complex_blas_level3(3) LAPACK complex_blas_level3(3)

complex_blas_level3 - complex


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

This is the group of complex LEVEL 3 BLAS routines.

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.

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 186 of file cgemm.f.

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.

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 chemm.f.

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.

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 196 of file cher2k.f.

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.

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 172 of file cherk.f.

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.

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 188 of file csymm.f.

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.

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 187 of file csyr2k.f.

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.

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 166 of file csyrk.f.

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.

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 176 of file ctrmm.f.

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.

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 ctrsm.f.

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