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complex_blas_level3(3) LAPACK complex_blas_level3(3)

NAME

complex_blas_level3

SYNOPSIS

Functions


subroutine cgemm (TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM subroutine chemm (SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CHEMM subroutine cher2k (UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CHER2K subroutine cherk (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
CHERK subroutine csymm (SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CSYMM subroutine csyr2k (UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CSYR2K subroutine csyrk (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
CSYRK subroutine ctrmm (SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRMM subroutine ctrsm (SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
CTRSM

Detailed Description

This is the group of complex LEVEL 3 BLAS routines.

Function Documentation

subroutine cgemm (character TRANSA, character TRANSB, integer M, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, complex BETA, complex, dimension(ldc,*) C, integer LDC)

CGEMM

Purpose:

 CGEMM  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**T   or   op( X ) = X**H,
 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

          TRANSA is 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**T.
              TRANSA = 'C' or 'c',  op( A ) = A**H.

TRANSB

          TRANSB is 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**T.
              TRANSB = 'C' or 'c',  op( B ) = B**H.

M

          M is 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.

N

          N is 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.

K

          K is 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.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, 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.

LDA

          LDA is 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 ).

B

          B is COMPLEX array, 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.

LDB

          LDB is 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 ).

BETA

          BETA is COMPLEX
           On entry,  BETA  specifies the scalar  beta.  When  BETA  is
           supplied as zero then C need not be set on input.

C

          C is COMPLEX array, 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

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

subroutine chemm (character SIDE, character UPLO, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, complex BETA, complex, dimension(ldc,*) C, integer LDC)

CHEMM

Purpose:

 CHEMM  performs one of the matrix-matrix operations
    C := alpha*A*B + beta*C,
 or
    C := alpha*B*A + beta*C,
 where alpha and beta are scalars, A is an hermitian matrix and  B and
 C are m by n matrices.

Parameters:

SIDE

          SIDE is CHARACTER*1
           On entry,  SIDE  specifies whether  the  hermitian matrix  A
           appears on the  left or right  in the  operation as follows:
              SIDE = 'L' or 'l'   C := alpha*A*B + beta*C,
              SIDE = 'R' or 'r'   C := alpha*B*A + beta*C,

UPLO

          UPLO is CHARACTER*1
           On  entry,   UPLO  specifies  whether  the  upper  or  lower
           triangular  part  of  the  hermitian  matrix   A  is  to  be
           referenced as follows:
              UPLO = 'U' or 'u'   Only the upper triangular part of the
                                  hermitian matrix is to be referenced.
              UPLO = 'L' or 'l'   Only the lower triangular part of the
                                  hermitian matrix is to be referenced.

M

          M is INTEGER
           On entry,  M  specifies the number of rows of the matrix  C.
           M  must be at least zero.

N

          N is INTEGER
           On entry, N specifies the number of columns of the matrix C.
           N  must be at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, dimension ( LDA, ka ), where ka is
           m  when  SIDE = 'L' or 'l'  and is n  otherwise.
           Before entry  with  SIDE = 'L' or 'l',  the  m by m  part of
           the array  A  must contain the  hermitian matrix,  such that
           when  UPLO = 'U' or 'u', the leading m by m upper triangular
           part of the array  A  must contain the upper triangular part
           of the  hermitian matrix and the  strictly  lower triangular
           part of  A  is not referenced,  and when  UPLO = 'L' or 'l',
           the leading  m by m  lower triangular part  of the  array  A
           must  contain  the  lower triangular part  of the  hermitian
           matrix and the  strictly upper triangular part of  A  is not
           referenced.
           Before entry  with  SIDE = 'R' or 'r',  the  n by n  part of
           the array  A  must contain the  hermitian matrix,  such that
           when  UPLO = 'U' or 'u', the leading n by n upper triangular
           part of the array  A  must contain the upper triangular part
           of the  hermitian matrix and the  strictly  lower triangular
           part of  A  is not referenced,  and when  UPLO = 'L' or 'l',
           the leading  n by n  lower triangular part  of the  array  A
           must  contain  the  lower triangular part  of the  hermitian
           matrix and the  strictly upper triangular part of  A  is not
           referenced.
           Note that the imaginary parts  of the diagonal elements need
           not be set, they are assumed to be zero.

LDA

          LDA is 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 ), otherwise  LDA must be at
           least max( 1, n ).

B

          B is COMPLEX array, dimension ( LDB, N )
           Before entry, the leading  m by n part of the array  B  must
           contain the matrix B.

