C THIS VERSION: 18/05/2004 AT 08:15:00 GMT.
C Updated 18/05/2004: depricated SNRM2 replaced by equivalent "streamlined" DNRM2
      REAL FUNCTION SDOT(N,SX,INCX,SY,INCY)
C
C     FORMS THE DOT PRODUCT OF TWO VECTORS.
C     USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO ONE.
C     JACK DONGARRA, LINPACK, 3/11/78.
C
      REAL SX(*),SY(*),STEMP
      INTEGER I,INCX,INCY,IX,IY,M,MP1,N
C
      STEMP = 0.0E0
      SDOT = 0.0E0
      IF(N.LE.0)RETURN
      IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20
C
C        CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS
C          NOT EQUAL TO 1
C
      IX = 1
      IY = 1
      IF(INCX.LT.0)IX = (-N+1)*INCX + 1
      IF(INCY.LT.0)IY = (-N+1)*INCY + 1
      DO 10 I = 1,N
        STEMP = STEMP + SX(IX)*SY(IY)
        IX = IX + INCX
        IY = IY + INCY
   10 CONTINUE
      SDOT = STEMP
      RETURN
C
C        CODE FOR BOTH INCREMENTS EQUAL TO 1
C
C
C        CLEAN-UP LOOP
C
   20 M = MOD(N,5)
      IF( M .EQ. 0 ) GO TO 40
      DO 30 I = 1,M
        STEMP = STEMP + SX(I)*SY(I)
   30 CONTINUE
      IF( N .LT. 5 ) GO TO 60
   40 MP1 = M + 1
      DO 50 I = MP1,N,5
        STEMP = STEMP + SX(I)*SY(I) + SX(I + 1)*SY(I + 1) +
     *   SX(I + 2)*SY(I + 2) + SX(I + 3)*SY(I + 3) + SX(I + 4)*SY(I + 4)
   50 CONTINUE
   60 SDOT = STEMP
      RETURN
      END
      double precision function ddot(n,dx,incx,dy,incy)
c
c     forms the dot product of two vectors.
c     uses unrolled loops for increments equal to one.
c     jack dongarra, linpack, 3/11/78.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      double precision dx(*),dy(*),dtemp
      integer i,incx,incy,ix,iy,m,mp1,n
c
      ddot = 0.0d0
      dtemp = 0.0d0
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c        code for unequal increments or equal increments
c          not equal to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        dtemp = dtemp + dx(ix)*dy(iy)
        ix = ix + incx
        iy = iy + incy
   10 continue
      ddot = dtemp
      return
c
c        code for both increments equal to 1
c
c
c        clean-up loop
c
   20 m = mod(n,5)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        dtemp = dtemp + dx(i)*dy(i)
   30 continue
      if( n .lt. 5 ) go to 60
   40 mp1 = m + 1
      do 50 i = mp1,n,5
        dtemp = dtemp + dx(i)*dy(i) + dx(i + 1)*dy(i + 1) +
     *   dx(i + 2)*dy(i + 2) + dx(i + 3)*dy(i + 3) + dx(i + 4)*dy(i + 4)
   50 continue
   60 ddot = dtemp
      return
      end
      SUBROUTINE  SCOPY(N,SX,INCX,SY,INCY)
C
C     COPIES A VECTOR, X, TO A VECTOR, Y.
C     USES UNROLLED LOOPS FOR INCREMENTS EQUAL TO 1.
C     JACK DONGARRA, LINPACK, 3/11/78.
C
      REAL SX(*),SY(*)
      INTEGER I,INCX,INCY,IX,IY,M,MP1,N
C
      IF(N.LE.0)RETURN
      IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20
C
C        CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS
C          NOT EQUAL TO 1
C
      IX = 1
      IY = 1
      IF(INCX.LT.0)IX = (-N+1)*INCX + 1
      IF(INCY.LT.0)IY = (-N+1)*INCY + 1
      DO 10 I = 1,N
        SY(IY) = SX(IX)
        IX = IX + INCX
        IY = IY + INCY
   10 CONTINUE
      RETURN
C
C        CODE FOR BOTH INCREMENTS EQUAL TO 1
C
C
C        CLEAN-UP LOOP
C
   20 M = MOD(N,7)
      IF( M .EQ. 0 ) GO TO 40
      DO 30 I = 1,M
        SY(I) = SX(I)
   30 CONTINUE
      IF( N .LT. 7 ) RETURN
   40 MP1 = M + 1
      DO 50 I = MP1,N,7
        SY(I) = SX(I)
        SY(I + 1) = SX(I + 1)
        SY(I + 2) = SX(I + 2)
        SY(I + 3) = SX(I + 3)
        SY(I + 4) = SX(I + 4)
        SY(I + 5) = SX(I + 5)
        SY(I + 6) = SX(I + 6)
   50 CONTINUE
      RETURN
      END

      subroutine  dcopy(n,dx,incx,dy,incy)
c
c     copies a vector, x, to a vector, y.
c     uses unrolled loops for increments equal to one.
c     jack dongarra, linpack, 3/11/78.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      double precision dx(*),dy(*)
      integer i,incx,incy,ix,iy,m,mp1,n
c
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c        code for unequal increments or equal increments
c          not equal to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        dy(iy) = dx(ix)
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c        code for both increments equal to 1
c
c
c        clean-up loop
c
   20 m = mod(n,7)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        dy(i) = dx(i)
   30 continue
      if( n .lt. 7 ) return
   40 mp1 = m + 1
      do 50 i = mp1,n,7
        dy(i) = dx(i)
        dy(i + 1) = dx(i + 1)
        dy(i + 2) = dx(i + 2)
        dy(i + 3) = dx(i + 3)
        dy(i + 4) = dx(i + 4)
        dy(i + 5) = dx(i + 5)
        dy(i + 6) = dx(i + 6)
   50 continue
      return
      end
      SUBROUTINE  SSCAL(N,SA,SX,INCX)
C
C     SCALES A VECTOR BY A CONSTANT.
C     USES UNROLLED LOOPS FOR INCREMENT EQUAL TO 1.
C     JACK DONGARRA, LINPACK, 3/11/78.
C
      REAL SA,SX(*)
      INTEGER I,INCX,M,MP1,N,NINCX
C
      IF(N.LE.0)RETURN
      IF(INCX.EQ.1)GO TO 20
C
C        CODE FOR INCREMENT NOT EQUAL TO 1
C
      NINCX = N*INCX
      DO 10 I = 1,NINCX,INCX
        SX(I) = SA*SX(I)
   10 CONTINUE
      RETURN
C
C        CODE FOR INCREMENT EQUAL TO 1
C
C
C        CLEAN-UP LOOP
C
   20 M = MOD(N,5)
      IF( M .EQ. 0 ) GO TO 40
      DO 30 I = 1,M
        SX(I) = SA*SX(I)
   30 CONTINUE
      IF( N .LT. 5 ) RETURN
   40 MP1 = M + 1
      DO 50 I = MP1,N,5
        SX(I) = SA*SX(I)
        SX(I + 1) = SA*SX(I + 1)
        SX(I + 2) = SA*SX(I + 2)
        SX(I + 3) = SA*SX(I + 3)
        SX(I + 4) = SA*SX(I + 4)
   50 CONTINUE
      RETURN
      END

      subroutine  dscal(n,da,dx,incx)
c
c     scales a vector by a constant.
c     uses unrolled loops for increment equal to one.
c     jack dongarra, linpack, 3/11/78.
c     modified 3/93 to return if incx .le. 0.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      double precision da,dx(*)
      integer i,incx,m,mp1,n,nincx
c
      if( n.le.0 .or. incx.le.0 )return
      if(incx.eq.1)go to 20
c
c        code for increment not equal to 1
c
      nincx = n*incx
      do 10 i = 1,nincx,incx
        dx(i) = da*dx(i)
   10 continue
      return
c
c        code for increment equal to 1
c
c
c        clean-up loop
c
   20 m = mod(n,5)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        dx(i) = da*dx(i)
   30 continue
      if( n .lt. 5 ) return
   40 mp1 = m + 1
      do 50 i = mp1,n,5
        dx(i) = da*dx(i)
        dx(i + 1) = da*dx(i + 1)
        dx(i + 2) = da*dx(i + 2)
        dx(i + 3) = da*dx(i + 3)
        dx(i + 4) = da*dx(i + 4)
   50 continue
      return
      end
      SUBROUTINE SAXPY(N,SA,SX,INCX,SY,INCY)
C
C     CONSTANT TIMES A VECTOR PLUS A VECTOR.
C     USES UNROLLED LOOP FOR INCREMENTS EQUAL TO ONE.
C     JACK DONGARRA, LINPACK, 3/11/78.
C
      REAL SX(*),SY(*),SA
      INTEGER I,INCX,INCY,IX,IY,M,MP1,N
C
      IF(N.LE.0)RETURN
      IF (SA .EQ. 0.0) RETURN
      IF(INCX.EQ.1.AND.INCY.EQ.1)GO TO 20
C
C        CODE FOR UNEQUAL INCREMENTS OR EQUAL INCREMENTS
C          NOT EQUAL TO 1
C
      IX = 1
      IY = 1
      IF(INCX.LT.0)IX = (-N+1)*INCX + 1
      IF(INCY.LT.0)IY = (-N+1)*INCY + 1
      DO 10 I = 1,N
        SY(IY) = SY(IY) + SA*SX(IX)
        IX = IX + INCX
        IY = IY + INCY
   10 CONTINUE
      RETURN
C
C        CODE FOR BOTH INCREMENTS EQUAL TO 1
C
C
C        CLEAN-UP LOOP
C
   20 M = MOD(N,4)
      IF( M .EQ. 0 ) GO TO 40
      DO 30 I = 1,M
        SY(I) = SY(I) + SA*SX(I)
   30 CONTINUE
      IF( N .LT. 4 ) RETURN
   40 MP1 = M + 1
      DO 50 I = MP1,N,4
        SY(I) = SY(I) + SA*SX(I)
        SY(I + 1) = SY(I + 1) + SA*SX(I + 1)
        SY(I + 2) = SY(I + 2) + SA*SX(I + 2)
        SY(I + 3) = SY(I + 3) + SA*SX(I + 3)
   50 CONTINUE
      RETURN
      END

      subroutine daxpy(n,da,dx,incx,dy,incy)
c
c     constant times a vector plus a vector.
c     uses unrolled loops for increments equal to one.
c     jack dongarra, linpack, 3/11/78.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      double precision dx(*),dy(*),da
      integer i,incx,incy,ix,iy,m,mp1,n
c
      if(n.le.0)return
      if (da .eq. 0.0d0) return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c        code for unequal increments or equal increments
c          not equal to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        dy(iy) = dy(iy) + da*dx(ix)
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c        code for both increments equal to 1
c
c
c        clean-up loop
c
   20 m = mod(n,4)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        dy(i) = dy(i) + da*dx(i)
   30 continue
      if( n .lt. 4 ) return
   40 mp1 = m + 1
      do 50 i = mp1,n,4
        dy(i) = dy(i) + da*dx(i)
        dy(i + 1) = dy(i + 1) + da*dx(i + 1)
        dy(i + 2) = dy(i + 2) + da*dx(i + 2)
        dy(i + 3) = dy(i + 3) + da*dx(i + 3)
   50 continue
      return
      end

      REAL FUNCTION SNRM2 ( N, X, INCX )
*     .. Scalar Arguments ..
      INTEGER                           INCX, N
*     .. Array Arguments ..
      REAL                 X( INCX * ( N - 1 ) + 1 )
*     ..
*
*  SNRM2 returns the euclidean norm of a vector via the function
*  name, so that
*
*     SNRM2 := sqrt( x'*x )
*
*
*
*  -- This version written on 25-October-1982.
*     Sven Hammarling, Nag Ltd.
*
*
*     .. Parameters ..
      REAL     ONE         , ZERO
      PARAMETER           ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     .. Local Scalars ..
      INTEGER               IX
      REAL                  ABSXI, NORM, SCALE, SSQ
*     .. Intrinsic Functions ..
      INTRINSIC             ABS, SQRT
*     ..
*     .. Executable Statements ..
      IF( N.LT.1 .OR. INCX.LT.1 )THEN
         NORM  = ZERO
      ELSE IF( N.EQ.1 )THEN
         NORM  = ABS( X( 1 ) )
      ELSE
         SCALE = ZERO
         SSQ   = ONE
         DO 10, IX = 1, 1 + ( N - 1 )*INCX, INCX
            IF( X( IX ).NE.ZERO )THEN
               ABSXI = ABS( X( IX ) )
               IF( SCALE.LT.ABSXI )THEN
                  SSQ   = ONE   + SSQ*( SCALE/ABSXI )**2
                  SCALE = ABSXI
               ELSE
                  SSQ   = SSQ   +     ( ABSXI/SCALE )**2
               END IF
            END IF
   10    CONTINUE
         NORM  = SCALE * SQRT( SSQ )
      END IF
*
      SNRM2 = NORM
      RETURN
*
*     End of SNRM2.
*
      END

      DOUBLE PRECISION FUNCTION DNRM2 ( N, X, INCX )
*     .. Scalar Arguments ..
      INTEGER                           INCX, N
*     .. Array Arguments ..
      DOUBLE PRECISION                  X( INCX * ( N - 1 ) + 1 )
*     ..
*
*  DNRM2 returns the euclidean norm of a vector via the function
*  name, so that
*
*     DNRM2 := sqrt( x'*x )
*
*
*
*  -- This version written on 25-October-1982.
*     Modified on 14-October-1993 to inline the call to DLASSQ.
*     Sven Hammarling, Nag Ltd.
*
*
*     .. Parameters ..
      DOUBLE PRECISION      ONE         , ZERO
      PARAMETER           ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     .. Local Scalars ..
      INTEGER               IX
      DOUBLE PRECISION      ABSXI, NORM, SCALE, SSQ
*     .. Intrinsic Functions ..
      INTRINSIC             ABS, SQRT
*     ..
*     .. Executable Statements ..
      IF( N.LT.1 .OR. INCX.LT.1 )THEN
         NORM  = ZERO
      ELSE IF( N.EQ.1 )THEN
         NORM  = ABS( X( 1 ) )
      ELSE
         SCALE = ZERO
         SSQ   = ONE
*        The following loop is equivalent to this call to the LAPACK
*        auxiliary routine:
*        CALL DLASSQ( N, X, INCX, SCALE, SSQ )
*
         DO 10, IX = 1, 1 + ( N - 1 )*INCX, INCX
            IF( X( IX ).NE.ZERO )THEN
               ABSXI = ABS( X( IX ) )
               IF( SCALE.LT.ABSXI )THEN
                  SSQ   = ONE   + SSQ*( SCALE/ABSXI )**2
                  SCALE = ABSXI
               ELSE
                  SSQ   = SSQ   +     ( ABSXI/SCALE )**2
               END IF
            END IF
   10    CONTINUE
         NORM  = SCALE * SQRT( SSQ )
      END IF
*
      DNRM2 = NORM
      RETURN
*
*     End of DNRM2.
*
      END

      subroutine  drot (n,dx,incx,dy,incy,c,s)
c
c     applies a plane rotation.
c     jack dongarra, linpack, 3/11/78.
c
      double precision dx(:),dy(:),dtemp,c,s
      integer i,incx,incy,ix,iy,n
c
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c       code for unequal increments or equal increments not equal
c         to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        dtemp = c*dx(ix) + s*dy(iy)
        dy(iy) = c*dy(iy) - s*dx(ix)
        dx(ix) = dtemp
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c       code for both increments equal to 1
c
   20 do 30 i = 1,n
        dtemp = c*dx(i) + s*dy(i)
        dy(i) = c*dy(i) - s*dx(i)
        dx(i) = dtemp
   30 continue
      return
      end

      subroutine drotg(da,db,c,s)
c
c     construct givens plane rotation.
c     jack dongarra, linpack, 3/11/78.
c
      double precision da,db,c,s,roe,scale,r,z
c
      roe = db
      if( dabs(da) .gt. dabs(db) ) roe = da
      scale = dabs(da) + dabs(db)
      if( scale .ne. 0.0d0 ) go to 10
         c = 1.0d0
         s = 0.0d0
         r = 0.0d0
         z = 0.0d0
         go to 20
   10 r = scale*dsqrt((da/scale)**2 + (db/scale)**2)
      r = dsign(1.0d0,roe)*r
      c = da/r
      s = db/r
      z = 1.0d0
      if( dabs(da) .gt. dabs(db) ) z = s
      if( dabs(db) .ge. dabs(da) .and. c .ne. 0.0d0 ) z = 1.0d0/c
   20 da = r
      db = z
      return
      end

      subroutine srot (n,sx,incx,sy,incy,c,s)
c
c     applies a plane rotation.
c     jack dongarra, linpack, 3/11/78.
c
      real sx(:),sy(:),stemp,c,s
      integer i,incx,incy,ix,iy,n
c
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c       code for unequal increments or equal increments not equal
c         to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        stemp = c*sx(ix) + s*sy(iy)
        sy(iy) = c*sy(iy) - s*sx(ix)
        sx(ix) = stemp
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c       code for both increments equal to 1
c
   20 do 30 i = 1,n
        stemp = c*sx(i) + s*sy(i)
        sy(i) = c*sy(i) - s*sx(i)
        sx(i) = stemp
   30 continue
      return
      end

      subroutine srotg(sa,sb,c,s)
c
c     construct givens plane rotation.
c     jack dongarra, linpack, 3/11/78.
c
      real sa,sb,c,s,roe,scale,r,z
c
      roe = sb
      if( abs(sa) .gt. abs(sb) ) roe = sa
      scale = abs(sa) + abs(sb)
      if( scale .ne. 0.0 ) go to 10
         c = 1.0
         s = 0.0
         r = 0.0
         z = 0.0
         go to 20
   10 r = scale*sqrt((sa/scale)**2 + (sb/scale)**2)
      r = sign(1.0,roe)*r
      c = sa/r
      s = sb/r
      z = 1.0
      if( abs(sa) .gt. abs(sb) ) z = s
      if( abs(sb) .ge. abs(sa) .and. c .ne. 0.0 ) z = 1.0/c
   20 sa = r
      sb = z
      return
      end

