
     [ -2.67709654e+301]
     [ -3.06763697e+301]
     [ -3.91108665e+301]
     [ -2.73507231e+301]
     [ -3.34316598e+301]
     [ -3.26378132e+301]
     [ -4.38588982e+301]
     [ -4.50502844e+301]
     [ -4.47596199e+301]
     [ -4.66631905e+301]
     [ -4.24011091e+301]
     [ -3.78470585e+301]
     [ -4.84796840e+301]
     [ -5.36703734e+301]
     [ -4.64507397e+301]
     [ -5.59066625e+301]
     [ -4.79476320e+301]
     [ -5.38310559e+301]
     [ -4.91808794e+301]
     [ -5.58382864e+301]
     [ -4.98675363e+301]
     [ -5.64699124e+301]
     [ -5.42348630e+301]
     [ -6.32972372e+301]
     [ -5.57526903e+301]
     [ -5.64055666e+301]
     [ -5.39300937e+301]
     [ -5.87636074e+301]
     [ -6.22444825e+301]
     [ -5.66720302e+301]
     [ -5.87504554e+301]
     [ -6.15809410e+301]
     [ -5.85203030e+301]
     [ -5.96719959e+301]
     [ -5.42415990e+301]
     [ -6.14170909e+301]]

    [[  9.59538399e+300]
     [  4.36573729e+301]
     [  2.94222498e+301]
     [  4.94212367e+301]
     [  6.68450544e+301]
     [  7.00725588e+301]
     [  1.12548227e+302]
     [  9.64429844e+301]
     [  1.34013980e+302]
     [  1.12401794e+302]
     [  1.89381871e+302]
     [  1.68974962e+302]
     [  1.24283705e+302]
     [  1.72952656e+302]
     [  1.78440591e+302]
     [  2.04471877e+302]
     [  2.60691613e+302]
     [  1.82304939e+302]
     [  2.22837131e+302]
     [  2.17545785e+302]
     [  2.92339391e+302]
     [  3.00280519e+302]
     [  2.98343109e+302]
     [  3.11031267e+302]
     [  2.82622568e+302]
     [  2.52267761e+302]
     [  3.23139018e+302]
     [  3.57737311e+302]
     [  3.09615187e+302]
     [  3.72643188e+302]
     [  3.19592651e+302]
     [  3.58808332e+302]
     [  3.27812803e+302]
     [  3.72187431e+302]
     [  3.32389681e+302]
     [  3.76397504e+302]
     [  3.61499889e+302]
     [  4.21904711e+302]
     [  3.71616894e+302]
     [  3.75968610e+302]
     [  3.59468464e+302]
     [  3.91686018e+302]
     [  4.14887625e+302]
     [  3.77744711e+302]
     [  3.91598354e+302]
     [  4.10464821e+302]
     [  3.90064285e+302]
     [  3.97740838e+302]
     [  3.61544787e+302]
     [  4.09372686e+302]]

    [[ -6.39575737e+301]
     [ -2.90996134e+302]
     [ -1.96112601e+302]
     [ -3.29414893e+302]
     [ -4.45552517e+302]
     [ -4.67065294e+302]
     [ -7.50184832e+302]
     [ -6.42836106e+302]
     [ -8.93263782e+302]
     [ -7.49208786e+302]
     [ -1.26231582e+303]
     [ -1.12629454e+303]
     [ -8.28407099e+302]
     [ -1.15280767e+303]
     [ -1.18938724e+303]
     [ -1.36289753e+303]
     [ -1.73762749e+303]
     [ -1.21514486e+303]
     [ -1.48531026e+303]
     [ -1.45004104e+303]
     [ -1.94857425e+303]
     [ -2.00150546e+303]
     [ -1.98859174e+303]
     [ -2.07316406e+303]
     [ -1.88380723e+303]
     [ -1.68147871e+303]
     [ -2.15386769e+303]
     [ -2.38448096e+303]
     [ -2.06372524e+303]
     [ -2.48383537e+303]
     [ -2.13022955e+303]
     [ -2.39161980e+303]
     [ -2.18502058e+303]
     [ -2.48079754e+303]
     [ -2.21552754e+303]
     [ -2.50885958e+303]
     [ -2.40956024e+303]
     [ -2.81218570e+303]
     [ -2.47699466e+303]
     [ -2.50600081e+303]
     [ -2.39601988e+303]
     [ -2.61076445e+303]
     [ -2.76541365e+303]
     [ -2.51783933e+303]
     [ -2.61018013e+303]
     [ -2.73593366e+303]
     [ -2.59995486e+303]
     [ -2.65112256e+303]
     [ -2.40985951e+303]
     [ -2.72865409e+303]]

    array must not contain infs or NaNs

```python
np.abs(A[0,:]).sum()
```

    21036

```python
A[0,0]
```

    20016

```python
N = A.shape[0]
D_inv = np.zeros((N,N))
D_inv[range(N),range(N)] = 1/A[range(N),range(N)]
R = A.copy()
R[range(N),range(N)] = 0
```

```python
D_inv
```

    array([[  4.99600320e-05,   0.00000000e+00,   0.00000000e+00, ...,
              0.00000000e+00,   0.00000000e+00,   0.00000000e+00],
           [  0.00000000e+00,   4.95933347e-05,   0.00000000e+00, ...,
              0.00000000e+00,   0.00000000e+00,   0.00000000e+00],
           [  0.00000000e+00,   0.00000000e+00,   4.98355427e-05, ...,
              0.00000000e+00,   0.00000000e+00,   0.00000000e+00],
           ...,
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00, ...,
              4.58631444e-05,   0.00000000e+00,   0.00000000e+00],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00, ...,
              0.00000000e+00,   4.61723151e-05,   0.00000000e+00],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00, ...,
              0.00000000e+00,   0.00000000e+00,   4.51854864e-05]])

```python
R
```

    array([[   0,   40,    4, ...,   28,   16,   12],
           [  40,    0,   66, ...,  150,   48,  110],
           [   4,   66,    0, ...,  105,   31,   97],
           ...,
           [  28,  150,  105, ...,    0, 1352, 1635],
           [  16,   48,   31, ..., 1352,    0, 1525],
           [  12,  110,   97, ..., 1635, 1525,    0]])

```python
D_invR = D_inv.dot(R)
D_invR
```

    array([[ 0.        ,  0.0019984 ,  0.00019984, ...,  0.00139888,
             0.00079936,  0.00059952],
           [ 0.00198373,  0.        ,  0.00327316, ...,  0.007439  ,
             0.00238048,  0.00545527],
           [ 0.00019934,  0.00328915,  0.        , ...,  0.00523273,
             0.0015449 ,  0.00483405],
           ...,
           [ 0.00128417,  0.00687947,  0.00481563, ...,  0.        ,
             0.06200697,  0.07498624],
           [ 0.00073876,  0.00221627,  0.00143134, ...,  0.06242497,
             0.        ,  0.07041278],
           [ 0.00054223,  0.0049704 ,  0.00438299, ...,  0.07387827,
             0.06890787,  0.        ]])

```python
eig = sp.linalg.eigvals(D_invR)
```

```python
eig.shape
```

    (50,)

```python
eig
```

    array([  1.46138930e+00+0.j,   1.38422414e-01+0.j,   3.34053619e-02+0.j,
             1.36009235e-02+0.j,  -8.00992956e-02+0.j,  -7.75198996e-02+0.j,
             4.86097750e-03+0.j,  -7.52190613e-02+0.j,   2.26808430e-03+0.j,
            -3.24259739e-04+0.j,  -7.10356170e-02+0.j,  -6.98061099e-02+0.j,
            -1.97000545e-03+0.j,  -6.66305109e-02+0.j,  -6.63882281e-02+0.j,
            -3.95245554e-03+0.j,  -5.81413995e-03+0.j,  -6.53925027e-03+0.j,
            -7.70665939e-03+0.j,  -1.00201453e-02+0.j,  -1.05857716e-02+0.j,
            -1.16879851e-02+0.j,  -1.33628818e-02+0.j,  -6.27296603e-02+0.j,
            -6.18518058e-02+0.j,  -5.99549818e-02+0.j,  -5.83651641e-02+0.j,
            -1.65421983e-02+0.j,  -1.85040101e-02+0.j,  -1.94203921e-02+0.j,
            -2.08643741e-02+0.j,  -2.22502339e-02+0.j,  -2.36202057e-02+0.j,
            -5.47104157e-02+0.j,  -5.42774334e-02+0.j,  -5.27933387e-02+0.j,
            -2.75694746e-02+0.j,  -2.92900384e-02+0.j,  -3.02246726e-02+0.j,
            -3.12431801e-02+0.j,  -3.39796164e-02+0.j,  -3.57691632e-02+0.j,
            -3.89296685e-02+0.j,  -5.09338538e-02+0.j,  -4.88809144e-02+0.j,
            -4.71045906e-02+0.j,  -4.52472441e-02+0.j,  -4.56075289e-02+0.j,
            -4.25669129e-02+0.j,  -4.20537146e-02+0.j])

```python
max_eig = np.abs(eig).max()
max_eig
```

    1.4613893032019745

```python
test = np.array([[1,2,3],[4,5,6]])
print(np.sum(test,axis = 0), end = '\n\n')
print(np.sum(test,axis = 1), end = '\n\n')
```

    [5 7 9]

