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.

Definition at line 189 of file cgemm.f.

  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.

Definition at line 193 of file chemm.f.

  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.

Definition at line 199 of file cher2k.f.

  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.

Definition at line 175 of file cherk.f.

  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.

Definition at line 191 of file csymm.f.

  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.

Definition at line 190 of file csyr2k.f.

  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.

Definition at line 169 of file csyrk.f.

  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.

Definition at line 179 of file ctrmm.f.

  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.

Definition at line 182 of file ctrsm.f.

Author

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