      SUBROUTINE DGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB,
     $                   BETA, C, LDC )
*     .. Scalar Arguments ..
      CHARACTER*1        TRANSA, TRANSB
      INTEGER            M, N, K, LDA, LDB, LDC
      DOUBLE PRECISION   ALPHA, BETA
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * )
*     ..
*
*  Purpose
*  =======
*
*  DGEMM  performs one of the matrix-matrix operations
*
*     C := alpha*op( A )*op( B ) + beta*C,
*
*  where  op( X ) is one of
*
*     op( X ) = X   or   op( X ) = X',
*
*  alpha and beta are scalars, and A, B and C are matrices, with op( A )
*  an m by k matrix,  op( B )  a  k by n matrix and  C an m by n matrix.
*
*  Parameters
*  ==========
*
*  TRANSA - CHARACTER*1.
*           On entry, TRANSA specifies the form of op( A ) to be used in
*           the matrix multiplication as follows:
*
*              TRANSA = 'N' or 'n',  op( A ) = A.
*
*              TRANSA = 'T' or 't',  op( A ) = A'.
*
*              TRANSA = 'C' or 'c',  op( A ) = A'.
*
*           Unchanged on exit.
*
*  TRANSB - CHARACTER*1.
*           On entry, TRANSB specifies the form of op( B ) to be used in
*           the matrix multiplication as follows:
*
*              TRANSB = 'N' or 'n',  op( B ) = B.
*
*              TRANSB = 'T' or 't',  op( B ) = B'.
*
*              TRANSB = 'C' or 'c',  op( B ) = B'.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry,  M  specifies  the number  of rows  of the  matrix
*           op( A )  and of the  matrix  C.  M  must  be at least  zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry,  N  specifies the number  of columns of the matrix
*           op( B ) and the number of columns of the matrix C. N must be
*           at least zero.
*           Unchanged on exit.
*
*  K      - INTEGER.
*           On entry,  K  specifies  the number of columns of the matrix
*           op( A ) and the number of rows of the matrix op( B ). K must
*           be at least  zero.
*           Unchanged on exit.
*
*  ALPHA  - DOUBLE PRECISION.
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is
*           k  when  TRANSA = 'N' or 'n',  and is  m  otherwise.
*           Before entry with  TRANSA = 'N' or 'n',  the leading  m by k
*           part of the array  A  must contain the matrix  A,  otherwise
*           the leading  k by m  part of the array  A  must contain  the
*           matrix A.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. When  TRANSA = 'N' or 'n' then
*           LDA must be at least  max( 1, m ), otherwise  LDA must be at
*           least  max( 1, k ).
*           Unchanged on exit.
*
*  B      - DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is
*           n  when  TRANSB = 'N' or 'n',  and is  k  otherwise.
*           Before entry with  TRANSB = 'N' or 'n',  the leading  k by n
*           part of the array  B  must contain the matrix  B,  otherwise
*           the leading  n by k  part of the array  B  must contain  the
*           matrix B.
*           Unchanged on exit.
*
*  LDB    - INTEGER.
*           On entry, LDB specifies the first dimension of B as declared
*           in the calling (sub) program. When  TRANSB = 'N' or 'n' then
*           LDB must be at least  max( 1, k ), otherwise  LDB must be at
*           least  max( 1, n ).
*           Unchanged on exit.
*
*  BETA   - DOUBLE PRECISION.
*           On entry,  BETA  specifies the scalar  beta.  When  BETA  is
*           supplied as zero then C need not be set on input.
*           Unchanged on exit.
*
*  C      - DOUBLE PRECISION array of DIMENSION ( LDC, n ).
*           Before entry, the leading  m by n  part of the array  C must
*           contain the matrix  C,  except when  beta  is zero, in which
*           case C need not be set on entry.
*           On exit, the array  C  is overwritten by the  m by n  matrix
*           ( alpha*op( A )*op( B ) + beta*C ).
*
*  LDC    - INTEGER.
*           On entry, LDC specifies the first dimension of C as declared
*           in  the  calling  (sub)  program.   LDC  must  be  at  least
*           max( 1, m ).
*           Unchanged on exit.
*
*
*  Level 3 Blas routine.
*
*  -- Written on 8-February-1989.
*     Jack Dongarra, Argonne National Laboratory.
*     Iain Duff, AERE Harwell.
*     Jeremy Du Croz, Numerical Algorithms Group Ltd.
*     Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     .. Local Scalars ..
      LOGICAL            NOTA, NOTB
      INTEGER            I, INFO, J, L, NROWA, NROWB
      DOUBLE PRECISION   TEMP
*     .. Parameters ..
      DOUBLE PRECISION   ONE         , ZERO
      PARAMETER        ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Executable Statements ..
*
*     Set  NOTA  and  NOTB  as  true if  A  and  B  respectively are not
*     transposed and set  NROWA, and  NROWB  as the number of rows
*     and  columns of  A  and the  number of  rows  of  B  respectively.
*
      NOTA  = LSAME( TRANSA, 'N' )
      NOTB  = LSAME( TRANSB, 'N' )
      IF( NOTA )THEN
         NROWA = M
      ELSE
         NROWA = K
      END IF
      IF( NOTB )THEN
         NROWB = K
      ELSE
         NROWB = N
      END IF
*
*     Test the input parameters.
*
      INFO = 0
      IF(      ( .NOT.NOTA                 ).AND.
     $         ( .NOT.LSAME( TRANSA, 'C' ) ).AND.
     $         ( .NOT.LSAME( TRANSA, 'T' ) )      )THEN
         INFO = 1
      ELSE IF( ( .NOT.NOTB                 ).AND.
     $         ( .NOT.LSAME( TRANSB, 'C' ) ).AND.
     $         ( .NOT.LSAME( TRANSB, 'T' ) )      )THEN
         INFO = 2
      ELSE IF( M  .LT.0               )THEN
         INFO = 3
      ELSE IF( N  .LT.0               )THEN
         INFO = 4
      ELSE IF( K  .LT.0               )THEN
         INFO = 5
      ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
         INFO = 8
      ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN
         INFO = 10
      ELSE IF( LDC.LT.MAX( 1, M     ) )THEN
         INFO = 13
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'DGEMM ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
     $    ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     And if  alpha.eq.zero.
*
      IF( ALPHA.EQ.ZERO )THEN
         IF( BETA.EQ.ZERO )THEN
            DO 20, J = 1, N
               DO 10, I = 1, M
                  C( I, J ) = ZERO
   10          CONTINUE
   20       CONTINUE
         ELSE
            DO 40, J = 1, N
               DO 30, I = 1, M
                  C( I, J ) = BETA*C( I, J )
   30          CONTINUE
   40       CONTINUE
         END IF
         RETURN
      END IF
*
*     Start the operations.
*
      IF( NOTB )THEN
         IF( NOTA )THEN
*
*           Form  C := alpha*A*B + beta*C.
*
            DO 90, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 50, I = 1, M
                     C( I, J ) = ZERO
   50             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 60, I = 1, M
                     C( I, J ) = BETA*C( I, J )
   60             CONTINUE
               END IF
               DO 80, L = 1, K
                  IF( B( L, J ).NE.ZERO )THEN
                     TEMP = ALPHA*B( L, J )
                     DO 70, I = 1, M
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
   70                CONTINUE
                  END IF
   80          CONTINUE
   90       CONTINUE
         ELSE
*
*           Form  C := alpha*A'*B + beta*C
*
            DO 120, J = 1, N
               DO 110, I = 1, M
                  TEMP = ZERO
                  DO 100, L = 1, K
                     TEMP = TEMP + A( L, I )*B( L, J )
  100             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  110          CONTINUE
  120       CONTINUE
         END IF
      ELSE
         IF( NOTA )THEN
*
*           Form  C := alpha*A*B' + beta*C
*
            DO 170, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 130, I = 1, M
                     C( I, J ) = ZERO
  130             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 140, I = 1, M
                     C( I, J ) = BETA*C( I, J )
  140             CONTINUE
               END IF
               DO 160, L = 1, K
                  IF( B( J, L ).NE.ZERO )THEN
                     TEMP = ALPHA*B( J, L )
                     DO 150, I = 1, M
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
  150                CONTINUE
                  END IF
  160          CONTINUE
  170       CONTINUE
         ELSE
*
*           Form  C := alpha*A'*B' + beta*C
*
            DO 200, J = 1, N
               DO 190, I = 1, M
                  TEMP = ZERO
                  DO 180, L = 1, K
                     TEMP = TEMP + A( L, I )*B( J, L )
  180             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  190          CONTINUE
  200       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DGEMM .
*
      END

      SUBROUTINE SGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB,
*
************************************************************************
*
*     File of the REAL             Level-3 BLAS.
*     ==========================================
*
*     SUBROUTINE SGEMM ( TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB,
*    $                   BETA, C, LDC )
*
*     SUBROUTINE SSYMM ( SIDE,   UPLO,   M, N,    ALPHA, A, LDA, B, LDB,
*    $                   BETA, C, LDC )
*
*     SUBROUTINE SSYRK ( UPLO,   TRANS,     N, K, ALPHA, A, LDA,
*    $                   BETA, C, LDC )
*
*     SUBROUTINE SSYR2K( UPLO,   TRANS,     N, K, ALPHA, A, LDA, B, LDB,
*    $                   BETA, C, LDC )
*
*     SUBROUTINE STRMM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA,
*    $                   B, LDB )
*
*     SUBROUTINE STRSM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA,
*    $                   B, LDB )
*
*     See:
*
*        Dongarra J. J.,   Du Croz J. J.,   Duff I.  and   Hammarling S.
*        A set of  Level 3  Basic Linear Algebra Subprograms.  Technical
*        Memorandum No.88 (Revision 1), Mathematics and Computer Science
*        Division,  Argonne National Laboratory, 9700 South Cass Avenue,
*        Argonne, Illinois 60439.
*
*
************************************************************************
*
     $                   BETA, C, LDC )
*     .. Scalar Arguments ..
      CHARACTER*1        TRANSA, TRANSB
      INTEGER            M, N, K, LDA, LDB, LDC
      REAL               ALPHA, BETA
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), C( LDC, * )
*     ..
*
*  Purpose
*  =======
*
*  SGEMM  performs one of the matrix-matrix operations
*
*     C := alpha*op( A )*op( B ) + beta*C,
*
*  where  op( X ) is one of
*
*     op( X ) = X   or   op( X ) = X',
*
*  alpha and beta are scalars, and A, B and C are matrices, with op( A )
*  an m by k matrix,  op( B )  a  k by n matrix and  C an m by n matrix.
*
*  Parameters
*  ==========
*
*  TRANSA - CHARACTER*1.
*           On entry, TRANSA specifies the form of op( A ) to be used in
*           the matrix multiplication as follows:
*
*              TRANSA = 'N' or 'n',  op( A ) = A.
*
*              TRANSA = 'T' or 't',  op( A ) = A'.
*
*              TRANSA = 'C' or 'c',  op( A ) = A'.
*
*           Unchanged on exit.
*
*  TRANSB - CHARACTER*1.
*           On entry, TRANSB specifies the form of op( B ) to be used in
*           the matrix multiplication as follows:
*
*              TRANSB = 'N' or 'n',  op( B ) = B.
*
*              TRANSB = 'T' or 't',  op( B ) = B'.
*
*              TRANSB = 'C' or 'c',  op( B ) = B'.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry,  M  specifies  the number  of rows  of the  matrix
*           op( A )  and of the  matrix  C.  M  must  be at least  zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry,  N  specifies the number  of columns of the matrix
*           op( B ) and the number of columns of the matrix C. N must be
*           at least zero.
*           Unchanged on exit.
*
*  K      - INTEGER.
*           On entry,  K  specifies  the number of columns of the matrix
*           op( A ) and the number of rows of the matrix op( B ). K must
*           be at least  zero.
*           Unchanged on exit.
*
*  ALPHA  - REAL            .
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - REAL             array of DIMENSION ( LDA, ka ), where ka is
*           k  when  TRANSA = 'N' or 'n',  and is  m  otherwise.
*           Before entry with  TRANSA = 'N' or 'n',  the leading  m by k
*           part of the array  A  must contain the matrix  A,  otherwise
*           the leading  k by m  part of the array  A  must contain  the
*           matrix A.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. When  TRANSA = 'N' or 'n' then
*           LDA must be at least  max( 1, m ), otherwise  LDA must be at
*           least  max( 1, k ).
*           Unchanged on exit.
*
*  B      - REAL             array of DIMENSION ( LDB, kb ), where kb is
*           n  when  TRANSB = 'N' or 'n',  and is  k  otherwise.
*           Before entry with  TRANSB = 'N' or 'n',  the leading  k by n
*           part of the array  B  must contain the matrix  B,  otherwise
*           the leading  n by k  part of the array  B  must contain  the
*           matrix B.
*           Unchanged on exit.
*
*  LDB    - INTEGER.
*           On entry, LDB specifies the first dimension of B as declared
*           in the calling (sub) program. When  TRANSB = 'N' or 'n' then
*           LDB must be at least  max( 1, k ), otherwise  LDB must be at
*           least  max( 1, n ).
*           Unchanged on exit.
*
*  BETA   - REAL            .
*           On entry,  BETA  specifies the scalar  beta.  When  BETA  is
*           supplied as zero then C need not be set on input.
*           Unchanged on exit.
*
*  C      - REAL             array of DIMENSION ( LDC, n ).
*           Before entry, the leading  m by n  part of the array  C must
*           contain the matrix  C,  except when  beta  is zero, in which
*           case C need not be set on entry.
*           On exit, the array  C  is overwritten by the  m by n  matrix
*           ( alpha*op( A )*op( B ) + beta*C ).
*
*  LDC    - INTEGER.
*           On entry, LDC specifies the first dimension of C as declared
*           in  the  calling  (sub)  program.   LDC  must  be  at  least
*           max( 1, m ).
*           Unchanged on exit.
*
*
*  Level 3 Blas routine.
*
*  -- Written on 8-February-1989.
*     Jack Dongarra, Argonne National Laboratory.
*     Iain Duff, AERE Harwell.
*     Jeremy Du Croz, Numerical Algorithms Group Ltd.
*     Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     .. Local Scalars ..
      LOGICAL            NOTA, NOTB
      INTEGER            I, INFO, J, L, NROWA, NROWB
      REAL               TEMP
*     .. Parameters ..
      REAL               ONE         , ZERO
      PARAMETER        ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     ..
*     .. Executable Statements ..
*
*     Set  NOTA  and  NOTB  as  true if  A  and  B  respectively are not
*     transposed and set  NROWA, and  NROWB  as the number of rows
*     and  columns of  A  and the  number of  rows  of  B  respectively.
*
      NOTA  = LSAME( TRANSA, 'N' )
      NOTB  = LSAME( TRANSB, 'N' )
      IF( NOTA )THEN
         NROWA = M
      ELSE
         NROWA = K
      END IF
      IF( NOTB )THEN
         NROWB = K
      ELSE
         NROWB = N
      END IF
*
*     Test the input parameters.
*
      INFO = 0
      IF(      ( .NOT.NOTA                 ).AND.
     $         ( .NOT.LSAME( TRANSA, 'C' ) ).AND.
     $         ( .NOT.LSAME( TRANSA, 'T' ) )      )THEN
         INFO = 1
      ELSE IF( ( .NOT.NOTB                 ).AND.
     $         ( .NOT.LSAME( TRANSB, 'C' ) ).AND.
     $         ( .NOT.LSAME( TRANSB, 'T' ) )      )THEN
         INFO = 2
      ELSE IF( M  .LT.0               )THEN
         INFO = 3
      ELSE IF( N  .LT.0               )THEN
         INFO = 4
      ELSE IF( K  .LT.0               )THEN
         INFO = 5
      ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
         INFO = 8
      ELSE IF( LDB.LT.MAX( 1, NROWB ) )THEN
         INFO = 10
      ELSE IF( LDC.LT.MAX( 1, M     ) )THEN
         INFO = 13
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'SGEMM ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
     $    ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     And if  alpha.eq.zero.
*
      IF( ALPHA.EQ.ZERO )THEN
         IF( BETA.EQ.ZERO )THEN
            DO 20, J = 1, N
               DO 10, I = 1, M
                  C( I, J ) = ZERO
   10          CONTINUE
   20       CONTINUE
         ELSE
            DO 40, J = 1, N
               DO 30, I = 1, M
                  C( I, J ) = BETA*C( I, J )
   30          CONTINUE
   40       CONTINUE
         END IF
         RETURN
      END IF
*
*     Start the operations.
*
      IF( NOTB )THEN
         IF( NOTA )THEN
*
*           Form  C := alpha*A*B + beta*C.
*
            DO 90, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 50, I = 1, M
                     C( I, J ) = ZERO
   50             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 60, I = 1, M
                     C( I, J ) = BETA*C( I, J )
   60             CONTINUE
               END IF
               DO 80, L = 1, K
                  IF( B( L, J ).NE.ZERO )THEN
                     TEMP = ALPHA*B( L, J )
                     DO 70, I = 1, M
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
   70                CONTINUE
                  END IF
   80          CONTINUE
   90       CONTINUE
         ELSE
*
*           Form  C := alpha*A'*B + beta*C
*
            DO 120, J = 1, N
               DO 110, I = 1, M
                  TEMP = ZERO
                  DO 100, L = 1, K
                     TEMP = TEMP + A( L, I )*B( L, J )
  100             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  110          CONTINUE
  120       CONTINUE
         END IF
      ELSE
         IF( NOTA )THEN
*
*           Form  C := alpha*A*B' + beta*C
*
            DO 170, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 130, I = 1, M
                     C( I, J ) = ZERO
  130             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 140, I = 1, M
                     C( I, J ) = BETA*C( I, J )
  140             CONTINUE
               END IF
               DO 160, L = 1, K
                  IF( B( J, L ).NE.ZERO )THEN
                     TEMP = ALPHA*B( J, L )
                     DO 150, I = 1, M
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
  150                CONTINUE
                  END IF
  160          CONTINUE
  170       CONTINUE
         ELSE
*
*           Form  C := alpha*A'*B' + beta*C
*
            DO 200, J = 1, N
               DO 190, I = 1, M
                  TEMP = ZERO
                  DO 180, L = 1, K
                     TEMP = TEMP + A( L, I )*B( J, L )
  180             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  190          CONTINUE
  200       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of SGEMM .
*
      END

C       Toolpack tool decs employed.
C       Arg dimension set to *.
C
      INTEGER FUNCTION IDAMAX(N,DX,INCX)
C
C     FINDS THE INDEX OF ELEMENT HAVING MAX. ABSOLUTE VALUE.
C     JACK DONGARRA, LINPACK, 3/11/78.
C
C     .. Scalar Arguments ..
      INTEGER INCX,N
C     ..
C     .. Array Arguments ..
      DOUBLE PRECISION DX(*)
C     ..
C     .. Local Scalars ..
      DOUBLE PRECISION DMAX
      INTEGER I,IX
C     ..
C     .. Intrinsic Functions ..
      INTRINSIC DABS
C     ..
C     .. Executable Statements ..
C
      IDAMAX = 0
      IF (N.LT.1) RETURN
      IDAMAX = 1
      IF (N.EQ.1) RETURN
      IF (INCX.EQ.1) GO TO 20
C
C        CODE FOR INCREMENT NOT EQUAL TO 1
C
      IX = 1
      DMAX = DABS(DX(1))
      IX = IX + INCX
      DO 10 I = 2,N
        IF (DABS(DX(IX)).LE.DMAX) GO TO 5
        IDAMAX = I
        DMAX = DABS(DX(IX))
    5   IX = IX + INCX
   10 CONTINUE
      RETURN
C
C        CODE FOR INCREMENT EQUAL TO 1
C
   20 DMAX = DABS(DX(1))
      DO 30 I = 2,N
        IF (DABS(DX(I)).LE.DMAX) GO TO 30
        IDAMAX = I
        DMAX = DABS(DX(I))
   30 CONTINUE
      RETURN