LDB

          LDB is 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 ).

BETA

          BETA is COMPLEX
           On entry,  BETA  specifies the scalar  beta.  When  BETA  is
           supplied as zero then C need not be set on input.

C

          C is COMPLEX array, 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 updated
           matrix.

LDC

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

subroutine cher2k (character UPLO, character TRANS, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, real BETA, complex, dimension(ldc,*) C, integer LDC)

CHER2K

Purpose:

 CHER2K  performs one of the hermitian rank 2k operations
    C := alpha*A*B**H + conjg( alpha )*B*A**H + beta*C,
 or
    C := alpha*A**H*B + conjg( alpha )*B**H*A + beta*C,
 where  alpha and beta  are scalars with  beta  real,  C is an  n by n
 hermitian matrix and  A and B  are  n by k matrices in the first case
 and  k by n  matrices in the second case.

Parameters:

UPLO

          UPLO is 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.

TRANS

          TRANS is CHARACTER*1
           On entry,  TRANS  specifies the operation to be performed as
           follows:
              TRANS = 'N' or 'n'    C := alpha*A*B**H          +
                                         conjg( alpha )*B*A**H +
                                         beta*C.
              TRANS = 'C' or 'c'    C := alpha*A**H*B          +
                                         conjg( alpha )*B**H*A +
                                         beta*C.

N

          N is INTEGER
           On entry,  N specifies the order of the matrix C.  N must be
           at least zero.

K

          K is INTEGER
           On entry with  TRANS = 'N' or 'n',  K  specifies  the number
           of  columns  of the  matrices  A and B,  and on  entry  with
           TRANS = 'C' or 'c',  K  specifies  the number of rows of the
           matrices  A and B.  K must be at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, 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.

LDA

          LDA is 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 ).

B

          B is COMPLEX array, dimension ( LDB, kb ), where kb 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  B  must contain the matrix  B,  otherwise
           the leading  k by n  part of the array  B  must contain  the
           matrix B.

LDB

          LDB is INTEGER
           On entry, LDB specifies the first dimension of B as declared
           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
           then  LDB must be at least  max( 1, n ), otherwise  LDB must
           be at least  max( 1, k ).

BETA

          BETA is REAL
           On entry, BETA specifies the scalar beta.

C

          C is COMPLEX array, 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  hermitian 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  hermitian 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.
           Note that the imaginary parts of the diagonal elements need
           not be set,  they are assumed to be zero,  and on exit they
           are set to zero.

LDC

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.
  -- Modified 8-Nov-93 to set C(J,J) to REAL( C(J,J) ) when BETA = 1.
     Ed Anderson, Cray Research Inc.

subroutine cherk (character UPLO, character TRANS, integer N, integer K, real ALPHA, complex, dimension(lda,*) A, integer LDA, real BETA, complex, dimension(ldc,*) C, integer LDC)

CHERK

Purpose:

 CHERK  performs one of the hermitian rank k operations
    C := alpha*A*A**H + beta*C,
 or
    C := alpha*A**H*A + beta*C,
 where  alpha and beta  are  real scalars,  C is an  n by n  hermitian
 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

          UPLO is 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.

TRANS

          TRANS is CHARACTER*1
           On entry,  TRANS  specifies the operation to be performed as
           follows:
              TRANS = 'N' or 'n'   C := alpha*A*A**H + beta*C.
              TRANS = 'C' or 'c'   C := alpha*A**H*A + beta*C.

N

          N is INTEGER
           On entry,  N specifies the order of the matrix C.  N must be
           at least zero.

K

          K is INTEGER
           On entry with  TRANS = 'N' or 'n',  K  specifies  the number
           of  columns   of  the   matrix   A,   and  on   entry   with
           TRANS = 'C' or 'c',  K  specifies  the number of rows of the
           matrix A.  K must be at least zero.

ALPHA

          ALPHA is REAL
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, 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.

LDA

          LDA is 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 ).

BETA

          BETA is REAL
           On entry, BETA specifies the scalar beta.

C

          C is COMPLEX array, 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  hermitian 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  hermitian 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.
           Note that the imaginary parts of the diagonal elements need
           not be set,  they are assumed to be zero,  and on exit they
           are set to zero.