    [ 6 15]

```python
A_row_sum = np.sum(A,axis = 1)
A_row_sum
```

    array([21036, 24774, 23142, 25320, 26978, 27342, 31660, 30144, 34140,
           31817, 39683, 37511, 32270, 37459, 37854, 40866, 46718, 38138,
           41997, 41774, 49916, 50907, 50341, 51847, 47742, 44471, 52538,
           56312, 50413, 57619, 51503, 56320, 51750, 56751, 52784, 57240,
           55945, 63139, 56551, 57553, 55252, 59475, 62511, 57795, 60179,
           61489, 59361, 60251, 55708, 62087])

```python
A_row_sum - A[range(A.shape[0]),range(A.shape[0])]
```

    array([ 1020,  4610,  3076,  5182,  6794,  7180, 11338,  9887, 13705,
           11532, 19054, 17025, 11999, 17009, 17375, 20269, 25809, 17688,
           21366, 21192, 28873, 29932, 29258, 30740, 26850, 23790, 31415,
           35017, 29291, 36177, 30405, 35053, 30630, 35266, 31594, 35747,
           34488, 41412, 35087, 36071, 33793, 37758, 40542, 36059, 38271,
           39870, 37608, 38447, 34050, 39956])

```python
A[49,:].sum() - A[49,49]
```

    39956

```python
A[49,:]
```

    array([   12,   110,    97,    92,   121,   143,   225,   225,   297,
             231,   460,   431,   313,   470,   446,   504,   669,   494,
             496,   514,   731,   798,   873,   870,   726,   722,   896,
            1014,   881,  1146,   968,  1074,  1070,  1253,   973,  1307,
            1210,  1457,  1269,  1305,  1189,  1357,  1511,  1456,  1438,
            1457,  1495,  1635,  1525, 22131])

```python
for i in range(A.shape[0]): #non-pithonic way
    ans = A[i,:].sum() - A[i,i]
    print('diagonal - sum(non-diagonal) = ', A[i,i] - ans)
```

    diagonal - sum(non-diagonal) =  18996
    diagonal - sum(non-diagonal) =  15554
    diagonal - sum(non-diagonal) =  16990
    diagonal - sum(non-diagonal) =  14956
    diagonal - sum(non-diagonal) =  13390
    diagonal - sum(non-diagonal) =  12982
    diagonal - sum(non-diagonal) =  8984
    diagonal - sum(non-diagonal) =  10370
    diagonal - sum(non-diagonal) =  6730
    diagonal - sum(non-diagonal) =  8753
    diagonal - sum(non-diagonal) =  1575
    diagonal - sum(non-diagonal) =  3461
    diagonal - sum(non-diagonal) =  8272
    diagonal - sum(non-diagonal) =  3441
    diagonal - sum(non-diagonal) =  3104
    diagonal - sum(non-diagonal) =  328
    diagonal - sum(non-diagonal) =  -4900
    diagonal - sum(non-diagonal) =  2762
    diagonal - sum(non-diagonal) =  -735
    diagonal - sum(non-diagonal) =  -610
    diagonal - sum(non-diagonal) =  -7830
    diagonal - sum(non-diagonal) =  -8957
    diagonal - sum(non-diagonal) =  -8175
    diagonal - sum(non-diagonal) =  -9633
    diagonal - sum(non-diagonal) =  -5958
    diagonal - sum(non-diagonal) =  -3109
    diagonal - sum(non-diagonal) =  -10292
    diagonal - sum(non-diagonal) =  -13722
    diagonal - sum(non-diagonal) =  -8169
    diagonal - sum(non-diagonal) =  -14735
    diagonal - sum(non-diagonal) =  -9307
    diagonal - sum(non-diagonal) =  -13786
    diagonal - sum(non-diagonal) =  -9510
    diagonal - sum(non-diagonal) =  -13781
    diagonal - sum(non-diagonal) =  -10404
    diagonal - sum(non-diagonal) =  -14254
    diagonal - sum(non-diagonal) =  -13031
    diagonal - sum(non-diagonal) =  -19685
    diagonal - sum(non-diagonal) =  -13623
    diagonal - sum(non-diagonal) =  -14589
    diagonal - sum(non-diagonal) =  -12334
    diagonal - sum(non-diagonal) =  -16041
    diagonal - sum(non-diagonal) =  -18573
    diagonal - sum(non-diagonal) =  -14323
    diagonal - sum(non-diagonal) =  -16363
    diagonal - sum(non-diagonal) =  -18251
    diagonal - sum(non-diagonal) =  -15855
    diagonal - sum(non-diagonal) =  -16643
    diagonal - sum(non-diagonal) =  -12392
    diagonal - sum(non-diagonal) =  -17825

# Gauss–Seidel method:  $\mathbf{x}^{(k+1)} = L_*^{-1} (\mathbf{b} - U \mathbf{x}^{(k)})$
## $A=L_*+U$

```python
def gauss_seidel(A,b, maxit = 200000):
    N = A.shape[0]
    x = np.ones((N,1))
    L = np.tril(A)
    U = np.triu(A, k = 1)
    for i in range(maxit):
        x_old = x
        #L x = rhs
        rhs = b - U.dot(x)
        x = forward_sub(L,rhs,N)
        diff = (((x-x_old)**2).sum())**0.5
        if diff < 1e-10:
            print('total iteration = ', i)
            print('diff = ', diff)
            SSE = ((A.dot(x) - b)**2).sum()
            print('SSE = ', SSE)
            break
    return x

def forward_sub(L,b,N):
    y = np.empty((N,1)) #(N,1) is needed here
    for i in range(N):  #to return column vector
        Sum = 0
        for k in range(i):
            Sum = Sum + L[i,k] * y[k]
        y[i] = (b[i] - Sum)/L[i,i]
    return y
```

```python
A = np.ones((5,5))
```

```python
L = np.tril(A)
L
```

    array([[ 1.,  0.,  0.,  0.,  0.],
           [ 1.,  1.,  0.,  0.,  0.],
           [ 1.,  1.,  1.,  0.,  0.],
           [ 1.,  1.,  1.,  1.,  0.],
           [ 1.,  1.,  1.,  1.,  1.]])

```python
U = np.triu(A,k=1)
U
```

    array([[ 0.,  1.,  1.,  1.,  1.],
           [ 0.,  0.,  1.,  1.,  1.],
           [ 0.,  0.,  0.,  1.,  1.],
           [ 0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  0.,  0.,  0.]])

```python
U.sum()
```

    10.0

```python
U.dot(U)
```

    array([[ 0.,  0.,  1.,  2.,  3.],
           [ 0.,  0.,  0.,  1.,  2.],
           [ 0.,  0.,  0.,  0.,  1.],
           [ 0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.]])

```python
np.random.seed(100)
A = np.ceil(100*np.random.random((5,5)))
A[range(5),range(5)] = A[range(5),range(5)] + 200
np.random.seed(1)
b = np.ceil(100*np.random.random((5,1)))
```

```python
L = np.tril(A)
L
```

    array([[ 255.,    0.,    0.,    0.,    0.],
           [  13.,  268.,    0.,    0.,    0.],
           [  90.,   21.,  219.,    0.,    0.],
           [  98.,   82.,   18.,  282.,    0.],
           [  44.,   95.,   82.,   34.,  218.]])

```python
b
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

```python
x = forward_sub(L,b,5)
x
```

    array([[ 0.16470588],
           [ 0.2643986 ],
           [-0.08847443],
           [-0.01854369],
           [-0.04348411]])

```python
L.dot(x)
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

```python
x.shape
```

    (5, 1)

```python
x = gauss_seidel(A,b)
x
```

    total iteration =  16
    diff =  2.90075908798e-11
    SSE =  8.46717808723e-19

    array([[ 0.15168993],
           [ 0.30401938],
           [-0.07977452],
           [-0.02002359],
           [-0.06116461]])

```python
A.dot(x)
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

```python
b
```

    array([[ 42.],
           [ 73.],
           [  1.],
           [ 31.],
           [ 15.]])

```python
np.random.seed(4)
a = np.ceil(10*np.random.random([4,4])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
np.random.seed(40)
b = np.ceil(10* np.random.random([4,1]))
```

```python
A
```

    array([[100,  70,  30,  90],
           [ 70,  58,  36,  93],
           [ 30,  36,  98,  93],
           [ 90,  93,  93, 221]])

```python
b
```

    array([[ 5.],
           [ 1.],
           [ 8.],
           [ 3.]])

```python
gs_ans = gauss_seidel(A,b)
gs_ans
```

    total iteration =  324
    diff =  9.54818421815e-11
    SSE =  2.30842019339e-17

    array([[ 0.52970679],
           [-0.89313271],
           [ 0.13773148],
           [ 0.11574074]])