      END

C       Toolpack tool decs employed.
C       Arg dimensions set to *.
C
      INTEGER FUNCTION ISAMAX(N,SX,INCX)
C
C     FINDS THE INDEX OF ELEMENT HAVING MAX. ABSOLUTE VALUE.
C     JACK DONGARRA, LINPACK, 3/11/78.
C
C     .. Scalar Arguments ..
      INTEGER INCX,N
C     ..
C     .. Array Arguments ..
      REAL SX(*)
C     ..
C     .. Local Scalars ..
      REAL SMAX
      INTEGER I,IX
C     ..
C     .. Intrinsic Functions ..
      INTRINSIC ABS
C     ..
C     .. Executable Statements ..
C
      ISAMAX = 0
      IF (N.LT.1) RETURN
      ISAMAX = 1
      IF (N.EQ.1) RETURN
      IF (INCX.EQ.1) GO TO 20
C
C        CODE FOR INCREMENT NOT EQUAL TO 1
C
      IX = 1
      SMAX = ABS(SX(1))
      IX = IX + INCX
      DO 10 I = 2,N
        IF (ABS(SX(IX)).LE.SMAX) GO TO 5
        ISAMAX = I
        SMAX = ABS(SX(IX))
    5   IX = IX + INCX
   10 CONTINUE
      RETURN
C
C        CODE FOR INCREMENT EQUAL TO 1
C
   20 SMAX = ABS(SX(1))
      DO 30 I = 2,N
        IF (ABS(SX(I)).LE.SMAX) GO TO 30
        ISAMAX = I
        SMAX = ABS(SX(I))
   30 CONTINUE
      RETURN

      END

      SUBROUTINE DTPSV ( UPLO, TRANS, DIAG, N, AP, X, INCX )
*
************************************************************************
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      CHARACTER*1        DIAG, TRANS, UPLO
*     .. Array Arguments ..
      DOUBLE PRECISION   AP( * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  DTPSV  solves one of the systems of equations
*
*     A*x = b,   or   A'*x = b,
*
*  where b and x are n element vectors and A is an n by n unit, or
*  non-unit, upper or lower triangular matrix, supplied in packed form.
*
*  No test for singularity or near-singularity is included in this
*  routine. Such tests must be performed before calling this routine.
*
*  Parameters
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the equations to be solved as
*           follows:
*
*              TRANS = 'N' or 'n'   A*x = b.
*
*              TRANS = 'T' or 't'   A'*x = b.
*
*              TRANS = 'C' or 'c'   A'*x = b.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit
*           triangular as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  AP     - DOUBLE PRECISION array of DIMENSION at least
*           ( ( n*( n + 1 ) )/2 ).
*           Before entry with  UPLO = 'U' or 'u', the array AP must
*           contain the upper triangular matrix packed sequentially,
*           column by column, so that AP( 1 ) contains a( 1, 1 ),
*           AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 )
*           respectively, and so on.
*           Before entry with UPLO = 'L' or 'l', the array AP must
*           contain the lower triangular matrix packed sequentially,
*           column by column, so that AP( 1 ) contains a( 1, 1 ),
*           AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 )
*           respectively, and so on.
*           Note that when  DIAG = 'U' or 'u', the diagonal elements of
*           A are not referenced, but are assumed to be unity.
*           Unchanged on exit.
*
*  X      - DOUBLE PRECISION array of dimension at least
*           ( 1 + ( n - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the n
*           element right-hand side vector b. On exit, X is overwritten
*           with the solution vector x.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER        ( ZERO = 0.0D+0 )
*     .. Local Scalars ..
      DOUBLE PRECISION   TEMP
      INTEGER            I, INFO, IX, J, JX, K, KK, KX
      LOGICAL            NOUNIT
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( UPLO , 'U' ).AND.
     $         .NOT.LSAME( UPLO , 'L' )      )THEN
         INFO = 1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 2
      ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND.
     $         .NOT.LSAME( DIAG , 'N' )      )THEN
         INFO = 3
      ELSE IF( N.LT.0 )THEN
         INFO = 4
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 7
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'DTPSV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Set up the start point in X if the increment is not unity. This
*     will be  ( N - 1 )*INCX  too small for descending loops.
*
      IF( INCX.LE.0 )THEN
         KX = 1 - ( N - 1 )*INCX
      ELSE IF( INCX.NE.1 )THEN
         KX = 1
      END IF
*
*     Start the operations. In this version the elements of AP are
*     accessed sequentially with one pass through AP.
*
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  x := inv( A )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            KK = ( N*( N + 1 ) )/2
            IF( INCX.EQ.1 )THEN
               DO 20, J = N, 1, -1
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/AP( KK )
                     TEMP = X( J )
                     K    = KK     - 1
                     DO 10, I = J - 1, 1, -1
                        X( I ) = X( I ) - TEMP*AP( K )
                        K      = K      - 1
   10                CONTINUE
                  END IF
                  KK = KK - J
   20          CONTINUE
            ELSE
               JX = KX + ( N - 1 )*INCX
               DO 40, J = N, 1, -1
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/AP( KK )
                     TEMP = X( JX )
                     IX   = JX
                     DO 30, K = KK - 1, KK - J + 1, -1
                        IX      = IX      - INCX
                        X( IX ) = X( IX ) - TEMP*AP( K )
   30                CONTINUE
                  END IF
                  JX = JX - INCX
                  KK = KK - J
   40          CONTINUE
            END IF
         ELSE
            KK = 1
            IF( INCX.EQ.1 )THEN
               DO 60, J = 1, N
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/AP( KK )
                     TEMP = X( J )
                     K    = KK     + 1
                     DO 50, I = J + 1, N
                        X( I ) = X( I ) - TEMP*AP( K )
                        K      = K      + 1
   50                CONTINUE
                  END IF
                  KK = KK + ( N - J + 1 )
   60          CONTINUE
            ELSE
               JX = KX
               DO 80, J = 1, N
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/AP( KK )
                     TEMP = X( JX )
                     IX   = JX
                     DO 70, K = KK + 1, KK + N - J
                        IX      = IX      + INCX
                        X( IX ) = X( IX ) - TEMP*AP( K )
   70                CONTINUE
                  END IF
                  JX = JX + INCX
                  KK = KK + ( N - J + 1 )
   80          CONTINUE
            END IF
         END IF
      ELSE
*
*        Form  x := inv( A' )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            KK = 1
            IF( INCX.EQ.1 )THEN
               DO 100, J = 1, N
                  TEMP = X( J )
                  K    = KK
                  DO 90, I = 1, J - 1
                     TEMP = TEMP - AP( K )*X( I )
                     K    = K    + 1
   90             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK + J - 1 )
                  X( J ) = TEMP
                  KK     = KK   + J
  100          CONTINUE
            ELSE
               JX = KX
               DO 120, J = 1, N
                  TEMP = X( JX )
                  IX   = KX
                  DO 110, K = KK, KK + J - 2
                     TEMP = TEMP - AP( K )*X( IX )
                     IX   = IX   + INCX
  110             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK + J - 1 )
                  X( JX ) = TEMP
                  JX      = JX   + INCX
                  KK      = KK   + J
  120          CONTINUE
            END IF
         ELSE
            KK = ( N*( N + 1 ) )/2
            IF( INCX.EQ.1 )THEN
               DO 140, J = N, 1, -1
                  TEMP = X( J )
                  K = KK
                  DO 130, I = N, J + 1, -1
                     TEMP = TEMP - AP( K )*X( I )
                     K    = K    - 1
  130             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK - N + J )
                  X( J ) = TEMP
                  KK     = KK   - ( N - J + 1 )
  140          CONTINUE
            ELSE
               KX = KX + ( N - 1 )*INCX
               JX = KX
               DO 160, J = N, 1, -1
                  TEMP = X( JX )
                  IX   = KX
                  DO 150, K = KK, KK - ( N - ( J + 1 ) ), -1
                     TEMP = TEMP - AP( K )*X( IX )
                     IX   = IX   - INCX
  150             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK - N + J )
                  X( JX ) = TEMP
                  JX      = JX   - INCX
                  KK      = KK   - (N - J + 1 )
  160          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of DTPSV .
*
      END

      SUBROUTINE STPSV ( UPLO, TRANS, DIAG, N, AP, X, INCX )
*
************************************************************************
*
*     .. Scalar Arguments ..
      INTEGER            INCX, N
      CHARACTER*1        DIAG, TRANS, UPLO
*     .. Array Arguments ..
      REAL               AP( * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  STPSV  solves one of the systems of equations
*
*     A*x = b,   or   A'*x = b,
*
*  where b and x are n element vectors and A is an n by n unit, or
*  non-unit, upper or lower triangular matrix, supplied in packed form.
*
*  No test for singularity or near-singularity is included in this
*  routine. Such tests must be performed before calling this routine.
*
*  Parameters
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the equations to be solved as
*           follows:
*
*              TRANS = 'N' or 'n'   A*x = b.
*
*              TRANS = 'T' or 't'   A'*x = b.
*
*              TRANS = 'C' or 'c'   A'*x = b.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit
*           triangular as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  AP     - REAL             array of DIMENSION at least
*           ( ( n*( n + 1 ) )/2 ).
*           Before entry with  UPLO = 'U' or 'u', the array AP must
*           contain the upper triangular matrix packed sequentially,
*           column by column, so that AP( 1 ) contains a( 1, 1 ),
*           AP( 2 ) and AP( 3 ) contain a( 1, 2 ) and a( 2, 2 )
*           respectively, and so on.
*           Before entry with UPLO = 'L' or 'l', the array AP must
*           contain the lower triangular matrix packed sequentially,
*           column by column, so that AP( 1 ) contains a( 1, 1 ),
*           AP( 2 ) and AP( 3 ) contain a( 2, 1 ) and a( 3, 1 )
*           respectively, and so on.
*           Note that when  DIAG = 'U' or 'u', the diagonal elements of
*           A are not referenced, but are assumed to be unity.
*           Unchanged on exit.
*
*  X      - REAL             array of dimension at least
*           ( 1 + ( n - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the n
*           element right-hand side vector b. On exit, X is overwritten
*           with the solution vector x.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      REAL               ZERO
      PARAMETER        ( ZERO = 0.0E+0 )
*     .. Local Scalars ..
      REAL               TEMP
      INTEGER            I, INFO, IX, J, JX, K, KK, KX
      LOGICAL            NOUNIT
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( UPLO , 'U' ).AND.
     $         .NOT.LSAME( UPLO , 'L' )      )THEN
         INFO = 1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 2
      ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND.
     $         .NOT.LSAME( DIAG , 'N' )      )THEN
         INFO = 3
      ELSE IF( N.LT.0 )THEN
         INFO = 4
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 7
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'STPSV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Set up the start point in X if the increment is not unity. This
*     will be  ( N - 1 )*INCX  too small for descending loops.
*
      IF( INCX.LE.0 )THEN
         KX = 1 - ( N - 1 )*INCX
      ELSE IF( INCX.NE.1 )THEN
         KX = 1
      END IF
*
*     Start the operations. In this version the elements of AP are
*     accessed sequentially with one pass through AP.
*
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  x := inv( A )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            KK = ( N*( N + 1 ) )/2
            IF( INCX.EQ.1 )THEN
               DO 20, J = N, 1, -1
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/AP( KK )
                     TEMP = X( J )
                     K    = KK     - 1
                     DO 10, I = J - 1, 1, -1
                        X( I ) = X( I ) - TEMP*AP( K )
                        K      = K      - 1
   10                CONTINUE
                  END IF
                  KK = KK - J
   20          CONTINUE
            ELSE
               JX = KX + ( N - 1 )*INCX
               DO 40, J = N, 1, -1
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/AP( KK )
                     TEMP = X( JX )
                     IX   = JX
                     DO 30, K = KK - 1, KK - J + 1, -1
                        IX      = IX      - INCX
                        X( IX ) = X( IX ) - TEMP*AP( K )
   30                CONTINUE
                  END IF
                  JX = JX - INCX
                  KK = KK - J
   40          CONTINUE
            END IF
         ELSE
            KK = 1
            IF( INCX.EQ.1 )THEN
               DO 60, J = 1, N
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/AP( KK )
                     TEMP = X( J )
                     K    = KK     + 1
                     DO 50, I = J + 1, N
                        X( I ) = X( I ) - TEMP*AP( K )
                        K      = K      + 1
   50                CONTINUE
                  END IF
                  KK = KK + ( N - J + 1 )
   60          CONTINUE
            ELSE
               JX = KX
               DO 80, J = 1, N
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/AP( KK )
                     TEMP = X( JX )
                     IX   = JX
                     DO 70, K = KK + 1, KK + N - J
                        IX      = IX      + INCX
                        X( IX ) = X( IX ) - TEMP*AP( K )
   70                CONTINUE
                  END IF
                  JX = JX + INCX
                  KK = KK + ( N - J + 1 )
   80          CONTINUE
            END IF
         END IF
      ELSE
*
*        Form  x := inv( A' )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            KK = 1
            IF( INCX.EQ.1 )THEN
               DO 100, J = 1, N
                  TEMP = X( J )
                  K    = KK
                  DO 90, I = 1, J - 1
                     TEMP = TEMP - AP( K )*X( I )
                     K    = K    + 1
   90             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK + J - 1 )
                  X( J ) = TEMP
                  KK     = KK   + J
  100          CONTINUE
            ELSE
               JX = KX
               DO 120, J = 1, N
                  TEMP = X( JX )
                  IX   = KX
                  DO 110, K = KK, KK + J - 2
                     TEMP = TEMP - AP( K )*X( IX )
                     IX   = IX   + INCX
  110             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK + J - 1 )
                  X( JX ) = TEMP
                  JX      = JX   + INCX
                  KK      = KK   + J
  120          CONTINUE
            END IF
         ELSE
            KK = ( N*( N + 1 ) )/2
            IF( INCX.EQ.1 )THEN
               DO 140, J = N, 1, -1
                  TEMP = X( J )
                  K = KK
                  DO 130, I = N, J + 1, -1
                     TEMP = TEMP - AP( K )*X( I )
                     K    = K    - 1
  130             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK - N + J )
                  X( J ) = TEMP
                  KK     = KK   - ( N - J + 1 )
  140          CONTINUE
            ELSE
               KX = KX + ( N - 1 )*INCX
               JX = KX
               DO 160, J = N, 1, -1
                  TEMP = X( JX )
                  IX   = KX
                  DO 150, K = KK, KK - ( N - ( J + 1 ) ), -1
                     TEMP = TEMP - AP( K )*X( IX )
                     IX   = IX   - INCX
  150             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/AP( KK - N + J )
                  X( JX ) = TEMP
                  JX      = JX   - INCX
                  KK      = KK   - (N - J + 1 )
  160          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of STPSV .
*
      END

      LOGICAL FUNCTION LSAME ( CA, CB )
*     .. Scalar Arguments ..
      CHARACTER*1            CA, CB
*     ..
*
*  Purpose
*  =======
*
*  LSAME  tests if CA is the same letter as CB regardless of case.
*
*  N.B. This version of the routine is only correct for ASCII code.
*       Installers must modify the routine for other character-codes.
*
*       For EBCDIC systems the constant IOFF must be changed to -64.
*       For CDC systems using 6-12 bit representations, the system-
*       specific code in comments must be activated.
*
*  Parameters
*  ==========
*
*  CA     - CHARACTER*1
*  CB     - CHARACTER*1
*           On entry, CA and CB specify characters to be compared.
*           Unchanged on exit.
*
*
*  Auxiliary routine for Level 2 Blas.
*
*  -- Written on 11-October-1988.
*     Richard Hanson, Sandia National Labs.
*     Jeremy Du Croz, Nag Central Office.
*
*     .. Parameters ..
      INTEGER                IOFF
      PARAMETER            ( IOFF=32 )
*     .. Intrinsic Functions ..
      INTRINSIC              ICHAR
*     .. Executable Statements ..
*
*     Test if the characters are equal
*
      LSAME = CA .EQ. CB
*
*     Now test for equivalence
*
      IF ( .NOT.LSAME ) THEN
         LSAME = ICHAR(CA) - IOFF .EQ. ICHAR(CB)
      END IF
      IF ( .NOT.LSAME ) THEN
         LSAME = ICHAR(CA) .EQ. ICHAR(CB) - IOFF
      END IF
*
      RETURN
*
*  The following comments contain code for CDC systems using 6-12 bit
*  representations.
*
*     .. Parameters ..
*     INTEGER                ICIRFX
*     PARAMETER            ( ICIRFX=62 )
*     .. Scalar Arguments ..
*     CHARACTER*1            CB
*     .. Array Arguments ..
*     CHARACTER*1            CA(*)
*     .. Local Scalars ..
*     INTEGER                IVAL
*     .. Intrinsic Functions ..
*     INTRINSIC              ICHAR, CHAR
*     .. Executable Statements ..
*
*     See if the first character in string CA equals string CB.
*
*     LSAME = CA(1) .EQ. CB .AND. CA(1) .NE. CHAR(ICIRFX)
*
*     IF (LSAME) RETURN
*
*     The characters are not identical. Now check them for equivalence.
*     Look for the 'escape' character, circumflex, followed by the
*     letter.
*
*     IVAL = ICHAR(CA(2))
*     IF (IVAL.GE.ICHAR('A') .AND. IVAL.LE.ICHAR('Z')) THEN
*        LSAME = CA(1) .EQ. CHAR(ICIRFX) .AND. CA(2) .EQ. CB
*     END IF
*
*     RETURN
*
*     End of LSAME.
*
      END

      subroutine  dswap (n,dx,incx,dy,incy)
c
c     interchanges two vectors.
c     uses unrolled loops for increments equal one.
c     jack dongarra, linpack, 3/11/78.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      double precision dx(*),dy(*),dtemp
      integer i,incx,incy,ix,iy,m,mp1,n
c
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c       code for unequal increments or equal increments not equal
c         to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        dtemp = dx(ix)
        dx(ix) = dy(iy)
        dy(iy) = dtemp
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c       code for both increments equal to 1
c
c
c       clean-up loop
c
   20 m = mod(n,3)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        dtemp = dx(i)
        dx(i) = dy(i)
        dy(i) = dtemp
   30 continue
      if( n .lt. 3 ) return
   40 mp1 = m + 1
      do 50 i = mp1,n,3
        dtemp = dx(i)
        dx(i) = dy(i)
        dy(i) = dtemp
        dtemp = dx(i + 1)
        dx(i + 1) = dy(i + 1)
        dy(i + 1) = dtemp
        dtemp = dx(i + 2)
        dx(i + 2) = dy(i + 2)
        dy(i + 2) = dtemp
   50 continue
      return
      end

      subroutine sswap (n,sx,incx,sy,incy)
c
c     interchanges two vectors.
c     uses unrolled loops for increments equal to 1.
c     jack dongarra, linpack, 3/11/78.
c     modified 12/3/93, array(1) declarations changed to array(*)
c
      real sx(*),sy(*),stemp
      integer i,incx,incy,ix,iy,m,mp1,n
c
      if(n.le.0)return
      if(incx.eq.1.and.incy.eq.1)go to 20
c
c       code for unequal increments or equal increments not equal
c         to 1
c
      ix = 1
      iy = 1
      if(incx.lt.0)ix = (-n+1)*incx + 1
      if(incy.lt.0)iy = (-n+1)*incy + 1
      do 10 i = 1,n
        stemp = sx(ix)
        sx(ix) = sy(iy)
        sy(iy) = stemp
        ix = ix + incx
        iy = iy + incy
   10 continue
      return
c
c       code for both increments equal to 1
c
c
c       clean-up loop
c
   20 m = mod(n,3)
      if( m .eq. 0 ) go to 40
      do 30 i = 1,m
        stemp = sx(i)
        sx(i) = sy(i)
        sy(i) = stemp
   30 continue
      if( n .lt. 3 ) return
   40 mp1 = m + 1
      do 50 i = mp1,n,3
        stemp = sx(i)
        sx(i) = sy(i)
        sy(i) = stemp
        stemp = sx(i + 1)
        sx(i + 1) = sy(i + 1)
        sy(i + 1) = stemp
        stemp = sx(i + 2)
        sx(i + 2) = sy(i + 2)
        sy(i + 2) = stemp
   50 continue
      return
      end