LDC

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.
  -- Modified 8-Nov-93 to set C(J,J) to REAL( C(J,J) ) when BETA = 1.
     Ed Anderson, Cray Research Inc.

subroutine csymm (character SIDE, character UPLO, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, complex BETA, complex, dimension(ldc,*) C, integer LDC)

CSYMM

Purpose:

 CSYMM  performs one of the matrix-matrix operations
    C := alpha*A*B + beta*C,
 or
    C := alpha*B*A + beta*C,
 where  alpha and beta are scalars, A is a symmetric matrix and  B and
 C are m by n matrices.

Parameters:

SIDE

          SIDE is CHARACTER*1
           On entry,  SIDE  specifies whether  the  symmetric matrix  A
           appears on the  left or right  in the  operation as follows:
              SIDE = 'L' or 'l'   C := alpha*A*B + beta*C,
              SIDE = 'R' or 'r'   C := alpha*B*A + beta*C,

UPLO

          UPLO is CHARACTER*1
           On  entry,   UPLO  specifies  whether  the  upper  or  lower
           triangular  part  of  the  symmetric  matrix   A  is  to  be
           referenced as follows:
              UPLO = 'U' or 'u'   Only the upper triangular part of the
                                  symmetric matrix is to be referenced.
              UPLO = 'L' or 'l'   Only the lower triangular part of the
                                  symmetric matrix is to be referenced.

M

          M is INTEGER
           On entry,  M  specifies the number of rows of the matrix  C.
           M  must be at least zero.

N

          N is INTEGER
           On entry, N specifies the number of columns of the matrix C.
           N  must be at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, dimension ( LDA, ka ), where ka is
           m  when  SIDE = 'L' or 'l'  and is n  otherwise.
           Before entry  with  SIDE = 'L' or 'l',  the  m by m  part of
           the array  A  must contain the  symmetric matrix,  such that
           when  UPLO = 'U' or 'u', the leading m by m upper triangular
           part of the array  A  must contain the upper triangular part
           of the  symmetric matrix and the  strictly  lower triangular
           part of  A  is not referenced,  and when  UPLO = 'L' or 'l',
           the leading  m by m  lower triangular part  of the  array  A
           must  contain  the  lower triangular part  of the  symmetric
           matrix and the  strictly upper triangular part of  A  is not
           referenced.
           Before entry  with  SIDE = 'R' or 'r',  the  n by n  part of
           the array  A  must contain the  symmetric matrix,  such that
           when  UPLO = 'U' or 'u', the leading n by n upper triangular
           part of the array  A  must contain the upper triangular part
           of the  symmetric matrix and the  strictly  lower triangular
           part of  A  is not referenced,  and when  UPLO = 'L' or 'l',
           the leading  n by n  lower triangular part  of the  array  A
           must  contain  the  lower triangular part  of the  symmetric
           matrix and the  strictly upper triangular part of  A  is not
           referenced.

LDA

          LDA is 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 ), otherwise  LDA must be at
           least max( 1, n ).

B

          B is COMPLEX array, dimension ( LDB, N )
           Before entry, the leading  m by n part of the array  B  must
           contain the matrix B.

LDB

          LDB is 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 ).

BETA

          BETA is COMPLEX
           On entry,  BETA  specifies the scalar  beta.  When  BETA  is
           supplied as zero then C need not be set on input.

C

          C is COMPLEX array, 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 updated
           matrix.

LDC

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

subroutine csyr2k (character UPLO, character TRANS, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB, complex BETA, complex, dimension(ldc,*) C, integer LDC)

CSYR2K

Purpose:

 CSYR2K  performs one of the symmetric rank 2k operations
    C := alpha*A*B**T + alpha*B*A**T + beta*C,
 or
    C := alpha*A**T*B + alpha*B**T*A + beta*C,
 where  alpha and beta  are scalars,  C is an  n by n symmetric matrix
 and  A and B  are  n by k  matrices  in the  first  case  and  k by n
 matrices in the second case.

Parameters:

UPLO

          UPLO is 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.

TRANS

          TRANS is CHARACTER*1
           On entry,  TRANS  specifies the operation to be performed as
           follows:
              TRANS = 'N' or 'n'    C := alpha*A*B**T + alpha*B*A**T +
                                         beta*C.
              TRANS = 'T' or 't'    C := alpha*A**T*B + alpha*B**T*A +
                                         beta*C.

N

          N is INTEGER
           On entry,  N specifies the order of the matrix C.  N must be
           at least zero.

K

          K is INTEGER
           On entry with  TRANS = 'N' or 'n',  K  specifies  the number
           of  columns  of the  matrices  A and B,  and on  entry  with
           TRANS = 'T' or 't',  K  specifies  the number of rows of the
           matrices  A and B.  K must be at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, 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.