```python
try:
    ans = jacobi(A,b)
except Exception as e:
    print(e)
```

    [[-1.85      ]
     [-3.4137931 ]
     [-1.54081633]
     [-1.23529412]]

    [[ 51.85088851]
     [ 75.92438803]
     [ 41.14473748]
     [ 36.79018941]]

    [[-1336.94933638]
     [-1983.2186419 ]
     [-1064.51135282]
     [ -952.58893314]]

    [[ 34691.21979451]
     [ 51431.71258188]
     [ 27617.97872464]
     [ 24712.53155695]]

    [[ -899915.64514228]
     [-1334207.46170767]
     [ -716435.03634274]
     [ -641066.8730261 ]]

    [[ 23344743.94262251]
     [ 34610690.4767043 ]
     [ 18585062.05663865]
     [ 16629932.76285485]]

    [[ -6.05586550e+08]
     [ -8.97836767e+08]
     [ -4.82115536e+08]
     [ -4.31397482e+08]]

    [[  1.57095352e+10]
     [  2.32908050e+10]
     [  1.25065706e+10]
     [  1.11908924e+10]]

    [[ -4.07521428e+11]
     [ -6.04187330e+11]
     [ -3.24433248e+11]
     [ -2.90303207e+11]]

    [[  1.05715231e+13]
     [  1.56732380e+13]
     [  8.41613064e+12]
     [  7.53076243e+12]]

    [[ -2.74236134e+14]
     [ -4.06579841e+14]
     [ -2.18323046e+14]
     [ -1.95355688e+14]]

    [[  7.11396610e+15]
     [  1.05470974e+16]
     [  5.66352336e+15]
     [  5.06772658e+15]]

    [[ -1.84543565e+17]
     [ -2.73602505e+17]
     [ -1.46917595e+17]
     [ -1.31462016e+17]]

    [[  4.78724902e+18]
     [  7.09752914e+18]
     [  3.81119285e+18]
     [  3.41025931e+18]]

    [[ -1.24186141e+20]
     [ -1.84117173e+20]
     [ -9.88662447e+19]
     [ -8.84656184e+19]]

    [[  3.22151566e+21]
     [  4.77618799e+21]
     [  2.56469162e+21]
     [  2.29488872e+21]]

    [[ -8.35694148e+22]
     [ -1.23899207e+23]
     [ -6.65307266e+22]
     [ -5.95317632e+22]]

    [[  2.16787619e+24]
     [  3.21407229e+24]
     [  1.72587517e+24]
     [  1.54431489e+24]]

    [[ -5.62369280e+25]
     [ -8.33763260e+25]
     [ -4.47709690e+25]
     [ -4.00611096e+25]]

    [[  1.45884350e+27]
     [  2.16286727e+27]
     [  1.16140478e+27]
     [  1.03922620e+27]]

    [[ -3.78438940e+28]
     [ -5.61069915e+28]
     [ -3.01280291e+28]
     [ -2.69585917e+28]]

    [[  9.81709359e+29]
     [  1.45547281e+30]
     [  7.81551924e+29]
     [  6.99333473e+29]]

    [[ -2.54665460e+31]
     [ -3.77564549e+31]
     [ -2.02742571e+31]
     [ -1.81414263e+31]]

    [[  6.60628277e+32]
     [  9.79441100e+32]
     [  5.25934990e+32]
     [  4.70607174e+32]]

    [[ -1.71373739e+34]
     [ -2.54077050e+34]
     [ -1.36432921e+34]
     [ -1.22080320e+34]]

    [[  4.44561027e+35]
     [  6.59101883e+35]
     [  3.53920968e+35]
     [  3.16688849e+35]]

    [[ -1.15323682e+37]
     [ -1.70977777e+37]
     [ -9.18107228e+36]
     [ -8.21523301e+36]]

    [[  2.99161440e+38]
     [  4.43533860e+38]
     [  2.38166416e+38]
     [  2.13111556e+38]]

    [[ -7.76055407e+39]
     [ -1.15057225e+40]
     [ -6.17828068e+39]
     [ -5.52833197e+39]]

    [[  2.01316719e+41]
     [  2.98470222e+41]
     [  1.60270927e+41]
     [  1.43410592e+41]]

    [[ -5.22236182e+42]
     [ -7.74262316e+42]
     [ -4.15759195e+42]
     [ -3.72021759e+42]]

    [[  1.35473413e+44]
     [  2.00851572e+44]
     [  1.07852192e+44]
     [  9.65062534e+43]]

    [[ -3.51431904e+45]
     [ -5.21029544e+45]
     [ -2.79779629e+45]
     [ -2.50347103e+45]]

    [[  9.11650342e+46]
     [  1.35160399e+47]
     [  7.25777006e+46]
     [  6.49426018e+46]]

    [[ -2.36491433e+48]
     [ -3.50619913e+48]
     [ -1.88273987e+48]
     [ -1.68467759e+48]]

    [[  6.13482993e+49]
     [  9.09543956e+49]
     [  4.88401999e+49]
     [  4.37022615e+49]]

    [[ -1.59143770e+51]
     [ -2.35945015e+51]
     [ -1.26696479e+51]
     [ -1.13368141e+51]]

    [[  4.12835234e+52]
     [  6.12065528e+52]
     [  3.28663640e+52]
     [  2.94088566e+52]]

    [[ -1.07093687e+54]
     [ -1.58776065e+54]
     [ -8.52587134e+53]
     [ -7.62895853e+53]]

    [[  2.77811989e+55]
     [  4.11881370e+55]
     [  2.21169832e+55]
     [  1.97902996e+55]]

    [[ -7.20672744e+56]
     [ -1.06846245e+57]
     [ -5.73737191e+56]
     [ -5.13380634e+56]]

    [[  1.86949889e+58]
     [  2.77170099e+58]
     [  1.48833302e+58]
     [  1.33176193e+58]]

    [[ -4.84967154e+59]
     [ -7.19007615e+59]
     [ -3.86088824e+59]
     [ -3.45472681e+59]]

    [[  1.25805445e+61]
     [  1.86517937e+61]
     [  1.00155394e+61]
     [  8.96191507e+60]]

    [[ -3.26352207e+62]
     [ -4.83846628e+62]
     [ -2.59813348e+62]
     [ -2.32481253e+62]]

    [[  8.46591040e+63]
     [  1.25514769e+64]
     [  6.73982427e+63]
     [  6.03080175e+63]]

    [[ -2.19614384e+65]
     [ -3.25598163e+65]
     [ -1.74837942e+65]
     [ -1.56445172e+65]]

    [[  5.69702199e+66]
     [  8.44634974e+66]
     [  4.53547525e+66]
     [  4.05834796e+66]]

    [[ -1.47786584e+68]
     [ -2.19106961e+68]
     [ -1.17654872e+68]
     [ -1.05277701e+68]]

    [[  3.83373530e+69]
     [  5.68385890e+69]
     [  3.05208784e+69]
     [  2.73101137e+69]]

    [[ -9.94510190e+70]
     [ -1.47445120e+71]
     [ -7.91742837e+70]
     [ -7.08452312e+70]]

    [[  2.57986126e+72]
     [  3.82487738e+72]
     [  2.05386199e+72]
     [  1.83779784e+72]]

    [[ -6.69242427e+73]
     [ -9.92212355e+73]
     [ -5.32792829e+73]
     [ -4.76743576e+73]]

    [[  1.73608338e+75]
     [  2.57390044e+75]
     [  1.38211915e+75]
     [  1.23672165e+75]]

    [[ -4.50357805e+76]
     [ -6.67696129e+76]
     [ -3.58535860e+76]
     [ -3.20818259e+76]]

    [[  1.16827426e+78]
     [  1.73207212e+78]
     [  9.30078731e+77]
     [  8.32235414e+77]]

    [[ -3.03062305e+79]
     [ -4.49317244e+79]
     [ -2.41271946e+79]
     [ -2.15890388e+79]]

    [[  7.86174649e+80]
     [  1.16557494e+81]
     [  6.25884132e+80]
     [  5.60041771e+80]]

    [[ -2.03941754e+82]
     [ -3.02362073e+82]
     [ -1.62360752e+82]
     [ -1.45280570e+82]]

    [[  5.29045791e+83]
     [  7.84358177e+83]
     [  4.21180416e+83]
     [  3.76872673e+83]]

    [[ -1.37239895e+85]
     [ -2.03470542e+85]
     [ -1.09258512e+85]
     [ -9.77646303e+84]]

    [[  3.56014340e+86]
     [  5.27823420e+86]
     [  2.83427768e+86]
     [  2.53611461e+86]]

    [[ -9.23537651e+87]
     [ -1.36922800e+88]
     [ -7.35240651e+87]
     [ -6.57894098e+87]]

    [[  2.39575123e+89]
     [  3.55191762e+89]
     [  1.90728953e+89]
     [  1.70664465e+89]]

    [[ -6.21482399e+90]
     [ -9.21403799e+90]
     [ -4.94770433e+90]
     [ -4.42721097e+90]]

    [[  1.61218898e+92]
     [  2.39021580e+92]
     [  1.28348516e+92]
     [  1.14846386e+92]]

    [[ -4.18218328e+93]
     [ -6.20046452e+93]
     [ -3.32949193e+93]
     [ -2.97923284e+93]]

    [[  1.08490117e+95]
     [  1.60846398e+95]
     [  8.63704299e+94]
     [  7.72843503e+94]]

    [[ -2.81434471e+96]
     [ -4.17252026e+96]
     [ -2.24053739e+96]
     [ -2.00483518e+96]]

    [[  7.30069834e+97]
     [  1.08239448e+98]
     [  5.81218339e+97]
     [  5.20074773e+97]]

    [[ -1.89387590e+99]
     [ -2.80784211e+99]
     [ -1.50773988e+99]
     [ -1.34912721e+99]]

    [[  4.91290800e+100]
     [  7.28382993e+100]
     [  3.91123164e+100]
     [  3.49977413e+100]]

    [[ -1.27445864e+102]
     [ -1.88950006e+102]
     [ -1.01461353e+102]
     [ -9.07877243e+101]]

    [[  3.30607620e+103]
     [  4.90155663e+103]
     [  2.63201140e+103]
     [  2.35512653e+103]]