      SUBROUTINE DGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX,
     $                   BETA, Y, INCY )
*     .. Scalar Arguments ..
      DOUBLE PRECISION   ALPHA, BETA
      INTEGER            INCX, INCY, LDA, M, N
      CHARACTER*1        TRANS
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), X( * ), Y( * )
*     ..
*
*  Purpose
*  =======
*
*  DGEMV  performs one of the matrix-vector operations
*
*     y := alpha*A*x + beta*y,   or   y := alpha*A'*x + beta*y,
*
*  where alpha and beta are scalars, x and y are vectors and A is an
*  m by n matrix.
*
*  Parameters
*  ==========
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the operation to be performed as
*           follows:
*
*              TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.
*
*              TRANS = 'T' or 't'   y := alpha*A'*x + beta*y.
*
*              TRANS = 'C' or 'c'   y := alpha*A'*x + beta*y.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry, M specifies the number of rows of the matrix A.
*           M must be at least zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the number of columns of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  ALPHA  - DOUBLE PRECISION.
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
*           Before entry, the leading m by n part of the array A must
*           contain the matrix of coefficients.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, m ).
*           Unchanged on exit.
*
*  X      - DOUBLE PRECISION array of DIMENSION at least
*           ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
*           and at least
*           ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
*           Before entry, the incremented array X must contain the
*           vector x.
*           Unchanged on exit.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*  BETA   - DOUBLE PRECISION.
*           On entry, BETA specifies the scalar beta. When BETA is
*           supplied as zero then Y need not be set on input.
*           Unchanged on exit.
*
*  Y      - DOUBLE PRECISION array of DIMENSION at least
*           ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
*           and at least
*           ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
*           Before entry with BETA non-zero, the incremented array Y
*           must contain the vector y. On exit, Y is overwritten by the
*           updated vector y.
*
*  INCY   - INTEGER.
*           On entry, INCY specifies the increment for the elements of
*           Y. INCY must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE         , ZERO
      PARAMETER        ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     .. Local Scalars ..
      DOUBLE PRECISION   TEMP
      INTEGER            I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 1
      ELSE IF( M.LT.0 )THEN
         INFO = 2
      ELSE IF( N.LT.0 )THEN
         INFO = 3
      ELSE IF( LDA.LT.MAX( 1, M ) )THEN
         INFO = 6
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 8
      ELSE IF( INCY.EQ.0 )THEN
         INFO = 11
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'DGEMV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
     $    ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     Set  LENX  and  LENY, the lengths of the vectors x and y, and set
*     up the start points in  X  and  Y.
*
      IF( LSAME( TRANS, 'N' ) )THEN
         LENX = N
         LENY = M
      ELSE
         LENX = M
         LENY = N
      END IF
      IF( INCX.GT.0 )THEN
         KX = 1
      ELSE
         KX = 1 - ( LENX - 1 )*INCX
      END IF
      IF( INCY.GT.0 )THEN
         KY = 1
      ELSE
         KY = 1 - ( LENY - 1 )*INCY
      END IF
*
*     Start the operations. In this version the elements of A are
*     accessed sequentially with one pass through A.
*
*     First form  y := beta*y.
*
      IF( BETA.NE.ONE )THEN
         IF( INCY.EQ.1 )THEN
            IF( BETA.EQ.ZERO )THEN
               DO 10, I = 1, LENY
                  Y( I ) = ZERO
   10          CONTINUE
            ELSE
               DO 20, I = 1, LENY
                  Y( I ) = BETA*Y( I )
   20          CONTINUE
            END IF
         ELSE
            IY = KY
            IF( BETA.EQ.ZERO )THEN
               DO 30, I = 1, LENY
                  Y( IY ) = ZERO
                  IY      = IY   + INCY
   30          CONTINUE
            ELSE
               DO 40, I = 1, LENY
                  Y( IY ) = BETA*Y( IY )
                  IY      = IY           + INCY
   40          CONTINUE
            END IF
         END IF
      END IF
      IF( ALPHA.EQ.ZERO )
     $   RETURN
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  y := alpha*A*x + y.
*
         JX = KX
         IF( INCY.EQ.1 )THEN
            DO 60, J = 1, N
               IF( X( JX ).NE.ZERO )THEN
                  TEMP = ALPHA*X( JX )
                  DO 50, I = 1, M
                     Y( I ) = Y( I ) + TEMP*A( I, J )
   50             CONTINUE
               END IF
               JX = JX + INCX
   60       CONTINUE
         ELSE
            DO 80, J = 1, N
               IF( X( JX ).NE.ZERO )THEN
                  TEMP = ALPHA*X( JX )
                  IY   = KY
                  DO 70, I = 1, M
                     Y( IY ) = Y( IY ) + TEMP*A( I, J )
                     IY      = IY      + INCY
   70             CONTINUE
               END IF
               JX = JX + INCX
   80       CONTINUE
         END IF
      ELSE
*
*        Form  y := alpha*A'*x + y.
*
         JY = KY
         IF( INCX.EQ.1 )THEN
            DO 100, J = 1, N
               TEMP = ZERO
               DO 90, I = 1, M
                  TEMP = TEMP + A( I, J )*X( I )
   90          CONTINUE
               Y( JY ) = Y( JY ) + ALPHA*TEMP
               JY      = JY      + INCY
  100       CONTINUE
         ELSE
            DO 120, J = 1, N
               TEMP = ZERO
               IX   = KX
               DO 110, I = 1, M
                  TEMP = TEMP + A( I, J )*X( IX )
                  IX   = IX   + INCX
  110          CONTINUE
               Y( JY ) = Y( JY ) + ALPHA*TEMP
               JY      = JY      + INCY
  120       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DGEMV .
*
      END

      SUBROUTINE SGEMV ( TRANS, M, N, ALPHA, A, LDA, X, INCX,
     $                   BETA, Y, INCY )
*     .. Scalar Arguments ..
      REAL               ALPHA, BETA
      INTEGER            INCX, INCY, LDA, M, N
      CHARACTER*1        TRANS
*     .. Array Arguments ..
      REAL               A( LDA, * ), X( * ), Y( * )
*     ..
*
*  Purpose
*  =======
*
*  SGEMV  performs one of the matrix-vector operations
*
*     y := alpha*A*x + beta*y,   or   y := alpha*A'*x + beta*y,
*
*  where alpha and beta are scalars, x and y are vectors and A is an
*  m by n matrix.
*
*  Parameters
*  ==========
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the operation to be performed as
*           follows:
*
*              TRANS = 'N' or 'n'   y := alpha*A*x + beta*y.
*
*              TRANS = 'T' or 't'   y := alpha*A'*x + beta*y.
*
*              TRANS = 'C' or 'c'   y := alpha*A'*x + beta*y.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry, M specifies the number of rows of the matrix A.
*           M must be at least zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the number of columns of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  ALPHA  - REAL            .
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - REAL             array of DIMENSION ( LDA, n ).
*           Before entry, the leading m by n part of the array A must
*           contain the matrix of coefficients.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, m ).
*           Unchanged on exit.
*
*  X      - REAL             array of DIMENSION at least
*           ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
*           and at least
*           ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
*           Before entry, the incremented array X must contain the
*           vector x.
*           Unchanged on exit.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*  BETA   - REAL            .
*           On entry, BETA specifies the scalar beta. When BETA is
*           supplied as zero then Y need not be set on input.
*           Unchanged on exit.
*
*  Y      - REAL             array of DIMENSION at least
*           ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
*           and at least
*           ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
*           Before entry with BETA non-zero, the incremented array Y
*           must contain the vector y. On exit, Y is overwritten by the
*           updated vector y.
*
*  INCY   - INTEGER.
*           On entry, INCY specifies the increment for the elements of
*           Y. INCY must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      REAL               ONE         , ZERO
      PARAMETER        ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     .. Local Scalars ..
      REAL               TEMP
      INTEGER            I, INFO, IX, IY, J, JX, JY, KX, KY, LENX, LENY
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 1
      ELSE IF( M.LT.0 )THEN
         INFO = 2
      ELSE IF( N.LT.0 )THEN
         INFO = 3
      ELSE IF( LDA.LT.MAX( 1, M ) )THEN
         INFO = 6
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 8
      ELSE IF( INCY.EQ.0 )THEN
         INFO = 11
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'SGEMV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( M.EQ.0 ).OR.( N.EQ.0 ).OR.
     $    ( ( ALPHA.EQ.ZERO ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     Set  LENX  and  LENY, the lengths of the vectors x and y, and set
*     up the start points in  X  and  Y.
*
      IF( LSAME( TRANS, 'N' ) )THEN
         LENX = N
         LENY = M
      ELSE
         LENX = M
         LENY = N
      END IF
      IF( INCX.GT.0 )THEN
         KX = 1
      ELSE
         KX = 1 - ( LENX - 1 )*INCX
      END IF
      IF( INCY.GT.0 )THEN
         KY = 1
      ELSE
         KY = 1 - ( LENY - 1 )*INCY
      END IF
*
*     Start the operations. In this version the elements of A are
*     accessed sequentially with one pass through A.
*
*     First form  y := beta*y.
*
      IF( BETA.NE.ONE )THEN
         IF( INCY.EQ.1 )THEN
            IF( BETA.EQ.ZERO )THEN
               DO 10, I = 1, LENY
                  Y( I ) = ZERO
   10          CONTINUE
            ELSE
               DO 20, I = 1, LENY
                  Y( I ) = BETA*Y( I )
   20          CONTINUE
            END IF
         ELSE
            IY = KY
            IF( BETA.EQ.ZERO )THEN
               DO 30, I = 1, LENY
                  Y( IY ) = ZERO
                  IY      = IY   + INCY
   30          CONTINUE
            ELSE
               DO 40, I = 1, LENY
                  Y( IY ) = BETA*Y( IY )
                  IY      = IY           + INCY
   40          CONTINUE
            END IF
         END IF
      END IF
      IF( ALPHA.EQ.ZERO )
     $   RETURN
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  y := alpha*A*x + y.
*
         JX = KX
         IF( INCY.EQ.1 )THEN
            DO 60, J = 1, N
               IF( X( JX ).NE.ZERO )THEN
                  TEMP = ALPHA*X( JX )
                  DO 50, I = 1, M
                     Y( I ) = Y( I ) + TEMP*A( I, J )
   50             CONTINUE
               END IF
               JX = JX + INCX
   60       CONTINUE
         ELSE
            DO 80, J = 1, N
               IF( X( JX ).NE.ZERO )THEN
                  TEMP = ALPHA*X( JX )
                  IY   = KY
                  DO 70, I = 1, M
                     Y( IY ) = Y( IY ) + TEMP*A( I, J )
                     IY      = IY      + INCY
   70             CONTINUE
               END IF
               JX = JX + INCX
   80       CONTINUE
         END IF
      ELSE
*
*        Form  y := alpha*A'*x + y.
*
         JY = KY
         IF( INCX.EQ.1 )THEN
            DO 100, J = 1, N
               TEMP = ZERO
               DO 90, I = 1, M
                  TEMP = TEMP + A( I, J )*X( I )
   90          CONTINUE
               Y( JY ) = Y( JY ) + ALPHA*TEMP
               JY      = JY      + INCY
  100       CONTINUE
         ELSE
            DO 120, J = 1, N
               TEMP = ZERO
               IX   = KX
               DO 110, I = 1, M
                  TEMP = TEMP + A( I, J )*X( IX )
                  IX   = IX   + INCX
  110          CONTINUE
               Y( JY ) = Y( JY ) + ALPHA*TEMP
               JY      = JY      + INCY
  120       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of SGEMV .
*
      END