LDA

          LDA is 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 ).

B

          B is COMPLEX array, dimension ( LDB, kb ), where kb 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  B  must contain the matrix  B,  otherwise
           the leading  k by n  part of the array  B  must contain  the
           matrix B.

LDB

          LDB is INTEGER
           On entry, LDB specifies the first dimension of B as declared
           in  the  calling  (sub)  program.   When  TRANS = 'N' or 'n'
           then  LDB must be at least  max( 1, n ), otherwise  LDB must
           be at least  max( 1, k ).

BETA

          BETA is COMPLEX
           On entry, BETA specifies the scalar beta.

C

          C is COMPLEX array, 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

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

subroutine csyrk (character UPLO, character TRANS, integer N, integer K, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex BETA, complex, dimension(ldc,*) C, integer LDC)

CSYRK

Purpose:

 CSYRK  performs one of the symmetric rank k operations
    C := alpha*A*A**T + beta*C,
 or
    C := alpha*A**T*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

          UPLO is 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.

TRANS

          TRANS is CHARACTER*1
           On entry,  TRANS  specifies the operation to be performed as
           follows:
              TRANS = 'N' or 'n'   C := alpha*A*A**T + beta*C.
              TRANS = 'T' or 't'   C := alpha*A**T*A + beta*C.

N

          N is INTEGER
           On entry,  N specifies the order of the matrix C.  N must be
           at least zero.

K

          K is 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',  K  specifies  the number of rows of the
           matrix A.  K must be at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.

A

          A is COMPLEX array, 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.

LDA

          LDA is 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 ).

BETA

          BETA is COMPLEX
           On entry, BETA specifies the scalar beta.

C

          C is COMPLEX array, 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

          LDC is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

subroutine ctrmm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB)

CTRMM

Purpose:

 CTRMM  performs one of the matrix-matrix operations
    B := alpha*op( A )*B,   or   B := alpha*B*op( A )
 where  alpha  is a scalar,  B  is an m by n matrix,  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**T   or   op( A ) = A**H.

Parameters:

SIDE

          SIDE is CHARACTER*1
           On entry,  SIDE specifies whether  op( A ) multiplies B from
           the left or right as follows:
              SIDE = 'L' or 'l'   B := alpha*op( A )*B.
              SIDE = 'R' or 'r'   B := alpha*B*op( A ).

UPLO

          UPLO is 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.

TRANSA

          TRANSA is 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**T.
              TRANSA = 'C' or 'c'   op( A ) = A**H.

DIAG

          DIAG is 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.

M

          M is INTEGER
           On entry, M specifies the number of rows of B. M must be at
           least zero.

N

          N is INTEGER
           On entry, N specifies the number of columns of B.  N must be
           at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
           zero then  A is not referenced and  B need not be set before
           entry.

A

          A is COMPLEX array, 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.

LDA

          LDA is 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 ).

B

          B is COMPLEX array, dimension ( LDB, N ).
           Before entry,  the leading  m by n part of the array  B must
           contain the matrix  B,  and  on exit  is overwritten  by the
           transformed matrix.

LDB

          LDB is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

subroutine ctrsm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, complex ALPHA, complex, dimension(lda,*) A, integer LDA, complex, dimension(ldb,*) B, integer LDB)

CTRSM

Purpose:

 CTRSM  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**T   or   op( A ) = A**H.
 The matrix X is overwritten on B.

Parameters:

SIDE

          SIDE is 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.

UPLO

          UPLO is 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.

TRANSA

          TRANSA is 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**T.
              TRANSA = 'C' or 'c'   op( A ) = A**H.

DIAG

          DIAG is 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.

M

          M is INTEGER
           On entry, M specifies the number of rows of B. M must be at
           least zero.

N

          N is INTEGER
           On entry, N specifies the number of columns of B.  N must be
           at least zero.

ALPHA

          ALPHA is COMPLEX
           On entry,  ALPHA specifies the scalar  alpha. When  alpha is
           zero then  A is not referenced and  B need not be set before
           entry.

A

          A is COMPLEX array, dimension ( LDA, k ),
           where k is m when SIDE = 'L' or 'l'
             and k 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.

LDA

          LDA is 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 ).

B

          B is COMPLEX array, 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

          LDB is 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 ).

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

December 2016

Further Details:

  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.

Author

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