    [[ -8.57630019e+104]
     [ -1.27151398e+105]
     [ -6.82770705e+104]
     [ -6.10943936e+104]]

    [[  2.22478008e+106]
     [  3.29843745e+106]
     [  1.77117712e+106]
     [  1.58485112e+106]]

    [[ -5.77130733e+107]
     [ -8.55648448e+107]
     [ -4.59461480e+107]
     [ -4.11126609e+107]]

    [[  1.49713622e+109]
     [  2.21963968e+109]
     [  1.19189012e+109]
     [  1.06650452e+109]]

    [[ -3.88372464e+110]
     [ -5.75797261e+110]
     [ -3.09188502e+110]
     [ -2.76662193e+110]]

    [[  1.00747794e+112]
     [  1.49367705e+112]
     [  8.02066633e+111]
     [  7.17690055e+111]]

    [[ -2.61350092e+113]
     [ -3.87475121e+113]
     [ -2.08064297e+113]
     [ -1.86176149e+113]]

    [[  6.77968900e+114]
     [  1.00515014e+115]
     [  5.39740090e+114]
     [  4.82959993e+114]]

    [[ -1.75872074e+116]
     [ -2.60746238e+116]
     [ -1.40014106e+116]
     [ -1.25284767e+116]]

    [[  4.56230169e+117]
     [  6.76402439e+117]
     [  3.63210927e+117]
     [  3.25001514e+117]]

    [[ -1.18350777e+119]
     [ -1.75465718e+119]
     [ -9.42206334e+118]
     [ -8.43087205e+118]]

    [[  3.07014034e+120]
     [  4.55176040e+120]
     [  2.44417970e+120]
     [  2.18705453e+120]]

    [[ -7.96425840e+121]
     [ -1.18077325e+122]
     [ -6.34045241e+121]
     [ -5.67344340e+121]]

    [[  2.06601017e+123]
     [  3.06304672e+123]
     [  1.64477828e+123]
     [  1.47174931e+123]]

    [[ -5.35944193e+124]
     [ -7.94585682e+124]
     [ -4.26672326e+124]
     [ -3.81786840e+124]]

    [[  1.39029411e+126]
     [  2.06123662e+126]
     [  1.10683170e+126]
     [  9.90394155e+125]]

    [[ -3.60656528e+127]
     [ -5.34705884e+127]
     [ -2.87123476e+127]
     [ -2.56918384e+127]]

    [[  9.35579963e+128]
     [  1.38708181e+129]
     [  7.44827696e+128]
     [  6.66472596e+128]]

    [[ -2.42699022e+130]
     [ -3.59823224e+130]
     [ -1.93215931e+130]
     [ -1.72889815e+130]]

    [[  6.29586113e+131]
     [  9.33418287e+131]
     [  5.01221908e+131]
     [  4.48493883e+131]]

    [[ -1.63321084e+133]
     [ -2.42138261e+133]
     [ -1.30022095e+133]
     [ -1.16343906e+133]]

    [[  4.23671615e+134]
     [  6.28131442e+134]
     [  3.37290628e+134]
     [  3.01808004e+134]]

    [[ -1.09904755e+136]
     [ -1.62943727e+136]
     [ -8.74966423e+135]
     [ -7.82920864e+135]]

    [[  2.85104188e+137]
     [  4.22692713e+137]
     [  2.26975249e+137]
     [  2.03097688e+137]]

    [[ -7.39589455e+138]
     [ -1.09650818e+139]
     [ -5.88797037e+138]
     [ -5.26856199e+138]]

    [[  1.91857077e+140]
     [  2.84445449e+140]
     [  1.52739980e+140]
     [  1.36671893e+140]]

    [[ -4.97696903e+141]
     [ -7.37880618e+141]
     [ -3.96223147e+141]
     [ -3.54540884e+141]]

    [[  1.29107673e+143]
     [  1.91413788e+143]
     [  1.02784341e+143]
     [  9.19715354e+142]]

    [[ -3.34918522e+144]
     [ -4.96546964e+144]
     [ -2.66633106e+144]
     [ -2.38583580e+144]]

    [[  8.68812939e+145]
     [  1.28809367e+146]
     [  6.91673577e+145]
     [  6.18910235e+145]]

    [[ -2.25378972e+147]
     [ -3.34144686e+147]
     [ -1.79427208e+147]
     [ -1.60551652e+147]]

    [[  5.84656131e+148]
     [  8.66805529e+148]
     [  4.65452550e+148]
     [  4.16487424e+148]]

    [[ -1.51665787e+150]
     [ -2.24858230e+150]
     [ -1.20743157e+150]
     [ -1.08041102e+150]]

    [[  3.93436580e+151]
     [  5.83305271e+151]
     [  3.13220113e+151]
     [  2.80269682e+151]]

    [[ -1.02061477e+153]
     [ -1.51315360e+153]
     [ -8.12525043e+152]
     [ -7.27048252e+152]]

    [[  2.64757922e+154]
     [  3.92527537e+154]
     [  2.10777316e+154]
     [  1.88603761e+154]]

    [[ -6.86809159e+155]
     [ -1.01825662e+156]
     [ -5.46777939e+155]
     [ -4.89257467e+155]]

    [[  1.78165328e+157]
     [  2.64146194e+157]
     [  1.41839796e+157]
     [  1.26918397e+157]]

    [[ -4.62179104e+158]
     [ -6.85222273e+158]
     [ -3.67946954e+158]
     [ -3.29239316e+158]]

    [[  1.19893992e+160]
     [  1.77753674e+160]
     [  9.54492072e+159]
     [  8.54080497e+159]]

    [[ -3.11017291e+161]
     [ -4.61111230e+161]
     [ -2.47605017e+161]
     [ -2.21557226e+161]]

    [[  8.06810698e+162]
     [  1.19616974e+163]
     [  6.42312766e+162]
     [  5.74742130e+162]]

    [[ -2.09294956e+164]
     [ -3.10298679e+164]
     [ -1.66622508e+164]
     [ -1.49093993e+164]]

    [[  5.42932546e+165]
     [  8.04946546e+165]
     [  4.32235847e+165]
     [  3.86765084e+165]]

    [[ -1.40842262e+167]
     [ -2.08811376e+167]
     [ -1.12126404e+167]
     [ -1.00330823e+167]]

    [[  3.65359248e+168]
     [  5.41678090e+168]
     [  2.90867374e+168]
     [  2.60268428e+168]]

    [[ -9.47779298e+169]
     [ -1.40516843e+170]
     [ -7.54539750e+169]
     [ -6.75162951e+169]]

    [[  2.45863654e+171]
     [  3.64515078e+171]
     [  1.95735337e+171]
     [  1.75144183e+171]]

    [[ -6.37795494e+172]
     [ -9.45589435e+172]
     [ -5.07757506e+172]
     [ -4.54341943e+172]]

    [[  1.65450682e+174]
     [  2.45295581e+174]
     [  1.31717496e+174]
     [  1.17860952e+174]]

    [[ -4.29196009e+175]
     [ -6.36321854e+175]
     [ -3.41688671e+175]
     [ -3.05743378e+175]]

    [[  1.11337839e+177]
     [  1.65068404e+177]
     [  8.86375399e+176]
     [  7.93129627e+176]]

    [[ -2.88821755e+178]
     [ -4.28204343e+178]
     [ -2.29934854e+178]
     [ -2.05745946e+178]]

    [[  7.49233205e+179]
     [  1.11080591e+180]
     [  5.96474555e+179]
     [  5.33726051e+179]]

    [[ -1.94358765e+181]
     [ -2.88154426e+181]
     [ -1.54731607e+181]
     [ -1.38454002e+181]]

    [[  5.04186537e+182]
     [  7.47502087e+182]
     [  4.01389632e+182]
     [  3.59163859e+182]]

    [[ -1.30791150e+184]
     [ -1.93909695e+184]
     [ -1.04124580e+184]
     [ -9.31707825e+183]]

    [[  3.39285635e+185]
     [  5.03021604e+185]
     [  2.70109823e+185]
     [  2.41694550e+185]]

    [[ -8.80141677e+186]
     [ -1.30488954e+187]
     [ -7.00692537e+186]
     [ -6.26980409e+186]]

    [[  2.28317763e+188]
     [  3.38501709e+188]
     [  1.81766819e+188]
     [  1.62645138e+188]]

    [[ -5.92279655e+189]
     [ -8.78108092e+189]
     [ -4.71521739e+189]
     [ -4.21918141e+189]]

    [[  1.53643407e+191]
     [  2.27790230e+191]
     [  1.22317567e+191]
     [  1.09449886e+191]]

    [[ -3.98566729e+192]
     [ -5.90911181e+192]
     [ -3.17304293e+192]
     [ -2.83924211e+192]]

    [[  1.03392291e+194]
     [  1.53288411e+194]
     [  8.23119825e+193]
     [  7.36528474e+193]]

    [[ -2.68210188e+195]
     [ -3.97645833e+195]
     [ -2.13525711e+195]
     [ -1.91063027e+195]]

    [[  6.95764690e+196]
     [  1.03153401e+197]
     [  5.53907557e+196]
     [  4.95637056e+196]]

    [[ -1.80488485e+198]
     [ -2.67590483e+198]
     [ -1.43689292e+198]
     [ -1.28573327e+198]]

    [[  4.68205610e+199]
     [  6.94157112e+199]
     [  3.72744735e+199]
     [  3.33532376e+199]]

    [[ -1.21457329e+201]
     [ -1.80071462e+201]
     [ -9.66938009e+200]
     [ -8.65217134e+200]]

    [[  3.15072748e+202]
     [  4.67123811e+202]
     [  2.50833620e+202]
     [  2.24446184e+202]]

    [[ -8.17330969e+203]
     [ -1.21176700e+204]
     [ -6.50688094e+203]
     [ -5.82236381e+203]]

    [[  2.12024022e+205]
     [  3.14344766e+205]
     [  1.68795154e+205]