      SUBROUTINE DTRSM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA,
*
************************************************************************
*
     $                   B, LDB )
*     .. Scalar Arguments ..
      CHARACTER*1        SIDE, UPLO, TRANSA, DIAG
      INTEGER            M, N, LDA, LDB
      DOUBLE PRECISION   ALPHA
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  Purpose
*  =======
*
*  DTRSM  solves one of the matrix equations
*
*     op( A )*X = alpha*B,   or   X*op( A ) = alpha*B,
*
*  where alpha is a scalar, X and B are m by n matrices, A is a unit, or
*  non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
*
*     op( A ) = A   or   op( A ) = A'.
*
*  The matrix X is overwritten on B.
*
*  Parameters
*  ==========
*
*  SIDE   - CHARACTER*1.
*           On entry, SIDE specifies whether op( A ) appears on the left
*           or right of X as follows:
*
*              SIDE = 'L' or 'l'   op( A )*X = alpha*B.
*
*              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
*
*           Unchanged on exit.
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix A is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANSA - CHARACTER*1.
*           On entry, TRANSA specifies the form of op( A ) to be used in
*           the matrix multiplication as follows:
*
*              TRANSA = 'N' or 'n'   op( A ) = A.
*
*              TRANSA = 'T' or 't'   op( A ) = A'.
*
*              TRANSA = 'C' or 'c'   op( A ) = A'.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit triangular
*           as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry, M specifies the number of rows of B. M must be at
*           least zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the number of columns of B.  N must be
*           at least zero.
*           Unchanged on exit.
*
*  ALPHA  - DOUBLE PRECISION.
*           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
*           zero then  A is not referenced and  B need not be set before
*           entry.
*           Unchanged on exit.
*
*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, k ), where k is m
*           when  SIDE = 'L' or 'l'  and is  n  when  SIDE = 'R' or 'r'.
*           Before entry  with  UPLO = 'U' or 'u',  the  leading  k by k
*           upper triangular part of the array  A must contain the upper
*           triangular matrix  and the strictly lower triangular part of
*           A is not referenced.
*           Before entry  with  UPLO = 'L' or 'l',  the  leading  k by k
*           lower triangular part of the array  A must contain the lower
*           triangular matrix  and the strictly upper triangular part of
*           A is not referenced.
*           Note that when  DIAG = 'U' or 'u',  the diagonal elements of
*           A  are not referenced either,  but are assumed to be  unity.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program.  When  SIDE = 'L' or 'l'  then
*           LDA  must be at least  max( 1, m ),  when  SIDE = 'R' or 'r'
*           then LDA must be at least max( 1, n ).
*           Unchanged on exit.
*
*  B      - DOUBLE PRECISION array of DIMENSION ( LDB, n ).
*           Before entry,  the leading  m by n part of the array  B must
*           contain  the  right-hand  side  matrix  B,  and  on exit  is
*           overwritten by the solution matrix  X.
*
*  LDB    - INTEGER.
*           On entry, LDB specifies the first dimension of B as declared
*           in  the  calling  (sub)  program.   LDB  must  be  at  least
*           max( 1, m ).
*           Unchanged on exit.
*
*
*  Level 3 Blas routine.
*
*
*  -- Written on 8-February-1989.
*     Jack Dongarra, Argonne National Laboratory.
*     Iain Duff, AERE Harwell.
*     Jeremy Du Croz, Numerical Algorithms Group Ltd.
*     Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     .. Local Scalars ..
      LOGICAL            LSIDE, NOUNIT, UPPER
      INTEGER            I, INFO, J, K, NROWA
      DOUBLE PRECISION   TEMP
*     .. Parameters ..
      DOUBLE PRECISION   ONE         , ZERO
      PARAMETER        ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      LSIDE  = LSAME( SIDE  , 'L' )
      IF( LSIDE )THEN
         NROWA = M
      ELSE
         NROWA = N
      END IF
      NOUNIT = LSAME( DIAG  , 'N' )
      UPPER  = LSAME( UPLO  , 'U' )
*
      INFO   = 0
      IF(      ( .NOT.LSIDE                ).AND.
     $         ( .NOT.LSAME( SIDE  , 'R' ) )      )THEN
         INFO = 1
      ELSE IF( ( .NOT.UPPER                ).AND.
     $         ( .NOT.LSAME( UPLO  , 'L' ) )      )THEN
         INFO = 2
      ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND.
     $         ( .NOT.LSAME( TRANSA, 'T' ) ).AND.
     $         ( .NOT.LSAME( TRANSA, 'C' ) )      )THEN
         INFO = 3
      ELSE IF( ( .NOT.LSAME( DIAG  , 'U' ) ).AND.
     $         ( .NOT.LSAME( DIAG  , 'N' ) )      )THEN
         INFO = 4
      ELSE IF( M  .LT.0               )THEN
         INFO = 5
      ELSE IF( N  .LT.0               )THEN
         INFO = 6
      ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
         INFO = 9
      ELSE IF( LDB.LT.MAX( 1, M     ) )THEN
         INFO = 11
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'DTRSM ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     And when  alpha.eq.zero.
*
      IF( ALPHA.EQ.ZERO )THEN
         DO 20, J = 1, N
            DO 10, I = 1, M
               B( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
         RETURN
      END IF
*
*     Start the operations.
*
      IF( LSIDE )THEN
         IF( LSAME( TRANSA, 'N' ) )THEN
*
*           Form  B := alpha*inv( A )*B.
*
            IF( UPPER )THEN
               DO 60, J = 1, N
                  IF( ALPHA.NE.ONE )THEN
                     DO 30, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
   30                CONTINUE
                  END IF
                  DO 50, K = M, 1, -1
                     IF( B( K, J ).NE.ZERO )THEN
                        IF( NOUNIT )
     $                     B( K, J ) = B( K, J )/A( K, K )
                        DO 40, I = 1, K - 1
                           B( I, J ) = B( I, J ) - B( K, J )*A( I, K )
   40                   CONTINUE
                     END IF
   50             CONTINUE
   60          CONTINUE
            ELSE
               DO 100, J = 1, N
                  IF( ALPHA.NE.ONE )THEN
                     DO 70, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
   70                CONTINUE
                  END IF
                  DO 90 K = 1, M
                     IF( B( K, J ).NE.ZERO )THEN
                        IF( NOUNIT )
     $                     B( K, J ) = B( K, J )/A( K, K )
                        DO 80, I = K + 1, M
                           B( I, J ) = B( I, J ) - B( K, J )*A( I, K )
   80                   CONTINUE
                     END IF
   90             CONTINUE
  100          CONTINUE
            END IF
         ELSE
*
*           Form  B := alpha*inv( A' )*B.
*
            IF( UPPER )THEN
               DO 130, J = 1, N
                  DO 120, I = 1, M
                     TEMP = ALPHA*B( I, J )
                     DO 110, K = 1, I - 1
                        TEMP = TEMP - A( K, I )*B( K, J )
  110                CONTINUE
                     IF( NOUNIT )
     $                  TEMP = TEMP/A( I, I )
                     B( I, J ) = TEMP
  120             CONTINUE
  130          CONTINUE
            ELSE
               DO 160, J = 1, N
                  DO 150, I = M, 1, -1
                     TEMP = ALPHA*B( I, J )
                     DO 140, K = I + 1, M
                        TEMP = TEMP - A( K, I )*B( K, J )
  140                CONTINUE
                     IF( NOUNIT )
     $                  TEMP = TEMP/A( I, I )
                     B( I, J ) = TEMP
  150             CONTINUE
  160          CONTINUE
            END IF
         END IF
      ELSE
         IF( LSAME( TRANSA, 'N' ) )THEN
*
*           Form  B := alpha*B*inv( A ).
*
            IF( UPPER )THEN
               DO 210, J = 1, N
                  IF( ALPHA.NE.ONE )THEN
                     DO 170, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
  170                CONTINUE
                  END IF
                  DO 190, K = 1, J - 1
                     IF( A( K, J ).NE.ZERO )THEN
                        DO 180, I = 1, M
                           B( I, J ) = B( I, J ) - A( K, J )*B( I, K )
  180                   CONTINUE
                     END IF
  190             CONTINUE
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( J, J )
                     DO 200, I = 1, M
                        B( I, J ) = TEMP*B( I, J )
  200                CONTINUE
                  END IF
  210          CONTINUE
            ELSE
               DO 260, J = N, 1, -1
                  IF( ALPHA.NE.ONE )THEN
                     DO 220, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
  220                CONTINUE
                  END IF
                  DO 240, K = J + 1, N
                     IF( A( K, J ).NE.ZERO )THEN
                        DO 230, I = 1, M
                           B( I, J ) = B( I, J ) - A( K, J )*B( I, K )
  230                   CONTINUE
                     END IF
  240             CONTINUE
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( J, J )
                     DO 250, I = 1, M
                       B( I, J ) = TEMP*B( I, J )
  250                CONTINUE
                  END IF
  260          CONTINUE
            END IF
         ELSE
*
*           Form  B := alpha*B*inv( A' ).
*
            IF( UPPER )THEN
               DO 310, K = N, 1, -1
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( K, K )
                     DO 270, I = 1, M
                        B( I, K ) = TEMP*B( I, K )
  270                CONTINUE
                  END IF
                  DO 290, J = 1, K - 1
                     IF( A( J, K ).NE.ZERO )THEN
                        TEMP = A( J, K )
                        DO 280, I = 1, M
                           B( I, J ) = B( I, J ) - TEMP*B( I, K )
  280                   CONTINUE
                     END IF
  290             CONTINUE
                  IF( ALPHA.NE.ONE )THEN
                     DO 300, I = 1, M
                        B( I, K ) = ALPHA*B( I, K )
  300                CONTINUE
                  END IF
  310          CONTINUE
            ELSE
               DO 360, K = 1, N
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( K, K )
                     DO 320, I = 1, M
                        B( I, K ) = TEMP*B( I, K )
  320                CONTINUE
                  END IF
                  DO 340, J = K + 1, N
                     IF( A( J, K ).NE.ZERO )THEN
                        TEMP = A( J, K )
                        DO 330, I = 1, M
                           B( I, J ) = B( I, J ) - TEMP*B( I, K )
  330                   CONTINUE
                     END IF
  340             CONTINUE
                  IF( ALPHA.NE.ONE )THEN
                     DO 350, I = 1, M
                        B( I, K ) = ALPHA*B( I, K )
  350                CONTINUE
                  END IF
  360          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of DTRSM .
*
      END
      SUBROUTINE DTRSV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
*
************************************************************************
*
*     .. Scalar Arguments ..
      INTEGER            INCX, LDA, N
      CHARACTER*1        DIAG, TRANS, UPLO
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  DTRSV  solves one of the systems of equations
*
*     A*x = b,   or   A'*x = b,
*
*  where b and x are n element vectors and A is an n by n unit, or
*  non-unit, upper or lower triangular matrix.
*
*  No test for singularity or near-singularity is included in this
*  routine. Such tests must be performed before calling this routine.
*
*  Parameters
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the equations to be solved as
*           follows:
*
*              TRANS = 'N' or 'n'   A*x = b.
*
*              TRANS = 'T' or 't'   A'*x = b.
*
*              TRANS = 'C' or 'c'   A'*x = b.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit
*           triangular as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
*           Before entry with  UPLO = 'U' or 'u', the leading n by n
*           upper triangular part of the array A must contain the upper
*           triangular matrix and the strictly lower triangular part of
*           A is not referenced.
*           Before entry with UPLO = 'L' or 'l', the leading n by n
*           lower triangular part of the array A must contain the lower
*           triangular matrix and the strictly upper triangular part of
*           A is not referenced.
*           Note that when  DIAG = 'U' or 'u', the diagonal elements of
*           A are not referenced either, but are assumed to be unity.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, n ).
*           Unchanged on exit.
*
*  X      - DOUBLE PRECISION array of dimension at least
*           ( 1 + ( n - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the n
*           element right-hand side vector b. On exit, X is overwritten
*           with the solution vector x.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO
      PARAMETER        ( ZERO = 0.0D+0 )
*     .. Local Scalars ..
      DOUBLE PRECISION   TEMP
      INTEGER            I, INFO, IX, J, JX, KX
      LOGICAL            NOUNIT
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( UPLO , 'U' ).AND.
     $         .NOT.LSAME( UPLO , 'L' )      )THEN
         INFO = 1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 2
      ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND.
     $         .NOT.LSAME( DIAG , 'N' )      )THEN
         INFO = 3
      ELSE IF( N.LT.0 )THEN
         INFO = 4
      ELSE IF( LDA.LT.MAX( 1, N ) )THEN
         INFO = 6
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 8
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'DTRSV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Set up the start point in X if the increment is not unity. This
*     will be  ( N - 1 )*INCX  too small for descending loops.
*
      IF( INCX.LE.0 )THEN
         KX = 1 - ( N - 1 )*INCX
      ELSE IF( INCX.NE.1 )THEN
         KX = 1
      END IF
*
*     Start the operations. In this version the elements of A are
*     accessed sequentially with one pass through A.
*
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  x := inv( A )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            IF( INCX.EQ.1 )THEN
               DO 20, J = N, 1, -1
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/A( J, J )
                     TEMP = X( J )
                     DO 10, I = J - 1, 1, -1
                        X( I ) = X( I ) - TEMP*A( I, J )
   10                CONTINUE
                  END IF
   20          CONTINUE
            ELSE
               JX = KX + ( N - 1 )*INCX
               DO 40, J = N, 1, -1
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/A( J, J )
                     TEMP = X( JX )
                     IX   = JX
                     DO 30, I = J - 1, 1, -1
                        IX      = IX      - INCX
                        X( IX ) = X( IX ) - TEMP*A( I, J )
   30                CONTINUE
                  END IF
                  JX = JX - INCX
   40          CONTINUE
            END IF
         ELSE
            IF( INCX.EQ.1 )THEN
               DO 60, J = 1, N
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/A( J, J )
                     TEMP = X( J )
                     DO 50, I = J + 1, N
                        X( I ) = X( I ) - TEMP*A( I, J )
   50                CONTINUE
                  END IF
   60          CONTINUE
            ELSE
               JX = KX
               DO 80, J = 1, N
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/A( J, J )
                     TEMP = X( JX )
                     IX   = JX
                     DO 70, I = J + 1, N
                        IX      = IX      + INCX
                        X( IX ) = X( IX ) - TEMP*A( I, J )
   70                CONTINUE
                  END IF
                  JX = JX + INCX
   80          CONTINUE
            END IF
         END IF
      ELSE
*
*        Form  x := inv( A' )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            IF( INCX.EQ.1 )THEN
               DO 100, J = 1, N
                  TEMP = X( J )
                  DO 90, I = 1, J - 1
                     TEMP = TEMP - A( I, J )*X( I )
   90             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( J ) = TEMP
  100          CONTINUE
            ELSE
               JX = KX
               DO 120, J = 1, N
                  TEMP = X( JX )
                  IX   = KX
                  DO 110, I = 1, J - 1
                     TEMP = TEMP - A( I, J )*X( IX )
                     IX   = IX   + INCX
  110             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( JX ) = TEMP
                  JX      = JX   + INCX
  120          CONTINUE
            END IF
         ELSE
            IF( INCX.EQ.1 )THEN
               DO 140, J = N, 1, -1
                  TEMP = X( J )
                  DO 130, I = N, J + 1, -1
                     TEMP = TEMP - A( I, J )*X( I )
  130             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( J ) = TEMP
  140          CONTINUE
            ELSE
               KX = KX + ( N - 1 )*INCX
               JX = KX
               DO 160, J = N, 1, -1
                  TEMP = X( JX )
                  IX   = KX
                  DO 150, I = N, J + 1, -1
                     TEMP = TEMP - A( I, J )*X( IX )
                     IX   = IX   - INCX
  150             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( JX ) = TEMP
                  JX      = JX   - INCX
  160          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of DTRSV .
*
      END

      SUBROUTINE STRSM ( SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA,
*
************************************************************************
*
     $                   B, LDB )
*     .. Scalar Arguments ..
      CHARACTER*1        SIDE, UPLO, TRANSA, DIAG
      INTEGER            M, N, LDA, LDB
      REAL               ALPHA
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * )
*     ..
*
*  Purpose
*  =======
*
*  STRSM  solves one of the matrix equations
*
*     op( A )*X = alpha*B,   or   X*op( A ) = alpha*B,
*
*  where alpha is a scalar, X and B are m by n matrices, A is a unit, or
*  non-unit,  upper or lower triangular matrix  and  op( A )  is one  of
*
*     op( A ) = A   or   op( A ) = A'.
*
*  The matrix X is overwritten on B.
*
*  Parameters
*  ==========
*
*  SIDE   - CHARACTER*1.
*           On entry, SIDE specifies whether op( A ) appears on the left
*           or right of X as follows:
*
*              SIDE = 'L' or 'l'   op( A )*X = alpha*B.
*
*              SIDE = 'R' or 'r'   X*op( A ) = alpha*B.
*
*           Unchanged on exit.
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix A is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANSA - CHARACTER*1.
*           On entry, TRANSA specifies the form of op( A ) to be used in
*           the matrix multiplication as follows:
*
*              TRANSA = 'N' or 'n'   op( A ) = A.
*
*              TRANSA = 'T' or 't'   op( A ) = A'.
*
*              TRANSA = 'C' or 'c'   op( A ) = A'.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit triangular
*           as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  M      - INTEGER.
*           On entry, M specifies the number of rows of B. M must be at
*           least zero.
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the number of columns of B.  N must be
*           at least zero.
*           Unchanged on exit.
*
*  ALPHA  - REAL            .
*           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
*           zero then  A is not referenced and  B need not be set before
*           entry.
*           Unchanged on exit.
*
*  A      - REAL             array of DIMENSION ( LDA, k ), where k is m
*           when  SIDE = 'L' or 'l'  and is  n  when  SIDE = 'R' or 'r'.
*           Before entry  with  UPLO = 'U' or 'u',  the  leading  k by k
*           upper triangular part of the array  A must contain the upper
*           triangular matrix  and the strictly lower triangular part of
*           A is not referenced.
*           Before entry  with  UPLO = 'L' or 'l',  the  leading  k by k
*           lower triangular part of the array  A must contain the lower
*           triangular matrix  and the strictly upper triangular part of
*           A is not referenced.
*           Note that when  DIAG = 'U' or 'u',  the diagonal elements of
*           A  are not referenced either,  but are assumed to be  unity.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program.  When  SIDE = 'L' or 'l'  then
*           LDA  must be at least  max( 1, m ),  when  SIDE = 'R' or 'r'
*           then LDA must be at least max( 1, n ).
*           Unchanged on exit.
*
*  B      - REAL             array of DIMENSION ( LDB, n ).
*           Before entry,  the leading  m by n part of the array  B must
*           contain  the  right-hand  side  matrix  B,  and  on exit  is
*           overwritten by the solution matrix  X.
*
*  LDB    - INTEGER.
*           On entry, LDB specifies the first dimension of B as declared
*           in  the  calling  (sub)  program.   LDB  must  be  at  least
*           max( 1, m ).
*           Unchanged on exit.
*
*
*  Level 3 Blas routine.
*
*
*  -- Written on 8-February-1989.
*     Jack Dongarra, Argonne National Laboratory.
*     Iain Duff, AERE Harwell.
*     Jeremy Du Croz, Numerical Algorithms Group Ltd.
*     Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     .. Local Scalars ..
      LOGICAL            LSIDE, NOUNIT, UPPER
      INTEGER            I, INFO, J, K, NROWA
      REAL               TEMP
*     .. Parameters ..
      REAL               ONE         , ZERO
      PARAMETER        ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      LSIDE  = LSAME( SIDE  , 'L' )
      IF( LSIDE )THEN
         NROWA = M
      ELSE
         NROWA = N
      END IF
      NOUNIT = LSAME( DIAG  , 'N' )
      UPPER  = LSAME( UPLO  , 'U' )
*
      INFO   = 0
      IF(      ( .NOT.LSIDE                ).AND.
     $         ( .NOT.LSAME( SIDE  , 'R' ) )      )THEN
         INFO = 1
      ELSE IF( ( .NOT.UPPER                ).AND.
     $         ( .NOT.LSAME( UPLO  , 'L' ) )      )THEN
         INFO = 2
      ELSE IF( ( .NOT.LSAME( TRANSA, 'N' ) ).AND.
     $         ( .NOT.LSAME( TRANSA, 'T' ) ).AND.
     $         ( .NOT.LSAME( TRANSA, 'C' ) )      )THEN
         INFO = 3
      ELSE IF( ( .NOT.LSAME( DIAG  , 'U' ) ).AND.
     $         ( .NOT.LSAME( DIAG  , 'N' ) )      )THEN
         INFO = 4
      ELSE IF( M  .LT.0               )THEN
         INFO = 5
      ELSE IF( N  .LT.0               )THEN
         INFO = 6
      ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
         INFO = 9
      ELSE IF( LDB.LT.MAX( 1, M     ) )THEN
         INFO = 11
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'STRSM ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     And when  alpha.eq.zero.
*
      IF( ALPHA.EQ.ZERO )THEN
         DO 20, J = 1, N
            DO 10, I = 1, M
               B( I, J ) = ZERO
   10       CONTINUE
   20    CONTINUE
         RETURN
      END IF
*
*     Start the operations.
*
      IF( LSIDE )THEN
         IF( LSAME( TRANSA, 'N' ) )THEN
*
*           Form  B := alpha*inv( A )*B.
*
            IF( UPPER )THEN
               DO 60, J = 1, N
                  IF( ALPHA.NE.ONE )THEN
                     DO 30, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
   30                CONTINUE
                  END IF
                  DO 50, K = M, 1, -1
                     IF( B( K, J ).NE.ZERO )THEN
                        IF( NOUNIT )
     $                     B( K, J ) = B( K, J )/A( K, K )
                        DO 40, I = 1, K - 1
                           B( I, J ) = B( I, J ) - B( K, J )*A( I, K )
   40                   CONTINUE
                     END IF
   50             CONTINUE
   60          CONTINUE
            ELSE
               DO 100, J = 1, N
                  IF( ALPHA.NE.ONE )THEN
                     DO 70, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
   70                CONTINUE
                  END IF
                  DO 90 K = 1, M
                     IF( B( K, J ).NE.ZERO )THEN
                        IF( NOUNIT )
     $                     B( K, J ) = B( K, J )/A( K, K )
                        DO 80, I = K + 1, M
                           B( I, J ) = B( I, J ) - B( K, J )*A( I, K )
   80                   CONTINUE
                     END IF
   90             CONTINUE
  100          CONTINUE
            END IF
         ELSE
*
*           Form  B := alpha*inv( A' )*B.
*
            IF( UPPER )THEN
               DO 130, J = 1, N
                  DO 120, I = 1, M
                     TEMP = ALPHA*B( I, J )
                     DO 110, K = 1, I - 1
                        TEMP = TEMP - A( K, I )*B( K, J )
  110                CONTINUE
                     IF( NOUNIT )
     $                  TEMP = TEMP/A( I, I )
                     B( I, J ) = TEMP
  120             CONTINUE
  130          CONTINUE
            ELSE
               DO 160, J = 1, N
                  DO 150, I = M, 1, -1
                     TEMP = ALPHA*B( I, J )
                     DO 140, K = I + 1, M
                        TEMP = TEMP - A( K, I )*B( K, J )
  140                CONTINUE
                     IF( NOUNIT )
     $                  TEMP = TEMP/A( I, I )
                     B( I, J ) = TEMP
  150             CONTINUE
  160          CONTINUE
            END IF
         END IF
      ELSE
         IF( LSAME( TRANSA, 'N' ) )THEN
*
*           Form  B := alpha*B*inv( A ).
*
            IF( UPPER )THEN
               DO 210, J = 1, N
                  IF( ALPHA.NE.ONE )THEN
                     DO 170, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
  170                CONTINUE
                  END IF
                  DO 190, K = 1, J - 1
                     IF( A( K, J ).NE.ZERO )THEN
                        DO 180, I = 1, M
                           B( I, J ) = B( I, J ) - A( K, J )*B( I, K )
  180                   CONTINUE
                     END IF
  190             CONTINUE
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( J, J )
                     DO 200, I = 1, M
                        B( I, J ) = TEMP*B( I, J )
  200                CONTINUE
                  END IF
  210          CONTINUE
            ELSE
               DO 260, J = N, 1, -1
                  IF( ALPHA.NE.ONE )THEN
                     DO 220, I = 1, M
                        B( I, J ) = ALPHA*B( I, J )
  220                CONTINUE
                  END IF
                  DO 240, K = J + 1, N
                     IF( A( K, J ).NE.ZERO )THEN
                        DO 230, I = 1, M
                           B( I, J ) = B( I, J ) - A( K, J )*B( I, K )
  230                   CONTINUE
                     END IF
  240             CONTINUE
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( J, J )
                     DO 250, I = 1, M
                       B( I, J ) = TEMP*B( I, J )
  250                CONTINUE
                  END IF
  260          CONTINUE
            END IF
         ELSE
*
*           Form  B := alpha*B*inv( A' ).
*
            IF( UPPER )THEN
               DO 310, K = N, 1, -1
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( K, K )
                     DO 270, I = 1, M
                        B( I, K ) = TEMP*B( I, K )
  270                CONTINUE
                  END IF
                  DO 290, J = 1, K - 1
                     IF( A( J, K ).NE.ZERO )THEN
                        TEMP = A( J, K )
                        DO 280, I = 1, M
                           B( I, J ) = B( I, J ) - TEMP*B( I, K )
  280                   CONTINUE
                     END IF
  290             CONTINUE
                  IF( ALPHA.NE.ONE )THEN
                     DO 300, I = 1, M
                        B( I, K ) = ALPHA*B( I, K )
  300                CONTINUE
                  END IF
  310          CONTINUE
            ELSE
               DO 360, K = 1, N
                  IF( NOUNIT )THEN
                     TEMP = ONE/A( K, K )
                     DO 320, I = 1, M
                        B( I, K ) = TEMP*B( I, K )
  320                CONTINUE
                  END IF
                  DO 340, J = K + 1, N
                     IF( A( J, K ).NE.ZERO )THEN
                        TEMP = A( J, K )
                        DO 330, I = 1, M
                           B( I, J ) = B( I, J ) - TEMP*B( I, K )
  330                   CONTINUE
                     END IF
  340             CONTINUE
                  IF( ALPHA.NE.ONE )THEN
                     DO 350, I = 1, M
                        B( I, K ) = ALPHA*B( I, K )
  350                CONTINUE
                  END IF
  360          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of STRSM .
*
      END