     [  1.51038079e+205]]

    [[ -5.50012023e+206]
     [ -8.15442509e+206]
     [ -4.37871912e+206]
     [ -3.91808242e+206]]

    [[  1.42678750e+208]
     [  2.11534136e+208]
     [  1.13588458e+208]
     [  1.01639070e+208]]

    [[ -3.70123287e+209]
     [ -5.48741209e+209]
     [ -2.94660089e+209]
     [ -2.63662154e+209]]

    [[  9.60137704e+210]
     [  1.42349088e+211]
     [  7.64378442e+210]
     [  6.83966623e+210]]

    [[ -2.49069552e+212]
     [ -3.69268110e+212]
     [ -1.98287594e+212]
     [ -1.77427945e+212]]

    [[  6.46111919e+213]
     [  9.57919287e+213]
     [  5.14378323e+213]
     [  4.60266257e+213]]

    [[ -1.67608047e+215]
     [ -2.48494071e+215]
     [ -1.33435004e+215]
     [ -1.19397780e+215]]

    [[  4.34792437e+216]
     [  6.44619065e+216]
     [  3.46144062e+216]
     [  3.09730066e+216]]

    [[ -1.12789610e+218]
     [ -1.67220786e+218]
     [ -8.97933140e+217]
     [ -8.03471505e+217]]

    [[  2.92587797e+219]
     [  4.33787840e+219]
     [  2.32933050e+219]
     [  2.08428735e+219]]

    [[ -7.59002703e+220]
     [ -1.12529007e+221]
     [ -6.04252183e+220]
     [ -5.40685481e+220]]

    [[  1.96893073e+222]
     [  2.91911767e+222]
     [  1.56749203e+222]
     [  1.40259350e+222]]

    [[ -5.10760791e+223]
     [ -7.57249012e+223]
     [ -4.06623483e+223]
     [ -3.63847115e+223]]

    [[  1.32496579e+225]
     [  1.96438147e+225]
     [  1.05482295e+225]
     [  9.43856670e+224]]

    [[ -3.43709692e+226]
     [ -5.09580668e+226]
     [ -2.73631874e+226]
     [ -2.44846085e+226]]

    [[  8.91618134e+227]
     [  1.32190443e+228]
     [  7.09829098e+227]
     [  6.35155813e+227]]

    [[ -2.31294873e+229]
     [ -3.42915544e+229]
     [ -1.84136936e+229]
     [ -1.64765921e+229]]

    [[  6.00002584e+230]
     [  8.89558033e+230]
     [  4.77670066e+230]
     [  4.27419671e+230]]

    [[ -1.55646814e+232]
     [ -2.30760462e+232]
     [ -1.23912506e+232]
     [ -1.10877039e+232]]

    [[  4.03763772e+233]
     [  5.98616266e+233]
     [  3.21441728e+233]
     [  2.87626392e+233]]

    [[ -1.04740457e+235]
     [ -1.55287189e+235]
     [ -8.33852756e+234]
     [ -7.46132311e+234]]

    [[  2.71707469e+236]
     [  4.02830868e+236]
     [  2.16309943e+236]
     [  1.93554361e+236]]

    [[ -7.04836995e+237]
     [ -1.04498452e+238]
     [ -5.61130140e+237]
     [ -5.02099831e+237]]

    [[  1.82841933e+239]
     [  2.71079683e+239]
     [  1.45562904e+239]
     [  1.30249837e+239]]

    [[ -4.74310697e+240]
     [ -7.03208455e+240]
     [ -3.77605077e+240]
     [ -3.37881414e+240]]

    [[  1.23041051e+242]
     [  1.82419473e+242]
     [  9.79546232e+241]
     [  8.76498985e+241]]

    [[ -3.19181085e+243]
     [ -4.73214792e+243]
     [ -2.54104322e+243]
     [ -2.27372811e+243]]

    [[  8.27988416e+244]
     [  1.22756762e+245]
     [  6.59172630e+244]
     [  5.89828353e+244]]

    [[ -2.14788673e+246]
     [ -3.18443611e+246]
     [ -1.70996130e+246]
     [ -1.53007514e+246]]

    [[  5.57183810e+247]
     [  8.26075332e+247]
     [  4.43581469e+247]
     [  3.96917158e+247]]

    [[ -1.44539185e+249]
     [ -2.14292400e+249]
     [ -1.15069575e+249]
     [ -1.02964375e+249]]

    [[  3.74949447e+250]
     [  5.55896426e+250]
     [  2.98502259e+250]
     [  2.67100131e+250]]

    [[ -9.72657256e+251]
     [ -1.44205225e+252]
     [ -7.74345424e+251]
     [ -6.92885089e+251]]

    [[  2.52317251e+253]
     [  3.74083118e+253]
     [  2.00873132e+253]
     [  1.79741487e+253]]

    [[ -6.54536785e+254]
     [ -9.70409912e+254]
     [ -5.21085471e+254]
     [ -4.66267820e+254]]

    [[  1.69793544e+256]
     [  2.51734267e+256]
     [  1.35174907e+256]
     [  1.20954647e+256]]

    [[ -4.40461838e+257]
     [ -6.53024465e+257]
     [ -3.50657549e+257]
     [ -3.13768738e+257]]

    [[  1.14260311e+259]
     [  1.69401233e+259]
     [  9.09641585e+258]
     [  8.13948235e+258]]

    [[ -2.96402945e+260]
     [ -4.39444142e+260]
     [ -2.35970341e+260]
     [ -2.11146506e+260]]

    [[  7.68899590e+261]
     [  1.13996310e+262]
     [  6.12131226e+261]
     [  5.47735657e+261]]

    [[ -1.99460426e+263]
     [ -2.95718101e+263]
     [ -1.58793108e+263]
     [ -1.42088238e+263]]

    [[  5.17420769e+264]
     [  7.67123032e+264]
     [  4.11925581e+264]
     [  3.68591437e+264]]

    [[ -1.34224245e+266]
     [ -1.98999569e+266]
     [ -1.06857713e+266]
     [ -9.56163928e+265]]

    [[  3.48191436e+267]
     [  5.16225258e+267]
     [  2.77199850e+267]
     [  2.48038713e+267]]

    [[ -9.03244237e+268]
     [ -1.33914118e+269]
     [ -7.19084793e+268]
     [ -6.43437819e+268]]

    [[  2.34310803e+270]
     [  3.47386933e+270]
     [  1.86537958e+270]
     [  1.66914359e+270]]

    [[ -6.07826215e+271]
     [ -9.01157273e+271]
     [ -4.83898563e+271]
     [ -4.32992937e+271]]

    [[  1.57676344e+273]
     [  2.33769424e+273]
     [  1.25528242e+273]
     [  1.12322801e+273]]

    [[ -4.09028581e+274]
     [ -6.06421821e+274]
     [ -3.25633113e+274]
     [ -2.91376847e+274]]

    [[  1.06106202e+276]
     [  1.57312030e+276]
     [  8.44725636e+275]
     [  7.55861376e+275]]

    [[ -2.75250352e+277]
     [ -4.08083512e+277]
     [ -2.19130479e+277]
     [ -1.96078180e+277]]

    [[  7.14027597e+278]
     [  1.05861041e+279]
     [  5.68446900e+278]
     [  5.08646876e+278]]

    [[ -1.85226070e+280]
     [ -2.74614381e+280]
     [ -1.47460946e+280]
     [ -1.31948208e+280]]

    [[  4.80495390e+281]
     [  7.12377822e+281]
     [  3.82528793e+281]
     [  3.42287161e+281]]

    [[ -1.24645424e+283]
     [ -1.84798101e+283]
     [ -9.92318858e+282]
     [ -8.87927944e+282]]

    [[  3.23342992e+284]
     [  4.79385196e+284]
     [  2.57417672e+284]
     [  2.30337600e+284]]

    [[ -8.38784830e+285]
     [ -1.24357428e+286]
     [ -6.67767799e+285]
     [ -5.97519319e+285]]

    [[  2.17589373e+287]
     [  3.22595902e+287]
     [  1.73225804e+287]
     [  1.55002630e+287]]

    [[ -5.64449114e+288]
     [ -8.36846801e+288]
     [ -4.49365473e+288]
     [ -4.02092692e+288]]

    [[  1.46423880e+290]
     [  2.17086629e+290]
     [  1.16570005e+290]
     [  1.04306961e+290]]

    [[ -3.79838536e+291]
     [ -5.63144943e+291]
     [ -3.02394528e+291]
     [ -2.70582938e+291]]

    [[  9.85340054e+292]
     [  1.46085564e+293]
     [  7.84442369e+292]
     [  7.01919846e+292]]

    [[ -2.55607299e+294]
     [ -3.78960911e+294]
     [ -2.03492383e+294]
     [ -1.82085195e+294]]

    [[  6.63071506e+295]
     [  9.83063406e+295]
     [  5.27880076e+295]
     [  4.72347640e+295]]

    [[ -1.72007538e+297]
     [ -2.55016713e+297]
     [ -1.36937497e+297]
     [ -1.22531814e+297]]

    [[  4.46205165e+298]
     [  6.61539466e+298]
     [  3.55229888e+298]
     [  3.17860072e+298]]

    [[ -1.15750189e+300]
     [ -1.71610111e+300]
     [ -9.21502702e+299]
     [ -8.24561574e+299]]

    [[  3.00267841e+301]
     [  4.45174199e+301]
     [  2.39047236e+301]
     [  2.13899715e+301]]

    [[ -7.78925525e+302]
     [ -1.15482746e+303]
     [ -6.20113007e+302]
     [ -5.54877762e+302]]

    [[  2.02061257e+304]
     [  2.99574066e+304]
     [  1.60863663e+304]
     [  1.43940973e+304]]

    [[ -5.24167589e+305]
     [ -7.77125802e+305]
     [ -4.17296814e+305]
     [ -3.73397622e+305]]

    array must not contain infs or NaNs

```python
A.dot(gs_ans)
```

    array([[ 5.],
           [ 1.],
           [ 8.],
           [ 3.]])

```python
b
```

    array([[ 5.],
           [ 1.],
           [ 8.],
           [ 3.]])

### Gauss-Seidel work for positive-definite matrix, but Jacobi does not

# Try Gauss-Seidel with larger matrix (dense)

```python
np.random.seed(46)
a = np.ceil(10*np.random.random([7,7])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
np.random.seed(406)
b = np.ceil(10* np.random.random([7,1]))
```

```python
A
```

    array([[ 64,  40,  72,  24,  24,  72,   8],
           [ 40, 125,  85,  25,  65,  55,  85],
           [ 72,  85, 101,  39,  49,  89,  43],
           [ 24,  25,  39,  35,  27,  57,  24],
           [ 24,  65,  49,  27,  53,  85,  59],
           [ 72,  55,  89,  57,  85, 236,  62],
           [  8,  85,  43,  24,  59,  62,  87]])

```python
b
```

    array([[ 10.],
           [  6.],
           [  4.],
           [  9.],
           [  2.],
           [  9.],
           [  7.]])

```python
gs_ans2 = gauss_seidel(A,b)
gs_ans2
```

    total iteration =  34006
    diff =  9.99998018289e-11
    SSE =  4.26448700686e-17

    array([[   6.21458333],
           [  79.92416659],
           [ -60.22916661],
           [  42.14166663],
           [-113.40833322],
           [  35.14999996],
           [  -8.57499999]])

```python
%%timeit -r 1 -n 1 -o
gs_ans2 = gauss_seidel(A,b)
```

    total iteration =  34006
    diff =  9.99998018289e-11
    SSE =  4.26448700686e-17
    1 loop, best of 1: 3.46 s per loop

    <TimeitResult : 1 loop, best of 1: 3.46 s per loop>

```python
gs_time = _
```

```python
def backward_sub(U,b,N):
    y = np.empty((N,1)) #(N,1) is needed here
    for i in range(N-1,-1,-1):  #to return column vector
        Sum = 0
        for k in range(i+1, N):
            Sum = Sum + U[i,k] * y[k]
        y[i] = (b[i] - Sum)/U[i,i]
    return y
```

```python
L = cholesky_user(A)
y = forward_sub(L,b,L.shape[0])
x = backward_sub(L.T,y,L.shape[0])
A.dot(x)-b
```

    array([[ -4.54747351e-13],
           [  1.81898940e-12],
           [ -9.09494702e-13],
           [  2.27373675e-13],
           [  0.00000000e+00],
           [  9.09494702e-13],
           [  4.54747351e-13]])

```python
x
```

    array([[   6.21458333],
           [  79.92416667],
           [ -60.22916667],
           [  42.14166667],
           [-113.40833333],
           [  35.15      ],
           [  -8.575     ]])

```python
x - gs_ans2
```

    array([[  1.01020081e-09],
           [  8.14847567e-08],
           [ -5.66639926e-08],
           [  4.14630748e-08],
           [ -1.17847847e-07],
           [  3.70258206e-08],
           [ -9.60257829e-09]])

```python
%%timeit -r 1 -n 1 -o
L = cholesky_user(A)
y = forward_sub(L,b,L.shape[0])
x = backward_sub(L.T,y,L.shape[0])
```

    1 loop, best of 1: 813 µs per loop

    <TimeitResult : 1 loop, best of 1: 813 µs per loop>

```python
cho_time = _
```

```python
cho_time.best
```

    0.0008132590000968776

```python
print('Cholesky decomposition is ',
      gs_time.best / cho_time.best,
     ' faster than Gaussian elimination',
     ' for this 7x7 symmetric positive-definite case')
```

    Cholesky decomposition is  4254.432930454434  faster than Gaussian elimination  for this 7x7 symmetric positive-definite case

## Asymmetric, non-positive definite case

```python
np.random.seed(46)
A = np.ceil(10*np.random.random([5,5])).astype(int)
np.random.seed(406)
b = np.ceil(10* np.random.random([5,1]))
```

```python
A
```

    array([[ 8,  7,  3,  8,  4],
           [10,  1,  5, 10,  5],
           [ 6,  1,  5,  8,  9],
           [ 4,  2,  9,  4,  7],
           [ 1,  3,  1,  4,  3]])

```python
b
```

    array([[ 10.],
           [  6.],
           [  4.],
           [  9.],
           [  2.]])

```python
try:
    L = cholesky_user(A)
    y = forward_sub(L,b,L.shape[0])
    x = backward_sub(L.T,y,L.shape[0])
except Exception as e:
    print(e)
```

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:11: RuntimeWarning: invalid value encountered in double_scalars
      # This is added back by InteractiveShellApp.init_path()

```python
x
```

    array([[ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan]])

```python
try:
    gs_ans3 = gauss_seidel(A,b, maxit = 400)
except Exception as e:
    print(e)
gs_ans3
```

    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:11: RuntimeWarning: overflow encountered in square
      # This is added back by InteractiveShellApp.init_path()
    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:26: RuntimeWarning: overflow encountered in subtract
    /home/me/anaconda3/lib/python3.6/site-packages/ipykernel_launcher.py:25: RuntimeWarning: invalid value encountered in add

    array([[ nan],
           [ nan],
           [ nan],
           [ nan],
           [ nan]])

```python
try:
    sp.linalg.cholesky(A)
except Exception as e:
    print(e)
```

    2-th leading minor not positive definite

In this case, both Cholesky and Gauss-Seidel fail

## Slightly off from positive definite case

```python
np.random.seed(46)
a = np.ceil(10*np.random.random([7,7])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
np.random.seed(406)
b = np.ceil(10* np.random.random([7,1]))
A[0,1] = A[0,1] + 5
```

```python
try:
    gs_ans4 = gauss_seidel(A,b, maxit = 10000)
except Exception as e:
    print(e)
gs_ans4
```

    total iteration =  9149
    diff =  9.99382550173e-11
    SSE =  1.9424032781e-17

    array([[-103.22931321],
           [  38.5563819 ],
           [  64.9451982 ],
           [  -2.03752094],
           [-114.74709657],
           [  43.18257955],
           [ -12.59128978]])

```python
A.dot(gs_ans4) - b
```

    array([[  3.81896825e-09],
           [  8.27640179e-11],
           [ -1.35241862e-09],
           [ -4.87943908e-10],
           [  1.61389835e-09],
           [ -4.01087163e-10],
           [ -4.54747351e-13]])

```python
try:
    L = cholesky_user(A)
    y = forward_sub(L,b,L.shape[0])
    x = backward_sub(L.T,y,L.shape[0])
except Exception as e:
    print(e)
x
```

    array([[   6.21458333],
           [  79.92416667],
           [ -60.22916667],
           [  42.14166667],
           [-113.40833333],
           [  35.15      ],
           [  -8.575     ]])

```python
np.set_printoptions(suppress=True)
A.dot(x)-b
```

    array([[ 399.62083333],
           [   0.        ],
           [  -0.        ],
           [   0.        ],
           [   0.        ],
           [   0.        ],
           [   0.        ]])

```python
try:
    sp.linalg.cholesky(A)
except Exception as e:
    print(e)
```

    6-th leading minor not positive definite

```python
A.dot(gs_ans4) - b
```

    array([[ 0.],
           [ 0.],
           [-0.],
           [-0.],
           [ 0.],
           [-0.],
           [-0.]])

## In this case where it is slightly off from the positive-definite case, Gauss-Seidel works, but Cholesky does not work

# SOR:   $\mathbf{x}^{(k+1)} = (D+\omega L)^{-1} \big(\omega \mathbf{b} - [\omega U + (\omega-1) D ] \mathbf{x}^{(k)}\big)$

```python
class sor_ans:
    def __init__(self, x, SSE, i, diff):
            self.x = x
            self.SSE = SSE
            self.i = i
            self.diff = diff
```

```python
def sor_method(A,b, omega = 1, maxit = 2000):
    N = A.shape[0]
    x = np.ones((N,1))
    L = np.tril(A, k = -1)
    U = np.triu(A, k = 1)
    D = np.zeros([N,N])
    D[range(N),range(N)]  = A[range(N),range(N)]
    for i in range(maxit):
        x_old = x
        #L x = rhs
        rhs = omega * b - (omega * U + (omega-1)*D).dot(x)
        x = forward_sub(D + omega * L,rhs,N)
        diff = (((x-x_old)**2).sum())**0.5
        if diff < 1e-10:
            print('total iteration = ', i)
            print('diff = ', diff)
            SSE = ((A.dot(x) - b)**2).sum()
            print('SSE = ', SSE)
            break
    if i < maxit-1:
        return sor_ans(x, SSE, i, diff)
    else:
        SSE = ((A.dot(x) - b)**2).sum()
        return sor_ans(x, SSE, i, diff)
```

```python
np.random.seed(46)
a = np.ceil(10*np.random.random([6,6])).astype(int)
b = np.tril(a)
A = b.dot(b.T)
np.random.seed(406)
b = np.ceil(10* np.random.random([6,1]))
```

```python
try:
    gs_ans5 = gauss_seidel(A,b, maxit = 10000)
except Exception as e:
    print(e)
gs_ans5
```

    total iteration =  476
    diff =  9.39128589046e-11
    SSE =  4.6035586437e-17

    array([[ 0.00198788],
           [ 1.38833931],
           [ 0.2764575 ],
           [ 0.47060577],
           [-1.2407382 ],
           [ 0.3032617 ]])

```python
try:
    ans = sor_method(A,b, omega = 1, maxit = 10000)
except Exception as e:
    print(e)
ans.x
```

    total iteration =  476
    diff =  9.39128589046e-11
    SSE =  4.6035586437e-17

    array([[ 0.00198788],
           [ 1.38833931],
           [ 0.2764575 ],
           [ 0.47060577],
           [-1.2407382 ],
           [ 0.3032617 ]])