      SUBROUTINE STRSV ( UPLO, TRANS, DIAG, N, A, LDA, X, INCX )
*
************************************************************************
*
*     .. Scalar Arguments ..
      INTEGER            INCX, LDA, N
      CHARACTER*1        DIAG, TRANS, UPLO
*     .. Array Arguments ..
      REAL               A( LDA, * ), X( * )
*     ..
*
*  Purpose
*  =======
*
*  STRSV  solves one of the systems of equations
*
*     A*x = b,   or   A'*x = b,
*
*  where b and x are n element vectors and A is an n by n unit, or
*  non-unit, upper or lower triangular matrix.
*
*  No test for singularity or near-singularity is included in this
*  routine. Such tests must be performed before calling this routine.
*
*  Parameters
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On entry, UPLO specifies whether the matrix is an upper or
*           lower triangular matrix as follows:
*
*              UPLO = 'U' or 'u'   A is an upper triangular matrix.
*
*              UPLO = 'L' or 'l'   A is a lower triangular matrix.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry, TRANS specifies the equations to be solved as
*           follows:
*
*              TRANS = 'N' or 'n'   A*x = b.
*
*              TRANS = 'T' or 't'   A'*x = b.
*
*              TRANS = 'C' or 'c'   A'*x = b.
*
*           Unchanged on exit.
*
*  DIAG   - CHARACTER*1.
*           On entry, DIAG specifies whether or not A is unit
*           triangular as follows:
*
*              DIAG = 'U' or 'u'   A is assumed to be unit triangular.
*
*              DIAG = 'N' or 'n'   A is not assumed to be unit
*                                  triangular.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry, N specifies the order of the matrix A.
*           N must be at least zero.
*           Unchanged on exit.
*
*  A      - REAL             array of DIMENSION ( LDA, n ).
*           Before entry with  UPLO = 'U' or 'u', the leading n by n
*           upper triangular part of the array A must contain the upper
*           triangular matrix and the strictly lower triangular part of
*           A is not referenced.
*           Before entry with UPLO = 'L' or 'l', the leading n by n
*           lower triangular part of the array A must contain the lower
*           triangular matrix and the strictly upper triangular part of
*           A is not referenced.
*           Note that when  DIAG = 'U' or 'u', the diagonal elements of
*           A are not referenced either, but are assumed to be unity.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in the calling (sub) program. LDA must be at least
*           max( 1, n ).
*           Unchanged on exit.
*
*  X      - REAL             array of dimension at least
*           ( 1 + ( n - 1 )*abs( INCX ) ).
*           Before entry, the incremented array X must contain the n
*           element right-hand side vector b. On exit, X is overwritten
*           with the solution vector x.
*
*  INCX   - INTEGER.
*           On entry, INCX specifies the increment for the elements of
*           X. INCX must not be zero.
*           Unchanged on exit.
*
*
*  Level 2 Blas routine.
*
*  -- Written on 22-October-1986.
*     Jack Dongarra, Argonne National Lab.
*     Jeremy Du Croz, Nag Central Office.
*     Sven Hammarling, Nag Central Office.
*     Richard Hanson, Sandia National Labs.
*
*
*     .. Parameters ..
      REAL               ZERO
      PARAMETER        ( ZERO = 0.0E+0 )
*     .. Local Scalars ..
      REAL               TEMP
      INTEGER            I, INFO, IX, J, JX, KX
      LOGICAL            NOUNIT
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      IF     ( .NOT.LSAME( UPLO , 'U' ).AND.
     $         .NOT.LSAME( UPLO , 'L' )      )THEN
         INFO = 1
      ELSE IF( .NOT.LSAME( TRANS, 'N' ).AND.
     $         .NOT.LSAME( TRANS, 'T' ).AND.
     $         .NOT.LSAME( TRANS, 'C' )      )THEN
         INFO = 2
      ELSE IF( .NOT.LSAME( DIAG , 'U' ).AND.
     $         .NOT.LSAME( DIAG , 'N' )      )THEN
         INFO = 3
      ELSE IF( N.LT.0 )THEN
         INFO = 4
      ELSE IF( LDA.LT.MAX( 1, N ) )THEN
         INFO = 6
      ELSE IF( INCX.EQ.0 )THEN
         INFO = 8
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'STRSV ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
      NOUNIT = LSAME( DIAG, 'N' )
*
*     Set up the start point in X if the increment is not unity. This
*     will be  ( N - 1 )*INCX  too small for descending loops.
*
      IF( INCX.LE.0 )THEN
         KX = 1 - ( N - 1 )*INCX
      ELSE IF( INCX.NE.1 )THEN
         KX = 1
      END IF
*
*     Start the operations. In this version the elements of A are
*     accessed sequentially with one pass through A.
*
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  x := inv( A )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            IF( INCX.EQ.1 )THEN
               DO 20, J = N, 1, -1
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/A( J, J )
                     TEMP = X( J )
                     DO 10, I = J - 1, 1, -1
                        X( I ) = X( I ) - TEMP*A( I, J )
   10                CONTINUE
                  END IF
   20          CONTINUE
            ELSE
               JX = KX + ( N - 1 )*INCX
               DO 40, J = N, 1, -1
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/A( J, J )
                     TEMP = X( JX )
                     IX   = JX
                     DO 30, I = J - 1, 1, -1
                        IX      = IX      - INCX
                        X( IX ) = X( IX ) - TEMP*A( I, J )
   30                CONTINUE
                  END IF
                  JX = JX - INCX
   40          CONTINUE
            END IF
         ELSE
            IF( INCX.EQ.1 )THEN
               DO 60, J = 1, N
                  IF( X( J ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( J ) = X( J )/A( J, J )
                     TEMP = X( J )
                     DO 50, I = J + 1, N
                        X( I ) = X( I ) - TEMP*A( I, J )
   50                CONTINUE
                  END IF
   60          CONTINUE
            ELSE
               JX = KX
               DO 80, J = 1, N
                  IF( X( JX ).NE.ZERO )THEN
                     IF( NOUNIT )
     $                  X( JX ) = X( JX )/A( J, J )
                     TEMP = X( JX )
                     IX   = JX
                     DO 70, I = J + 1, N
                        IX      = IX      + INCX
                        X( IX ) = X( IX ) - TEMP*A( I, J )
   70                CONTINUE
                  END IF
                  JX = JX + INCX
   80          CONTINUE
            END IF
         END IF
      ELSE
*
*        Form  x := inv( A' )*x.
*
         IF( LSAME( UPLO, 'U' ) )THEN
            IF( INCX.EQ.1 )THEN
               DO 100, J = 1, N
                  TEMP = X( J )
                  DO 90, I = 1, J - 1
                     TEMP = TEMP - A( I, J )*X( I )
   90             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( J ) = TEMP
  100          CONTINUE
            ELSE
               JX = KX
               DO 120, J = 1, N
                  TEMP = X( JX )
                  IX   = KX
                  DO 110, I = 1, J - 1
                     TEMP = TEMP - A( I, J )*X( IX )
                     IX   = IX   + INCX
  110             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( JX ) = TEMP
                  JX      = JX   + INCX
  120          CONTINUE
            END IF
         ELSE
            IF( INCX.EQ.1 )THEN
               DO 140, J = N, 1, -1
                  TEMP = X( J )
                  DO 130, I = N, J + 1, -1
                     TEMP = TEMP - A( I, J )*X( I )
  130             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( J ) = TEMP
  140          CONTINUE
            ELSE
               KX = KX + ( N - 1 )*INCX
               JX = KX
               DO 160, J = N, 1, -1
                  TEMP = X( JX )
                  IX   = KX
                  DO 150, I = N, J + 1, -1
                     TEMP = TEMP - A( I, J )*X( IX )
                     IX   = IX   - INCX
  150             CONTINUE
                  IF( NOUNIT )
     $               TEMP = TEMP/A( J, J )
                  X( JX ) = TEMP
                  JX      = JX   - INCX
  160          CONTINUE
            END IF
         END IF
      END IF
*
      RETURN
*
*     End of STRSV .
*
      END

      SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  DPOTRF computes the Cholesky factorization of a real symmetric
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U**T * U,  if UPLO = 'U', or
*     A = L  * L**T,  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the block version of the algorithm, calling Level 3 BLAS.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U**T*U or A = L*L**T.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, the leading minor of order i is not
*                positive definite, and the factorization could not be
*                completed.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, JB, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMM, DPOTF2, DSYRK, DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'DPOTRF', UPLO, N, -1, -1, -1 )
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code.
*
         CALL DPOTF2( UPLO, N, A, LDA, INFO )
      ELSE
*
*        Use blocked code.
*
         IF( UPPER ) THEN
*
*           Compute the Cholesky factorization A = U'*U.
*
            DO 10 J = 1, N, NB
*
*              Update and factorize the current diagonal block and test
*              for non-positive-definiteness.
*
               JB = MIN( NB, N-J+1 )
               CALL DSYRK( 'Upper', 'Transpose', JB, J-1, -ONE,
     $                     A( 1, J ), LDA, ONE, A( J, J ), LDA )
               CALL DPOTF2( 'Upper', JB, A( J, J ), LDA, INFO )
               IF( INFO.NE.0 )
     $            GO TO 30
               IF( J+JB.LE.N ) THEN
*
*                 Compute the current block row.
*
                  CALL DGEMM( 'Transpose', 'No transpose', JB, N-J-JB+1,
     $                        J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ),
     $                        LDA, ONE, A( J, J+JB ), LDA )
                  CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit',
     $                        JB, N-J-JB+1, ONE, A( J, J ), LDA,
     $                        A( J, J+JB ), LDA )
               END IF
   10       CONTINUE
*
         ELSE
*
*           Compute the Cholesky factorization A = L*L'.
*
            DO 20 J = 1, N, NB
*
*              Update and factorize the current diagonal block and test
*              for non-positive-definiteness.
*
               JB = MIN( NB, N-J+1 )
               CALL DSYRK( 'Lower', 'No transpose', JB, J-1, -ONE,
     $                     A( J, 1 ), LDA, ONE, A( J, J ), LDA )
               CALL DPOTF2( 'Lower', JB, A( J, J ), LDA, INFO )
               IF( INFO.NE.0 )
     $            GO TO 30
               IF( J+JB.LE.N ) THEN
*
*                 Compute the current block column.
*
                  CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
     $                        J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ),
     $                        LDA, ONE, A( J+JB, J ), LDA )
                  CALL DTRSM( 'Right', 'Lower', 'Transpose', 'Non-unit',
     $                        N-J-JB+1, JB, ONE, A( J, J ), LDA,
     $                        A( J+JB, J ), LDA )
               END IF
   20       CONTINUE
         END IF
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = INFO + J - 1
*
   40 CONTINUE
      RETURN
*
*     End of DPOTRF
*
      END

      SUBROUTINE SPOTRF( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  SPOTRF computes the Cholesky factorization of a real symmetric
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U**T * U,  if UPLO = 'U', or
*     A = L  * L**T,  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the block version of the algorithm, calling Level 3 BLAS.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) REAL array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U**T*U or A = L*L**T.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, the leading minor of order i is not
*                positive definite, and the factorization could not be
*                completed.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      PARAMETER          ( ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J, JB, NB
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      EXTERNAL           LSAME, ILAENV
*     ..
*     .. External Subroutines ..
      EXTERNAL           SGEMM, SPOTF2, SSYRK, STRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SPOTRF', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Determine the block size for this environment.
*
      NB = ILAENV( 1, 'SPOTRF', UPLO, N, -1, -1, -1 )
      IF( NB.LE.1 .OR. NB.GE.N ) THEN
*
*        Use unblocked code.
*
         CALL SPOTF2( UPLO, N, A, LDA, INFO )
      ELSE
*
*        Use blocked code.
*
         IF( UPPER ) THEN
*
*           Compute the Cholesky factorization A = U'*U.
*
            DO 10 J = 1, N, NB
*
*              Update and factorize the current diagonal block and test
*              for non-positive-definiteness.
*
               JB = MIN( NB, N-J+1 )
               CALL SSYRK( 'Upper', 'Transpose', JB, J-1, -ONE,
     $                     A( 1, J ), LDA, ONE, A( J, J ), LDA )
               CALL SPOTF2( 'Upper', JB, A( J, J ), LDA, INFO )
               IF( INFO.NE.0 )
     $            GO TO 30
               IF( J+JB.LE.N ) THEN
*
*                 Compute the current block row.
*
                  CALL SGEMM( 'Transpose', 'No transpose', JB, N-J-JB+1,
     $                        J-1, -ONE, A( 1, J ), LDA, A( 1, J+JB ),
     $                        LDA, ONE, A( J, J+JB ), LDA )
                  CALL STRSM( 'Left', 'Upper', 'Transpose', 'Non-unit',
     $                        JB, N-J-JB+1, ONE, A( J, J ), LDA,
     $                        A( J, J+JB ), LDA )
               END IF
   10       CONTINUE
*
         ELSE
*
*           Compute the Cholesky factorization A = L*L'.
*
            DO 20 J = 1, N, NB
*
*              Update and factorize the current diagonal block and test
*              for non-positive-definiteness.
*
               JB = MIN( NB, N-J+1 )
               CALL SSYRK( 'Lower', 'No transpose', JB, J-1, -ONE,
     $                     A( J, 1 ), LDA, ONE, A( J, J ), LDA )
               CALL SPOTF2( 'Lower', JB, A( J, J ), LDA, INFO )
               IF( INFO.NE.0 )
     $            GO TO 30
               IF( J+JB.LE.N ) THEN
*
*                 Compute the current block column.
*
                  CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
     $                        J-1, -ONE, A( J+JB, 1 ), LDA, A( J, 1 ),
     $                        LDA, ONE, A( J+JB, J ), LDA )
                  CALL STRSM( 'Right', 'Lower', 'Transpose', 'Non-unit',
     $                        N-J-JB+1, JB, ONE, A( J, J ), LDA,
     $                        A( J+JB, J ), LDA )
               END IF
   20       CONTINUE
         END IF
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = INFO + J - 1
*
   40 CONTINUE
      RETURN
*
*     End of SPOTRF
*
      END

      SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * )
*     ..
*
*  Purpose
*  =======
*
*  DPOTRS solves a system of linear equations A*X = B with a symmetric
*  positive definite matrix A using the Cholesky factorization
*  A = U**T*U or A = L*L**T computed by DPOTRF.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
*          The triangular factor U or L from the Cholesky factorization
*          A = U**T*U or A = L*L**T, as computed by DPOTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          On entry, the right hand side matrix B.
*          On exit, the solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           DTRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B where A = U'*U.
*
*        Solve U'*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS,
     $               ONE, A, LDA, B, LDB )
*
*        Solve U*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
      ELSE
*
*        Solve A*X = B where A = L*L'.
*
*        Solve L*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
*
*        Solve L'*X = B, overwriting B with X.
*
         CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N, NRHS,
     $               ONE, A, LDA, B, LDB )
      END IF
*
      RETURN
*
*     End of DPOTRS
*
      END

      SUBROUTINE SPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     March 31, 1993
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * )
*     ..
*
*  Purpose
*  =======
*
*  SPOTRS solves a system of linear equations A*X = B with a symmetric
*  positive definite matrix A using the Cholesky factorization
*  A = U**T*U or A = L*L**T computed by SPOTRF.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input) REAL array, dimension (LDA,N)
*          The triangular factor U or L from the Cholesky factorization
*          A = U**T*U or A = L*L**T, as computed by SPOTRF.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  B       (input/output) REAL array, dimension (LDB,NRHS)
*          On entry, the right hand side matrix B.
*          On exit, the solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE
      PARAMETER          ( ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
      EXTERNAL           STRSM, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -7
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SPOTRS', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 .OR. NRHS.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Solve A*X = B where A = U'*U.
*
*        Solve U'*X = B, overwriting B with X.
*
         CALL STRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS,
     $               ONE, A, LDA, B, LDB )
*
*        Solve U*X = B, overwriting B with X.
*
         CALL STRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
      ELSE
*
*        Solve A*X = B where A = L*L'.
*
*        Solve L*X = B, overwriting B with X.
*
         CALL STRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', N,
     $               NRHS, ONE, A, LDA, B, LDB )
*
*        Solve L'*X = B, overwriting B with X.
*
         CALL STRSM( 'Left', 'Lower', 'Transpose', 'Non-unit', N, NRHS,
     $               ONE, A, LDA, B, LDB )
      END IF
*
      RETURN
*
*     End of SPOTRS
*
      END

      SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  DPOTF2 computes the Cholesky factorization of a real symmetric
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U' * U ,  if UPLO = 'U', or
*     A = L  * L',  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the unblocked version of the algorithm, calling Level 2 BLAS.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored.
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          n by n upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading n by n lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U'*U  or A = L*L'.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -k, the k-th argument had an illegal value
*          > 0: if INFO = k, the leading minor of order k is not
*               positive definite, and the factorization could not be
*               completed.
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ONE, ZERO
      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J
      DOUBLE PRECISION   AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      DOUBLE PRECISION   DDOT
      EXTERNAL           LSAME, DDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEMV, DSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DPOTF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization A = U'*U.
*
         DO 10 J = 1, N
*
*           Compute U(J,J) and test for non-positive-definiteness.
*
            AJJ = A( J, J ) - DDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 )
            IF( AJJ.LE.ZERO ) THEN
               A( J, J ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of row J.
*
            IF( J.LT.N ) THEN
               CALL DGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ),
     $                     LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
               CALL DSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
            END IF
   10    CONTINUE
      ELSE
*
*        Compute the Cholesky factorization A = L*L'.
*
         DO 20 J = 1, N
*
*           Compute L(J,J) and test for non-positive-definiteness.
*
            AJJ = A( J, J ) - DDOT( J-1, A( J, 1 ), LDA, A( J, 1 ),
     $            LDA )
            IF( AJJ.LE.ZERO ) THEN
               A( J, J ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of column J.
*
            IF( J.LT.N ) THEN
               CALL DGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1, 1 ),
     $                     LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
               CALL DSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
            END IF
   20    CONTINUE
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = J
*
   40 CONTINUE
      RETURN
*
*     End of DPOTF2
*
      END
      SUBROUTINE SPOTF2( UPLO, N, A, LDA, INFO )
*
*  -- LAPACK routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     February 29, 1992
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  SPOTF2 computes the Cholesky factorization of a real symmetric
*  positive definite matrix A.
*
*  The factorization has the form
*     A = U' * U ,  if UPLO = 'U', or
*     A = L  * L',  if UPLO = 'L',
*  where U is an upper triangular matrix and L is lower triangular.
*
*  This is the unblocked version of the algorithm, calling Level 2 BLAS.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored.
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) REAL array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          n by n upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading n by n lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, if INFO = 0, the factor U or L from the Cholesky
*          factorization A = U'*U  or A = L*L'.
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -k, the k-th argument had an illegal value
*          > 0: if INFO = k, the leading minor of order k is not
*               positive definite, and the factorization could not be
*               completed.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            J
      REAL               AJJ
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SDOT
      EXTERNAL           LSAME, SDOT
*     ..
*     .. External Subroutines ..
      EXTERNAL           SGEMV, SSCAL, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
      UPPER = LSAME( UPLO, 'U' )
      IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SPOTF2', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( UPPER ) THEN
*
*        Compute the Cholesky factorization A = U'*U.
*
         DO 10 J = 1, N
*
*           Compute U(J,J) and test for non-positive-definiteness.
*
            AJJ = A( J, J ) - SDOT( J-1, A( 1, J ), 1, A( 1, J ), 1 )
            IF( AJJ.LE.ZERO ) THEN
               A( J, J ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of row J.
*
            IF( J.LT.N ) THEN
               CALL SGEMV( 'Transpose', J-1, N-J, -ONE, A( 1, J+1 ),
     $                     LDA, A( 1, J ), 1, ONE, A( J, J+1 ), LDA )
               CALL SSCAL( N-J, ONE / AJJ, A( J, J+1 ), LDA )
            END IF
   10    CONTINUE
      ELSE
*
*        Compute the Cholesky factorization A = L*L'.
*
         DO 20 J = 1, N
*
*           Compute L(J,J) and test for non-positive-definiteness.
*
            AJJ = A( J, J ) - SDOT( J-1, A( J, 1 ), LDA, A( J, 1 ),
     $            LDA )
            IF( AJJ.LE.ZERO ) THEN
               A( J, J ) = AJJ
               GO TO 30
            END IF
            AJJ = SQRT( AJJ )
            A( J, J ) = AJJ
*
*           Compute elements J+1:N of column J.
*
            IF( J.LT.N ) THEN
               CALL SGEMV( 'No transpose', N-J, J-1, -ONE, A( J+1, 1 ),
     $                     LDA, A( J, 1 ), LDA, ONE, A( J+1, J ), 1 )
               CALL SSCAL( N-J, ONE / AJJ, A( J+1, J ), 1 )
            END IF
   20    CONTINUE
      END IF
      GO TO 40
*
   30 CONTINUE
      INFO = J
*
   40 CONTINUE
      RETURN
*
*     End of SPOTF2
*
      END

      SUBROUTINE DSYRK ( UPLO, TRANS, N, K, ALPHA, A, LDA,
     $                   BETA, C, LDC )
*     .. Scalar Arguments ..
      CHARACTER*1        UPLO, TRANS
      INTEGER            N, K, LDA, LDC
      DOUBLE PRECISION   ALPHA, BETA
*     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * )
*     ..
*
*  Purpose
*  =======
*
*  DSYRK  performs one of the symmetric rank k operations
*
*     C := alpha*A*A' + beta*C,
*
*  or
*
*     C := alpha*A'*A + beta*C,
*
*  where  alpha and beta  are scalars, C is an  n by n  symmetric matrix
*  and  A  is an  n by k  matrix in the first case and a  k by n  matrix
*  in the second case.
*
*  Parameters
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On  entry,   UPLO  specifies  whether  the  upper  or  lower
*           triangular  part  of the  array  C  is to be  referenced  as
*           follows:
*
*              UPLO = 'U' or 'u'   Only the  upper triangular part of  C
*                                  is to be referenced.
*
*              UPLO = 'L' or 'l'   Only the  lower triangular part of  C
*                                  is to be referenced.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry,  TRANS  specifies the operation to be performed as
*           follows:
*
*              TRANS = 'N' or 'n'   C := alpha*A*A' + beta*C.
*
*              TRANS = 'T' or 't'   C := alpha*A'*A + beta*C.
*
*              TRANS = 'C' or 'c'   C := alpha*A'*A + beta*C.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry,  N specifies the order of the matrix C.  N must be
*           at least zero.
*           Unchanged on exit.
*
*  K      - INTEGER.
*           On entry with  TRANS = 'N' or 'n',  K  specifies  the number
*           of  columns   of  the   matrix   A,   and  on   entry   with
*           TRANS = 'T' or 't' or 'C' or 'c',  K  specifies  the  number
*           of rows of the matrix  A.  K must be at least zero.
*           Unchanged on exit.
*
*  ALPHA  - DOUBLE PRECISION.
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is
*           k  when  TRANS = 'N' or 'n',  and is  n  otherwise.
*           Before entry with  TRANS = 'N' or 'n',  the  leading  n by k
*           part of the array  A  must contain the matrix  A,  otherwise
*           the leading  k by n  part of the array  A  must contain  the
*           matrix A.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
*           then  LDA must be at least  max( 1, n ), otherwise  LDA must
*           be at least  max( 1, k ).
*           Unchanged on exit.
*
*  BETA   - DOUBLE PRECISION.
*           On entry, BETA specifies the scalar beta.
*           Unchanged on exit.
*
*  C      - DOUBLE PRECISION array of DIMENSION ( LDC, n ).
*           Before entry  with  UPLO = 'U' or 'u',  the leading  n by n
*           upper triangular part of the array C must contain the upper
*           triangular part  of the  symmetric matrix  and the strictly
*           lower triangular part of C is not referenced.  On exit, the
*           upper triangular part of the array  C is overwritten by the
*           upper triangular part of the updated matrix.
*           Before entry  with  UPLO = 'L' or 'l',  the leading  n by n
*           lower triangular part of the array C must contain the lower
*           triangular part  of the  symmetric matrix  and the strictly
*           upper triangular part of C is not referenced.  On exit, the
*           lower triangular part of the array  C is overwritten by the
*           lower triangular part of the updated matrix.
*
*  LDC    - INTEGER.
*           On entry, LDC specifies the first dimension of C as declared
*           in  the  calling  (sub)  program.   LDC  must  be  at  least
*           max( 1, n ).
*           Unchanged on exit.
*
*
*  Level 3 Blas routine.
*
*  -- Written on 8-February-1989.
*     Jack Dongarra, Argonne National Laboratory.
*     Iain Duff, AERE Harwell.
*     Jeremy Du Croz, Numerical Algorithms Group Ltd.
*     Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, INFO, J, L, NROWA
      DOUBLE PRECISION   TEMP
*     .. Parameters ..
      DOUBLE PRECISION   ONE ,         ZERO
      PARAMETER        ( ONE = 1.0D+0, ZERO = 0.0D+0 )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      IF( LSAME( TRANS, 'N' ) )THEN
         NROWA = N
      ELSE
         NROWA = K
      END IF
      UPPER = LSAME( UPLO, 'U' )
*
      INFO = 0
      IF(      ( .NOT.UPPER               ).AND.
     $         ( .NOT.LSAME( UPLO , 'L' ) )      )THEN
         INFO = 1
      ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND.
     $         ( .NOT.LSAME( TRANS, 'T' ) ).AND.
     $         ( .NOT.LSAME( TRANS, 'C' ) )      )THEN
         INFO = 2
      ELSE IF( N  .LT.0               )THEN
         INFO = 3
      ELSE IF( K  .LT.0               )THEN
         INFO = 4
      ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
         INFO = 7
      ELSE IF( LDC.LT.MAX( 1, N     ) )THEN
         INFO = 10
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'DSYRK ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( N.EQ.0 ).OR.
     $    ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     And when  alpha.eq.zero.
*
      IF( ALPHA.EQ.ZERO )THEN
         IF( UPPER )THEN
            IF( BETA.EQ.ZERO )THEN
               DO 20, J = 1, N
                  DO 10, I = 1, J
                     C( I, J ) = ZERO
   10             CONTINUE
   20          CONTINUE
            ELSE
               DO 40, J = 1, N
                  DO 30, I = 1, J
                     C( I, J ) = BETA*C( I, J )
   30             CONTINUE
   40          CONTINUE
            END IF
         ELSE
            IF( BETA.EQ.ZERO )THEN
               DO 60, J = 1, N
                  DO 50, I = J, N
                     C( I, J ) = ZERO
   50             CONTINUE
   60          CONTINUE
            ELSE
               DO 80, J = 1, N
                  DO 70, I = J, N
                     C( I, J ) = BETA*C( I, J )
   70             CONTINUE
   80          CONTINUE
            END IF
         END IF
         RETURN
      END IF
*
*     Start the operations.
*
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  C := alpha*A*A' + beta*C.
*
         IF( UPPER )THEN
            DO 130, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 90, I = 1, J
                     C( I, J ) = ZERO
   90             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 100, I = 1, J
                     C( I, J ) = BETA*C( I, J )
  100             CONTINUE
               END IF
               DO 120, L = 1, K
                  IF( A( J, L ).NE.ZERO )THEN
                     TEMP = ALPHA*A( J, L )
                     DO 110, I = 1, J
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
  110                CONTINUE
                  END IF
  120          CONTINUE
  130       CONTINUE
         ELSE
            DO 180, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 140, I = J, N
                     C( I, J ) = ZERO
  140             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 150, I = J, N
                     C( I, J ) = BETA*C( I, J )
  150             CONTINUE
               END IF
               DO 170, L = 1, K
                  IF( A( J, L ).NE.ZERO )THEN
                     TEMP      = ALPHA*A( J, L )
                     DO 160, I = J, N
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
  160                CONTINUE
                  END IF
  170          CONTINUE
  180       CONTINUE
         END IF
      ELSE
*
*        Form  C := alpha*A'*A + beta*C.
*
         IF( UPPER )THEN
            DO 210, J = 1, N
               DO 200, I = 1, J
                  TEMP = ZERO
                  DO 190, L = 1, K
                     TEMP = TEMP + A( L, I )*A( L, J )
  190             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  200          CONTINUE
  210       CONTINUE
         ELSE
            DO 240, J = 1, N
               DO 230, I = J, N
                  TEMP = ZERO
                  DO 220, L = 1, K
                     TEMP = TEMP + A( L, I )*A( L, J )
  220             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  230          CONTINUE
  240       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of DSYRK .
*
      END

      SUBROUTINE SSYRK ( UPLO, TRANS, N, K, ALPHA, A, LDA,
     $                   BETA, C, LDC )
*     .. Scalar Arguments ..
      CHARACTER*1        UPLO, TRANS
      INTEGER            N, K, LDA, LDC
      REAL               ALPHA, BETA
*     .. Array Arguments ..
      REAL               A( LDA, * ), C( LDC, * )
*     ..
*
*  Purpose
*  =======
*
*  SSYRK  performs one of the symmetric rank k operations
*
*     C := alpha*A*A' + beta*C,
*
*  or
*
*     C := alpha*A'*A + beta*C,
*
*  where  alpha and beta  are scalars, C is an  n by n  symmetric matrix
*  and  A  is an  n by k  matrix in the first case and a  k by n  matrix
*  in the second case.
*
*  Parameters
*  ==========
*
*  UPLO   - CHARACTER*1.
*           On  entry,   UPLO  specifies  whether  the  upper  or  lower
*           triangular  part  of the  array  C  is to be  referenced  as
*           follows:
*
*              UPLO = 'U' or 'u'   Only the  upper triangular part of  C
*                                  is to be referenced.
*
*              UPLO = 'L' or 'l'   Only the  lower triangular part of  C
*                                  is to be referenced.
*
*           Unchanged on exit.
*
*  TRANS  - CHARACTER*1.
*           On entry,  TRANS  specifies the operation to be performed as
*           follows:
*
*              TRANS = 'N' or 'n'   C := alpha*A*A' + beta*C.
*
*              TRANS = 'T' or 't'   C := alpha*A'*A + beta*C.
*
*              TRANS = 'C' or 'c'   C := alpha*A'*A + beta*C.
*
*           Unchanged on exit.
*
*  N      - INTEGER.
*           On entry,  N specifies the order of the matrix C.  N must be
*           at least zero.
*           Unchanged on exit.
*
*  K      - INTEGER.
*           On entry with  TRANS = 'N' or 'n',  K  specifies  the number
*           of  columns   of  the   matrix   A,   and  on   entry   with
*           TRANS = 'T' or 't' or 'C' or 'c',  K  specifies  the  number
*           of rows of the matrix  A.  K must be at least zero.
*           Unchanged on exit.
*
*  ALPHA  - REAL            .
*           On entry, ALPHA specifies the scalar alpha.
*           Unchanged on exit.
*
*  A      - REAL             array of DIMENSION ( LDA, ka ), where ka is
*           k  when  TRANS = 'N' or 'n',  and is  n  otherwise.
*           Before entry with  TRANS = 'N' or 'n',  the  leading  n by k
*           part of the array  A  must contain the matrix  A,  otherwise
*           the leading  k by n  part of the array  A  must contain  the
*           matrix A.
*           Unchanged on exit.
*
*  LDA    - INTEGER.
*           On entry, LDA specifies the first dimension of A as declared
*           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
*           then  LDA must be at least  max( 1, n ), otherwise  LDA must
*           be at least  max( 1, k ).
*           Unchanged on exit.
*
*  BETA   - REAL            .
*           On entry, BETA specifies the scalar beta.
*           Unchanged on exit.
*
*  C      - REAL             array of DIMENSION ( LDC, n ).
*           Before entry  with  UPLO = 'U' or 'u',  the leading  n by n
*           upper triangular part of the array C must contain the upper
*           triangular part  of the  symmetric matrix  and the strictly
*           lower triangular part of C is not referenced.  On exit, the
*           upper triangular part of the array  C is overwritten by the
*           upper triangular part of the updated matrix.
*           Before entry  with  UPLO = 'L' or 'l',  the leading  n by n
*           lower triangular part of the array C must contain the lower
*           triangular part  of the  symmetric matrix  and the strictly
*           upper triangular part of C is not referenced.  On exit, the
*           lower triangular part of the array  C is overwritten by the
*           lower triangular part of the updated matrix.
*
*  LDC    - INTEGER.
*           On entry, LDC specifies the first dimension of C as declared
*           in  the  calling  (sub)  program.   LDC  must  be  at  least
*           max( 1, n ).
*           Unchanged on exit.
*
*
*  Level 3 Blas routine.
*
*  -- Written on 8-February-1989.
*     Jack Dongarra, Argonne National Laboratory.
*     Iain Duff, AERE Harwell.
*     Jeremy Du Croz, Numerical Algorithms Group Ltd.
*     Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           LSAME
*     .. External Subroutines ..
      EXTERNAL           XERBLA
*     .. Intrinsic Functions ..
      INTRINSIC          MAX
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, INFO, J, L, NROWA
      REAL               TEMP
*     .. Parameters ..
      REAL               ONE ,         ZERO
      PARAMETER        ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      IF( LSAME( TRANS, 'N' ) )THEN
         NROWA = N
      ELSE
         NROWA = K
      END IF
      UPPER = LSAME( UPLO, 'U' )
*
      INFO = 0
      IF(      ( .NOT.UPPER               ).AND.
     $         ( .NOT.LSAME( UPLO , 'L' ) )      )THEN
         INFO = 1
      ELSE IF( ( .NOT.LSAME( TRANS, 'N' ) ).AND.
     $         ( .NOT.LSAME( TRANS, 'T' ) ).AND.
     $         ( .NOT.LSAME( TRANS, 'C' ) )      )THEN
         INFO = 2
      ELSE IF( N  .LT.0               )THEN
         INFO = 3
      ELSE IF( K  .LT.0               )THEN
         INFO = 4
      ELSE IF( LDA.LT.MAX( 1, NROWA ) )THEN
         INFO = 7
      ELSE IF( LDC.LT.MAX( 1, N     ) )THEN
         INFO = 10
      END IF
      IF( INFO.NE.0 )THEN
         CALL XERBLA( 'SSYRK ', INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( ( N.EQ.0 ).OR.
     $    ( ( ( ALPHA.EQ.ZERO ).OR.( K.EQ.0 ) ).AND.( BETA.EQ.ONE ) ) )
     $   RETURN
*
*     And when  alpha.eq.zero.
*
      IF( ALPHA.EQ.ZERO )THEN
         IF( UPPER )THEN
            IF( BETA.EQ.ZERO )THEN
               DO 20, J = 1, N
                  DO 10, I = 1, J
                     C( I, J ) = ZERO
   10             CONTINUE
   20          CONTINUE
            ELSE
               DO 40, J = 1, N
                  DO 30, I = 1, J
                     C( I, J ) = BETA*C( I, J )
   30             CONTINUE
   40          CONTINUE
            END IF
         ELSE
            IF( BETA.EQ.ZERO )THEN
               DO 60, J = 1, N
                  DO 50, I = J, N
                     C( I, J ) = ZERO
   50             CONTINUE
   60          CONTINUE
            ELSE
               DO 80, J = 1, N
                  DO 70, I = J, N
                     C( I, J ) = BETA*C( I, J )
   70             CONTINUE
   80          CONTINUE
            END IF
         END IF
         RETURN
      END IF
*
*     Start the operations.
*
      IF( LSAME( TRANS, 'N' ) )THEN
*
*        Form  C := alpha*A*A' + beta*C.
*
         IF( UPPER )THEN
            DO 130, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 90, I = 1, J
                     C( I, J ) = ZERO
   90             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 100, I = 1, J
                     C( I, J ) = BETA*C( I, J )
  100             CONTINUE
               END IF
               DO 120, L = 1, K
                  IF( A( J, L ).NE.ZERO )THEN
                     TEMP = ALPHA*A( J, L )
                     DO 110, I = 1, J
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
  110                CONTINUE
                  END IF
  120          CONTINUE
  130       CONTINUE
         ELSE
            DO 180, J = 1, N
               IF( BETA.EQ.ZERO )THEN
                  DO 140, I = J, N
                     C( I, J ) = ZERO
  140             CONTINUE
               ELSE IF( BETA.NE.ONE )THEN
                  DO 150, I = J, N
                     C( I, J ) = BETA*C( I, J )
  150             CONTINUE
               END IF
               DO 170, L = 1, K
                  IF( A( J, L ).NE.ZERO )THEN
                     TEMP      = ALPHA*A( J, L )
                     DO 160, I = J, N
                        C( I, J ) = C( I, J ) + TEMP*A( I, L )
  160                CONTINUE
                  END IF
  170          CONTINUE
  180       CONTINUE
         END IF
      ELSE
*
*        Form  C := alpha*A'*A + beta*C.
*
         IF( UPPER )THEN
            DO 210, J = 1, N
               DO 200, I = 1, J
                  TEMP = ZERO
                  DO 190, L = 1, K
                     TEMP = TEMP + A( L, I )*A( L, J )
  190             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  200          CONTINUE
  210       CONTINUE
         ELSE
            DO 240, J = 1, N
               DO 230, I = J, N
                  TEMP = ZERO
                  DO 220, L = 1, K
                     TEMP = TEMP + A( L, I )*A( L, J )
  220             CONTINUE
                  IF( BETA.EQ.ZERO )THEN
                     C( I, J ) = ALPHA*TEMP
                  ELSE
                     C( I, J ) = ALPHA*TEMP + BETA*C( I, J )
                  END IF
  230          CONTINUE
  240       CONTINUE
         END IF
      END IF
*
      RETURN
*
*     End of SSYRK .
*
      END