```python
try:
    ans = sor_method(A,b, omega = 1.1, maxit = 10000)
except Exception as e:
    print(e)
ans.x
```

    total iteration =  457
    diff =  9.6279266756e-11
    SSE =  5.97252009536e-17

    array([[ 0.00198788],
           [ 1.38833931],
           [ 0.2764575 ],
           [ 0.47060577],
           [-1.2407382 ],
           [ 0.3032617 ]])

```python
dat = [(j,sor_method(A,b, omega = j, maxit = 10000).i) for j in np.arange(0.1,2.0,0.1)]
```

    total iteration =  7360
    diff =  9.98746026931e-11
    SSE =  9.90919019875e-15
    total iteration =  3643
    diff =  9.9735919303e-11
    SSE =  2.01345432715e-15
    total iteration =  2348
    diff =  9.99914141769e-11
    SSE =  7.18569115455e-16
    total iteration =  1677
    diff =  9.96869796507e-11
    SSE =  3.13170000409e-16
    total iteration =  1257
    diff =  9.93262881044e-11
    SSE =  1.50427150655e-16
    total iteration =  958
    diff =  9.87749464294e-11
    SSE =  7.50908016768e-17
    total iteration =  701
    diff =  9.86262179718e-11
    SSE =  3.75735705042e-17
    total iteration =  571
    diff =  9.79913809859e-11
    SSE =  4.18909907815e-17
    total iteration =  513
    diff =  9.4095334482e-11
    SSE =  3.90344119874e-17
    total iteration =  476
    diff =  9.39128589046e-11
    SSE =  4.6035586437e-17
    total iteration =  457
    diff =  9.6279266756e-11
    SSE =  5.97252009536e-17
    total iteration =  477
    diff =  9.52289381172e-11
    SSE =  2.57478160426e-17
    total iteration =  508
    diff =  9.74482881038e-11
    SSE =  2.25425876879e-17
    total iteration =  556
    diff =  9.79444714235e-11
    SSE =  5.68303613885e-17
    total iteration =  667
    diff =  9.66550117045e-11
    SSE =  2.7488277615e-17
    total iteration =  830
    diff =  9.54596124626e-11
    SSE =  4.18280854715e-17
    total iteration =  1134
    diff =  9.73376890901e-11
    SSE =  2.82304745617e-17
    total iteration =  1743
    diff =  9.78149412914e-11
    SSE =  4.36048684875e-17
    total iteration =  3602
    diff =  9.78913569573e-11
    SSE =  5.79638759874e-17

```python

```

```python
x = [i[0] for i in dat]
y = [i[1] for i in dat]
```

```python
plt.figure()
plt.plot(x,y,'ro')
plt.xlabel('omega')
plt.ylabel('# iteration')
plt.show()
```

```python
dat2 = [(j,sor_method(A,b, omega = j, maxit = 10000).i) for j in np.linspace(0.7,1.3,15)]
```

    total iteration =  701
    diff =  9.86262048882e-11
    SSE =  3.75734271673e-17
    total iteration =  632
    diff =  9.95852253027e-11
    SSE =  3.90352756799e-17
    total iteration =  593
    diff =  9.5718827768e-11
    SSE =  3.0554969864e-17
    total iteration =  541
    diff =  9.83838694745e-11
    SSE =  6.79008327114e-17
    total iteration =  538
    diff =  9.87193745242e-11
    SSE =  2.56788398837e-17
    total iteration =  498
    diff =  9.82894267441e-11
    SSE =  6.67750254964e-17
    total iteration =  500
    diff =  9.9725461057e-11
    SSE =  2.09860710417e-17
    total iteration =  476
    diff =  9.39128589046e-11
    SSE =  4.6035586437e-17
    total iteration =  478
    diff =  9.98118509541e-11
    SSE =  1.82425388378e-17
    total iteration =  462
    diff =  9.51814041678e-11
    SSE =  5.09843406492e-17
    total iteration =  472
    diff =  9.57024150573e-11
    SSE =  2.00739660654e-17
    total iteration =  461
    diff =  9.84587724858e-11
    SSE =  5.0568285984e-17
    total iteration =  474
    diff =  9.30208635468e-11
    SSE =  4.25633161774e-17
    total iteration =  490
    diff =  9.71650586629e-11
    SSE =  2.89511698969e-17
    total iteration =  508
    diff =  9.74472104447e-11
    SSE =  2.25427673552e-17

```python
x = [i[0] for i in dat2]
y = [i[1] for i in dat2]
plt.figure()
plt.plot(x,y,'bo')
plt.xlabel('omega')
plt.ylabel('# iteration')
plt.show()
```

# Thomas Algorighm: Creating tridiagonal matrix

```python
N = 10
A = np.zeros((N,N))
for i in range(N):
    for j in range(N):
        if i == j:
            A[i,j] = 6
        if j == i+1:
            A[i,j] = -2
        if j == i-1:
            A[i,j] = -1
print(A)
```

    [[ 6. -2.  0.  0.  0.  0.  0.  0.  0.  0.]
     [-1.  6. -2.  0.  0.  0.  0.  0.  0.  0.]
     [ 0. -1.  6. -2.  0.  0.  0.  0.  0.  0.]
     [ 0.  0. -1.  6. -2.  0.  0.  0.  0.  0.]
     [ 0.  0.  0. -1.  6. -2.  0.  0.  0.  0.]
     [ 0.  0.  0.  0. -1.  6. -2.  0.  0.  0.]
     [ 0.  0.  0.  0.  0. -1.  6. -2.  0.  0.]
     [ 0.  0.  0.  0.  0.  0. -1.  6. -2.  0.]
     [ 0.  0.  0.  0.  0.  0.  0. -1.  6. -2.]
     [ 0.  0.  0.  0.  0.  0.  0.  0. -1.  6.]]

```python
N = 10
a = np.array(N-1)
a = -1
b = np.array(N)
b = 6
c = np.array(N-1)
c = -2
A2 = sp.sparse.diags(diagonals = [a,b,c], offsets = [-1,0,1], shape = (N,N))
A2.todense()
```

    matrix([[ 6., -2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
            [-1.,  6., -2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
            [ 0., -1.,  6., -2.,  0.,  0.,  0.,  0.,  0.,  0.],
            [ 0.,  0., -1.,  6., -2.,  0.,  0.,  0.,  0.,  0.],
            [ 0.,  0.,  0., -1.,  6., -2.,  0.,  0.,  0.,  0.],
            [ 0.,  0.,  0.,  0., -1.,  6., -2.,  0.,  0.,  0.],
            [ 0.,  0.,  0.,  0.,  0., -1.,  6., -2.,  0.,  0.],
            [ 0.,  0.,  0.,  0.,  0.,  0., -1.,  6., -2.,  0.],
            [ 0.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  6., -2.],
            [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  6.]])

```python
A2
```

    <10x10 sparse matrix of type '<class 'numpy.float64'>'
    	with 28 stored elements (3 diagonals) in DIAgonal format>

```python
A2.data
```

    array([[-1., -1., -1., -1., -1., -1., -1., -1., -1.,  0.],
           [ 6.,  6.,  6.,  6.,  6.,  6.,  6.,  6.,  6.,  6.],
           [ 0., -2., -2., -2., -2., -2., -2., -2., -2., -2.]])

```python
A2.offsets
```

    array([-1,  0,  1], dtype=int32)

```python
A2.data[0]
```

    array([-1., -1., -1., -1., -1., -1., -1., -1., -1.,  0.])

```python
A3 = sp.sparse.diags([[40]*6,[-1 * i for i in range(1,6)],[range(-20,-10,2)]],[0,1,-1],(6,6))
A3.todense()
```

    matrix([[ 40.,  -1.,   0.,   0.,   0.,   0.],
            [-20.,  40.,  -2.,   0.,   0.,   0.],
            [  0., -18.,  40.,  -3.,   0.,   0.],
            [  0.,   0., -16.,  40.,  -4.,   0.],
            [  0.,   0.,   0., -14.,  40.,  -5.],
            [  0.,   0.,   0.,   0., -12.,  40.]])

```python
A3.offsets
```

    array([ 0,  1, -1], dtype=int32)

```python
A3.data
```

    array([[ 40.,  40.,  40.,  40.,  40.,  40.],
           [  0.,  -1.,  -2.,  -3.,  -4.,  -5.],
           [-20., -18., -16., -14., -12.,   0.]])

```python
A3.data[1,0]
```

    0.0

```python
A3_sort = sorted(zip(A3.data,A3.offsets), key = lambda x: x[1])
A4 = np.array([i[0] for i in A3_sort])
A4
```

    array([[-20., -18., -16., -14., -12.,   0.],
           [ 40.,  40.,  40.,  40.,  40.,  40.],
           [  0.,  -1.,  -2.,  -3.,  -4.,  -5.]])

```python
a = A4[0] #A should start with 0, but it does not
b = A4[1]
c = A4[2] #C should not start with 0, but it does
print(a,'\n',b,'\n',c,'\n')
```

    [-20. -18. -16. -14. -12.   0.]
     [ 40.  40.  40.  40.  40.  40.]
     [ 0. -1. -2. -3. -4. -5.]

# Thomas method coding

```python
def thomas(A,d):
    #imput: A = scipy.sparse.diags
    #input: d = array shape (N,)
    #output: x from Ax = d where x.shape is (N,)
    N = d.shape[0]
    #sorting array data based on the offsets
    A_sort = sorted(zip(A.data,A.offsets), key = lambda x: x[1])
    A_dat = np.array([i[0] for i in A_sort])
    a = np.empty(N)
    b = np.empty(N)
    c = np.empty(N)
    a[0] = 0
    a[1:] = A_dat[0,:-1]
    b = A_dat[1]
    c[-1] = 0
    c[:-1]= A_dat[2,1:]
    #move 0 to the first cell of A
    #move 0 to the last cell of C
    cp = np.empty(N)
    dp = np.empty(N)
    x = np.empty(N)
    cp[0] = c[0] / b[0]
    dp[0] = d[0] / b[0]
    for i in range(1,N-1):
        cp[i] = c[i] / (b[i]-a[i]*cp[i-1])
        dp[i] = (d[i] - a[i] * dp[i-1]) / (b[i]-a[i] * cp[i-1])
    i = N-1
    x[N-1] = (d[i] - a[i] * dp[i-1]) / (b[i]-a[i] * cp[i-1])
    for i in range(N-2,-1,-1):
        x[i] = dp[i] - cp[i] * x[i+1]
    return x
```

```python
np.random.seed(1)
d = 10 * np.random.random(10)
d
```

    array([ 4.17022005,  7.20324493,  0.00114375,  3.02332573,  1.46755891,
            0.92338595,  1.86260211,  3.45560727,  3.96767474,  5.38816734])

```python
x = thomas(sp.sparse.dia_matrix(A),d)
```

```python
A
```

    array([[ 6., -2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [-1.,  6., -2.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0., -1.,  6., -2.,  0.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0., -1.,  6., -2.,  0.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0., -1.,  6., -2.,  0.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0., -1.,  6., -2.,  0.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0., -1.,  6., -2.,  0.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0., -1.,  6., -2.,  0.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  6., -2.],
           [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0., -1.,  6.]])

```python
A.dot(x)
```

    array([ 4.17022005,  7.20324493,  0.00114375,  3.02332573,  1.46755891,
            0.92338595,  1.86260211,  3.45560727,  3.96767474,  5.38816734])

```python
d
```

    array([ 4.17022005,  7.20324493,  0.00114375,  3.02332573,  1.46755891,
            0.92338595,  1.86260211,  3.45560727,  3.96767474,  5.38816734])

```python
A.dot(x) - d
```

    array([ 0.,  0.,  0.,  0.,  0., -0., -0., -0., -0.,  0.])

# Test function with other matrix A and D

```python
A3
```

    <6x6 sparse matrix of type '<class 'numpy.float64'>'
    	with 16 stored elements (3 diagonals) in DIAgonal format>

```python
A3.todense()
```

    matrix([[ 40.,  -1.,   0.,   0.,   0.,   0.],
            [-20.,  40.,  -2.,   0.,   0.,   0.],
            [  0., -18.,  40.,  -3.,   0.,   0.],
            [  0.,   0., -16.,  40.,  -4.,   0.],
            [  0.,   0.,   0., -14.,  40.,  -5.],
            [  0.,   0.,   0.,   0., -12.,  40.]])

```python
np.random.seed(22)
d3 = np.random.random(6) * 3
d3
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

```python
x3 = thomas(A3,d3)
```

```python
x3
```

    array([ 0.01682291,  0.04753462,  0.05994181,  0.09347838,  0.05063002,
            0.0406038 ])

```python
A3.dot(x3) - d3
```

    array([ 0.,  0., -0.,  0., -0.,  0.])

```python
A3.dot(x3)
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

```python
d3
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

```python
from scipy.sparse import linalg
```

```python
A3.todense()
```

    matrix([[ 40.,  -1.,   0.,   0.,   0.,   0.],
            [-20.,  40.,  -2.,   0.,   0.,   0.],
            [  0., -18.,  40.,  -3.,   0.,   0.],
            [  0.,   0., -16.,  40.,  -4.,   0.],
            [  0.,   0.,   0., -14.,  40.,  -5.],
            [  0.,   0.,   0.,   0., -12.,  40.]])

```python
sp_ans3 = linalg.spsolve(A3,d3)
```

    /home/me/anaconda3/lib/python3.6/site-packages/scipy/sparse/linalg/dsolve/linsolve.py:102: SparseEfficiencyWarning: spsolve requires A be CSC or CSR matrix format
      SparseEfficiencyWarning)

```python
A3.dot(sp_ans3)
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

```python
d3
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

# Re-create A3 in CSC sparse matrix format
```Compressed Sparse Column Format
Compressed Sparse Row Format (CSR) is not covered (but it is similar)```
### Important attributes of csc matrix
data
<br>indices
<br>indptr

```python
A3csc = A3.tocsc()
```

```python
A3csc.data
```

    array([ 40., -20.,  40.,  -1., -18.,  40.,  -2., -16.,  40.,  -3., -14.,
            40.,  -4., -12.,  40.,  -5.])

```python
A3csc.data.shape
```

    (16,)

```python
A3csc.indices
```

    array([0, 1, 1, 0, 2, 2, 1, 3, 3, 2, 4, 4, 3, 5, 5, 4], dtype=int32)

```python
A3csc.indices.shape
```

    (16,)

```python
A3csc.indptr
```

    array([ 0,  2,  5,  8, 11, 14, 16], dtype=int32)

```python
A3csc.indptr.shape
```

    (7,)

```python
A3csc.todense()
```

    matrix([[ 40.,  -1.,   0.,   0.,   0.,   0.],
            [-20.,  40.,  -2.,   0.,   0.,   0.],
            [  0., -18.,  40.,  -3.,   0.,   0.],
            [  0.,   0., -16.,  40.,  -4.,   0.],
            [  0.,   0.,   0., -14.,  40.,  -5.],
            [  0.,   0.,   0.,   0., -12.,  40.]])

## data (`.data`) is the non-zero value in sparse matrix
A3csc.data = [ 40., -20.,  40.,  -1., -18.,  40.,  -2., -16.,  40.,  -3., -14.,
        40.,  -4., -12.,  40.,  -5.]
## indices (`.indicies`) tell the row that the data belong
A3csc.indices = [0, 1, 1, 0, 2, 2, 1, 3, 3, 2, 4, 4, 3, 5, 5, 4]
## index pointer (`.indptr`) tell how to split the data before applying the row index
A3csc.indptr = [ 0,  2,  5,  8, 11, 14, 16] means

## Explanation
### For the first column, look at A3csc.indptr
<br>1st column pointer is [0:2] (start from 0 to 2 but exclude 2)
<br>Use this index for the data
<br>A3csc.data[0:2] = [40, -20]
<br>The corresponding row is
<br>A3csc.indices[0:2] = [0, 1]
<br>This is for the first column
<br>Thus, for the first column we have 40 in row 0, and -20 in row 1
### For the second column, look at A3csc.indptr
<br>2nd column pointer is [2:5] (start from 2 to 5 but exclude 5)
<br>Use this index for the data
<br>A3csc.data[2:5] = [ 40.,  -1., -18.]
<br>The corresponding row is
<br>A3csc.indices[2:5] = [1, 0, 2]
<br>This is for the first column
<br>Thus, for the first column we have 40 in row 1, and -1 in row 0, and -18 in row 2

```python
ptr = A3csc.indptr
dat = A3csc.data
rix = A3csc.indices

print('data in each column')
for i in range(ptr.shape[0]-1):
    print('column = {:1d} '.format(i), dat[ptr[i]:ptr[i+1]])
    print('row index =', rix[ptr[i]:ptr[i+1]])
```

    data in each column
    column = 0  [ 40. -20.]
    row index = [0 1]
    column = 1  [ 40.  -1. -18.]
    row index = [1 0 2]
    column = 2  [ 40.  -2. -16.]
    row index = [2 1 3]
    column = 3  [ 40.  -3. -14.]
    row index = [3 2 4]
    column = 4  [ 40.  -4. -12.]
    row index = [4 3 5]
    column = 5  [ 40.  -5.]
    row index = [5 4]

```python
A3csc.todense()
```

    matrix([[ 40.,  -1.,   0.,   0.,   0.,   0.],
            [-20.,  40.,  -2.,   0.,   0.,   0.],
            [  0., -18.,  40.,  -3.,   0.,   0.],
            [  0.,   0., -16.,  40.,  -4.,   0.],
            [  0.,   0.,   0., -14.,  40.,  -5.],
            [  0.,   0.,   0.,   0., -12.,  40.]])

# sparse matrix solver

```python
sparse_ans = linalg.spsolve(A3csc,d3)
```

```python
sparse_ans
```

    array([ 0.01682291,  0.04753462,  0.05994181,  0.09347838,  0.05063002,
            0.0406038 ])

```python
A3csc.dot(sparse_ans)
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

```python
d3
```

    array([ 0.62538161,  1.44504319,  1.26161411,  2.577546  ,  0.51348466,
            1.01659188])

# User Thomas algorithm

```python
thomas(A3,d3)
```

    array([ 0.01682291,  0.04753462,  0.05994181,  0.09347838,  0.05063002,
            0.0406038 ])

# Solve as dense matrix

```python
A3_dense = A3.todense()
sp.linalg.solve(A3_dense,d3)
```

    array([ 0.01682291,  0.04753462,  0.05994181,  0.09347838,  0.05063002,
            0.0406038 ])

# Solve time comparison

```python
%%timeit -n 5
thomas(A3,d3)
```

    5 loops, best of 3: 38.5 µs per loop

```python
%%timeit -n 5
linalg.spsolve(A3csc,d3)
```

    5 loops, best of 3: 52.1 µs per loop

```python
%%timeit -n 5
sp.linalg.solve(A3_dense,d3)
```

    5 loops, best of 3: 63.5 µs per loop

Note that spsolve can solve any sparse matrix, but thomas just solve tri-diagonal matrix (with the input in diagonal sparse matrix format.