      INTEGER          FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3,
     $                 N4 )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1999
*
*     .. Scalar Arguments ..
      CHARACTER*( * )    NAME, OPTS
      INTEGER            ISPEC, N1, N2, N3, N4
*     ..
*
*  Purpose
*  =======
*
*  ILAENV is called from the LAPACK routines to choose problem-dependent
*  parameters for the local environment.  See ISPEC for a description of
*  the parameters.
*
*  This version provides a set of parameters which should give good,
*  but not optimal, performance on many of the currently available
*  computers.  Users are encouraged to modify this subroutine to set
*  the tuning parameters for their particular machine using the option
*  and problem size information in the arguments.
*
*  This routine will not function correctly if it is converted to all
*  lower case.  Converting it to all upper case is allowed.
*
*  Arguments
*  =========
*
*  ISPEC   (input) INTEGER
*          Specifies the parameter to be returned as the value of
*          ILAENV.
*          = 1: the optimal blocksize; if this value is 1, an unblocked
*               algorithm will give the best performance.
*          = 2: the minimum block size for which the block routine
*               should be used; if the usable block size is less than
*               this value, an unblocked routine should be used.
*          = 3: the crossover point (in a block routine, for N less
*               than this value, an unblocked routine should be used)
*          = 4: the number of shifts, used in the nonsymmetric
*               eigenvalue routines
*          = 5: the minimum column dimension for blocking to be used;
*               rectangular blocks must have dimension at least k by m,
*               where k is given by ILAENV(2,...) and m by ILAENV(5,...)
*          = 6: the crossover point for the SVD (when reducing an m by n
*               matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
*               this value, a QR factorization is used first to reduce
*               the matrix to a triangular form.)
*          = 7: the number of processors
*          = 8: the crossover point for the multishift QR and QZ methods
*               for nonsymmetric eigenvalue problems.
*          = 9: maximum size of the subproblems at the bottom of the
*               computation tree in the divide-and-conquer algorithm
*               (used by xGELSD and xGESDD)
*          =10: ieee NaN arithmetic can be trusted not to trap
*          =11: infinity arithmetic can be trusted not to trap
*
*  NAME    (input) CHARACTER*(*)
*          The name of the calling subroutine, in either upper case or
*          lower case.
*
*  OPTS    (input) CHARACTER*(*)
*          The character options to the subroutine NAME, concatenated
*          into a single character string.  For example, UPLO = 'U',
*          TRANS = 'T', and DIAG = 'N' for a triangular routine would
*          be specified as OPTS = 'UTN'.
*
*  N1      (input) INTEGER
*  N2      (input) INTEGER
*  N3      (input) INTEGER
*  N4      (input) INTEGER
*          Problem dimensions for the subroutine NAME; these may not all
*          be required.
*
* (ILAENV) (output) INTEGER
*          >= 0: the value of the parameter specified by ISPEC
*          < 0:  if ILAENV = -k, the k-th argument had an illegal value.
*
*  Further Details
*  ===============
*
*  The following conventions have been used when calling ILAENV from the
*  LAPACK routines:
*  1)  OPTS is a concatenation of all of the character options to
*      subroutine NAME, in the same order that they appear in the
*      argument list for NAME, even if they are not used in determining
*      the value of the parameter specified by ISPEC.
*  2)  The problem dimensions N1, N2, N3, N4 are specified in the order
*      that they appear in the argument list for NAME.  N1 is used
*      first, N2 second, and so on, and unused problem dimensions are
*      passed a value of -1.
*  3)  The parameter value returned by ILAENV is checked for validity in
*      the calling subroutine.  For example, ILAENV is used to retrieve
*      the optimal blocksize for STRTRI as follows:
*
*      NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
*      IF( NB.LE.1 ) NB = MAX( 1, N )
*
*  =====================================================================
*
*     .. Local Scalars ..
      LOGICAL            CNAME, SNAME
      CHARACTER*1        C1
      CHARACTER*2        C2, C4
      CHARACTER*3        C3
      CHARACTER*6        SUBNAM
      INTEGER            I, IC, IZ, NB, NBMIN, NX
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          CHAR, ICHAR, INT, MIN, REAL
*     ..
*     .. External Functions ..
      INTEGER            IEEECK
      EXTERNAL           IEEECK
*     ..
*     .. Executable Statements ..
*
      GO TO ( 100, 100, 100, 400, 500, 600, 700, 800, 900, 1000,
     $        1100 ) ISPEC
*
*     Invalid value for ISPEC
*
      ILAENV = -1
      RETURN
*
  100 CONTINUE
*
*     Convert NAME to upper case if the first character is lower case.
*
      ILAENV = 1
      SUBNAM = NAME
      IC = ICHAR( SUBNAM( 1:1 ) )
      IZ = ICHAR( 'Z' )
      IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
*        ASCII character set
*
         IF( IC.GE.97 .AND. IC.LE.122 ) THEN
            SUBNAM( 1:1 ) = CHAR( IC-32 )
            DO 10 I = 2, 6
               IC = ICHAR( SUBNAM( I:I ) )
               IF( IC.GE.97 .AND. IC.LE.122 )
     $            SUBNAM( I:I ) = CHAR( IC-32 )
   10       CONTINUE
         END IF
*
      ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
*        EBCDIC character set
*
         IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $       ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $       ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
            SUBNAM( 1:1 ) = CHAR( IC+64 )
            DO 20 I = 2, 6
               IC = ICHAR( SUBNAM( I:I ) )
               IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
     $             ( IC.GE.145 .AND. IC.LE.153 ) .OR.
     $             ( IC.GE.162 .AND. IC.LE.169 ) )
     $            SUBNAM( I:I ) = CHAR( IC+64 )
   20       CONTINUE
         END IF
*
      ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
*        Prime machines:  ASCII+128
*
         IF( IC.GE.225 .AND. IC.LE.250 ) THEN
            SUBNAM( 1:1 ) = CHAR( IC-32 )
            DO 30 I = 2, 6
               IC = ICHAR( SUBNAM( I:I ) )
               IF( IC.GE.225 .AND. IC.LE.250 )
     $            SUBNAM( I:I ) = CHAR( IC-32 )
   30       CONTINUE
         END IF
      END IF
*
      C1 = SUBNAM( 1:1 )
      SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
      CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
      IF( .NOT.( CNAME .OR. SNAME ) )
     $   RETURN
      C2 = SUBNAM( 2:3 )
      C3 = SUBNAM( 4:6 )
      C4 = C3( 2:3 )
*
      GO TO ( 110, 200, 300 ) ISPEC
*
  110 CONTINUE
*
*     ISPEC = 1:  block size
*
*     In these examples, separate code is provided for setting NB for
*     real and complex.  We assume that NB will take the same value in
*     single or double precision.
*
      NB = 1
*
      IF( C2.EQ.'GE' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
     $            C3.EQ.'QLF' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         ELSE IF( C3.EQ.'HRD' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         ELSE IF( C3.EQ.'BRD' ) THEN
            IF( SNAME ) THEN
               NB = 32
            ELSE
               NB = 32
            END IF
         ELSE IF( C3.EQ.'TRI' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         END IF
      ELSE IF( C2.EQ.'PO' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         END IF
      ELSE IF( C2.EQ.'SY' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
            NB = 32
         ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
            NB = 64
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            NB = 64
         ELSE IF( C3.EQ.'TRD' ) THEN
            NB = 32
         ELSE IF( C3.EQ.'GST' ) THEN
            NB = 64
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
         IF( C3( 1:1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NB = 32
            END IF
         ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NB = 32
            END IF
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
         IF( C3( 1:1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NB = 32
            END IF
         ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NB = 32
            END IF
         END IF
      ELSE IF( C2.EQ.'GB' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               IF( N4.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            ELSE
               IF( N4.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            END IF
         END IF
      ELSE IF( C2.EQ.'PB' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               IF( N2.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            ELSE
               IF( N2.LE.64 ) THEN
                  NB = 1
               ELSE
                  NB = 32
               END IF
            END IF
         END IF
      ELSE IF( C2.EQ.'TR' ) THEN
         IF( C3.EQ.'TRI' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         END IF
      ELSE IF( C2.EQ.'LA' ) THEN
         IF( C3.EQ.'UUM' ) THEN
            IF( SNAME ) THEN
               NB = 64
            ELSE
               NB = 64
            END IF
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
         IF( C3.EQ.'EBZ' ) THEN
            NB = 1
         END IF
      END IF
      ILAENV = NB
      RETURN
*
  200 CONTINUE
*
*     ISPEC = 2:  minimum block size
*
      NBMIN = 2
      IF( C2.EQ.'GE' ) THEN
         IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
     $       C3.EQ.'QLF' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( C3.EQ.'HRD' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( C3.EQ.'BRD' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         ELSE IF( C3.EQ.'TRI' ) THEN
            IF( SNAME ) THEN
               NBMIN = 2
            ELSE
               NBMIN = 2
            END IF
         END IF
      ELSE IF( C2.EQ.'SY' ) THEN
         IF( C3.EQ.'TRF' ) THEN
            IF( SNAME ) THEN
               NBMIN = 8
            ELSE
               NBMIN = 8
            END IF
         ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
            NBMIN = 2
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
         IF( C3.EQ.'TRD' ) THEN
            NBMIN = 2
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
         IF( C3( 1:1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NBMIN = 2
            END IF
         ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NBMIN = 2
            END IF
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
         IF( C3( 1:1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NBMIN = 2
            END IF
         ELSE IF( C3( 1:1 ).EQ.'M' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NBMIN = 2
            END IF
         END IF
      END IF
      ILAENV = NBMIN
      RETURN
*
  300 CONTINUE
*
*     ISPEC = 3:  crossover point
*
      NX = 0
      IF( C2.EQ.'GE' ) THEN
         IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
     $       C3.EQ.'QLF' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         ELSE IF( C3.EQ.'HRD' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         ELSE IF( C3.EQ.'BRD' ) THEN
            IF( SNAME ) THEN
               NX = 128
            ELSE
               NX = 128
            END IF
         END IF
      ELSE IF( C2.EQ.'SY' ) THEN
         IF( SNAME .AND. C3.EQ.'TRD' ) THEN
            NX = 32
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
         IF( C3.EQ.'TRD' ) THEN
            NX = 32
         END IF
      ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
         IF( C3( 1:1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NX = 128
            END IF
         END IF
      ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
         IF( C3( 1:1 ).EQ.'G' ) THEN
            IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR.
     $          C4.EQ.'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR.
     $          C4.EQ.'BR' ) THEN
               NX = 128
            END IF
         END IF
      END IF
      ILAENV = NX
      RETURN
*
  400 CONTINUE
*
*     ISPEC = 4:  number of shifts (used by xHSEQR)
*
      ILAENV = 6
      RETURN
*
  500 CONTINUE
*
*     ISPEC = 5:  minimum column dimension (not used)
*
      ILAENV = 2
      RETURN
*
  600 CONTINUE
*
*     ISPEC = 6:  crossover point for SVD (used by xGELSS and xGESVD)
*
      ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
      RETURN
*
  700 CONTINUE
*
*     ISPEC = 7:  number of processors (not used)
*
      ILAENV = 1
      RETURN
*
  800 CONTINUE
*
*     ISPEC = 8:  crossover point for multishift (used by xHSEQR)
*
      ILAENV = 50
      RETURN
*
  900 CONTINUE
*
*     ISPEC = 9:  maximum size of the subproblems at the bottom of the
*                 computation tree in the divide-and-conquer algorithm
*                 (used by xGELSD and xGESDD)
*
      ILAENV = 25
      RETURN
*
 1000 CONTINUE
*
*     ISPEC = 10: ieee NaN arithmetic can be trusted not to trap
*
C     ILAENV = 0
      ILAENV = 1
      IF( ILAENV.EQ.1 ) THEN
         ILAENV = IEEECK( 0, 0.0, 1.0 )
      END IF
      RETURN
*
 1100 CONTINUE
*
*     ISPEC = 11: infinity arithmetic can be trusted not to trap
*
C     ILAENV = 0
      ILAENV = 1
      IF( ILAENV.EQ.1 ) THEN
         ILAENV = IEEECK( 1, 0.0, 1.0 )
      END IF
      RETURN
*
*     End of ILAENV
*
      END

      INTEGER          FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
*  -- LAPACK auxiliary routine (version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     June 30, 1998
*
*     .. Scalar Arguments ..
      INTEGER            ISPEC
      REAL               ONE, ZERO
*     ..
*
*  Purpose
*  =======
*
*  IEEECK is called from the ILAENV to verify that Infinity and
*  possibly NaN arithmetic is safe (i.e. will not trap).
*
*  Arguments
*  =========
*
*  ISPEC   (input) INTEGER
*          Specifies whether to test just for inifinity arithmetic
*          or whether to test for infinity and NaN arithmetic.
*          = 0: Verify infinity arithmetic only.
*          = 1: Verify infinity and NaN arithmetic.
*
*  ZERO    (input) REAL
*          Must contain the value 0.0
*          This is passed to prevent the compiler from optimizing
*          away this code.
*
*  ONE     (input) REAL
*          Must contain the value 1.0
*          This is passed to prevent the compiler from optimizing
*          away this code.
*
*  RETURN VALUE:  INTEGER
*          = 0:  Arithmetic failed to produce the correct answers
*          = 1:  Arithmetic produced the correct answers
*
*     .. Local Scalars ..
      REAL               NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF,
     $                   NEGZRO, NEWZRO, POSINF
*     ..
*     .. Executable Statements ..
      IEEECK = 1
*
      POSINF = ONE / ZERO
      IF( POSINF.LE.ONE ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGINF = -ONE / ZERO
      IF( NEGINF.GE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGZRO = ONE / ( NEGINF+ONE )
      IF( NEGZRO.NE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGINF = ONE / NEGZRO
      IF( NEGINF.GE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEWZRO = NEGZRO + ZERO
      IF( NEWZRO.NE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      POSINF = ONE / NEWZRO
      IF( POSINF.LE.ONE ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      NEGINF = NEGINF*POSINF
      IF( NEGINF.GE.ZERO ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      POSINF = POSINF*POSINF
      IF( POSINF.LE.ONE ) THEN
         IEEECK = 0
         RETURN
      END IF
*
*
*
*
*     Return if we were only asked to check infinity arithmetic
*
      IF( ISPEC.EQ.0 )
     $   RETURN
*
      NAN1 = POSINF + NEGINF
*
      NAN2 = POSINF / NEGINF
*
      NAN3 = POSINF / POSINF
*
      NAN4 = POSINF*ZERO
*
      NAN5 = NEGINF*NEGZRO
*
      NAN6 = NAN5*0.0
*
      IF( NAN1.EQ.NAN1 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN2.EQ.NAN2 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN3.EQ.NAN3 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN4.EQ.NAN4 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN5.EQ.NAN5 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      IF( NAN6.EQ.NAN6 ) THEN
         IEEECK = 0
         RETURN
      END IF
*
      RETURN
      END

      SUBROUTINE XERBLA( SRNAME, INFO )
*
*  -- LAPACK auxiliary routine (version 2.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     September 30, 1994
*
*     .. Scalar Arguments ..
      CHARACTER*6        SRNAME
      INTEGER            INFO
*     ..
*
*  Purpose
*  =======
*
*  XERBLA  is an error handler for the LAPACK routines.
*  It is called by an LAPACK routine if an input parameter has an
*  invalid value.  A message is printed and execution stops.
*
*  Installers may consider modifying the STOP statement in order to
*  call system-specific exception-handling facilities.
*
*  Arguments
*  =========
*
*  SRNAME  (input) CHARACTER*6
*          The name of the routine which called XERBLA.
*
*  INFO    (input) INTEGER
*          The position of the invalid parameter in the parameter list
*          of the calling routine.
*
* =====================================================================
*
*     .. Executable Statements ..
*
      WRITE( *, FMT = 9999 )SRNAME, INFO
*
      STOP
*
 9999 FORMAT( ' ** On entry to ', A6, ' parameter number ', I2, ' had ',
     $      'an illegal value' )
*
*     End of XERBLA
*
      END