.TH "complex16_blas_level3" 3 "Sun Nov 27 2022" "Version 3.11.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
complex16_blas_level3 \- complex16
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBzgemm\fP (TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)"
.br
.RI "\fBZGEMM\fP "
.ti -1c
.RI "subroutine \fBzhemm\fP (SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)"
.br
.RI "\fBZHEMM\fP "
.ti -1c
.RI "subroutine \fBzher2k\fP (UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)"
.br
.RI "\fBZHER2K\fP "
.ti -1c
.RI "subroutine \fBzherk\fP (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)"
.br
.RI "\fBZHERK\fP "
.ti -1c
.RI "subroutine \fBzsymm\fP (SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)"
.br
.RI "\fBZSYMM\fP "
.ti -1c
.RI "subroutine \fBzsyr2k\fP (UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)"
.br
.RI "\fBZSYR2K\fP "
.ti -1c
.RI "subroutine \fBzsyrk\fP (UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)"
.br
.RI "\fBZSYRK\fP "
.ti -1c
.RI "subroutine \fBztrmm\fP (SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)"
.br
.RI "\fBZTRMM\fP "
.ti -1c
.RI "subroutine \fBztrsm\fP (SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)"
.br
.RI "\fBZTRSM\fP "
.in -1c
.SH "Detailed Description"
.PP
This is the group of complex16 LEVEL 3 BLAS routines\&.
.SH "Function Documentation"
.PP
.SS "subroutine zgemm (character TRANSA, character TRANSB, integer M, integer N, integer K, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB, complex*16 BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZGEMM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZGEMM 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fITRANSA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANSB\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then C need not be set on input\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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 )\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zhemm (character SIDE, character UPLO, integer M, integer N, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB, complex*16 BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZHEMM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHEMM 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
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,
.fi
.PP
.br
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of the matrix C\&.
M must be at least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of the matrix C\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 array, dimension ( LDB, N )
Before entry, the leading m by n part of the array B must
contain the matrix B\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then C need not be set on input\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zher2k (character UPLO, character TRANS, integer N, integer K, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB, double precision BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZHER2K\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHER2K 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix C\&. N must be
at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16 \&.
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
Unchanged on exit\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is DOUBLE PRECISION \&.
On entry, BETA specifies the scalar beta\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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 DBLE( C(J,J) ) when BETA = 1\&.
Ed Anderson, Cray Research Inc\&.
.fi
.PP
.RE
.PP
.SS "subroutine zherk (character UPLO, character TRANS, integer N, integer K, double precision ALPHA, complex*16, dimension(lda,*) A, integer LDA, double precision BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZHERK\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZHERK 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix C\&. N must be
at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is DOUBLE PRECISION \&.
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is DOUBLE PRECISION\&.
On entry, BETA specifies the scalar beta\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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 DBLE( C(J,J) ) when BETA = 1\&.
Ed Anderson, Cray Research Inc\&.
.fi
.PP
.RE
.PP
.SS "subroutine zsymm (character SIDE, character UPLO, integer M, integer N, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB, complex*16 BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZSYMM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZSYMM 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
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,
.fi
.PP
.br
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of the matrix C\&.
M must be at least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of the matrix C\&.
N must be at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 array, dimension ( LDB, N )
Before entry, the leading m by n part of the array B must
contain the matrix B\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&. When BETA is
supplied as zero then C need not be set on input\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zsyr2k (character UPLO, character TRANS, integer N, integer K, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB, complex*16 BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZSYR2K\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZSYR2K 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix C\&. N must be
at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine zsyrk (character UPLO, character TRANS, integer N, integer K, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16 BETA, complex*16, dimension(ldc,*) C, integer LDC)"
.PP
\fBZSYRK\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZSYRK 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANS\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the order of the matrix C\&. N must be
at least zero\&.
.fi
.PP
.br
\fIK\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIBETA\fP
.PP
.nf
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta\&.
.fi
.PP
.br
\fIC\fP
.PP
.nf
C is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDC\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztrmm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB)"
.PP
\fBZTRMM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTRMM 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANSA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of B\&. M must be at
least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of B\&. N must be
at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&. When alpha is
zero then A is not referenced and B need not be set before
entry\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztrsm (character SIDE, character UPLO, character TRANSA, character DIAG, integer M, integer N, complex*16 ALPHA, complex*16, dimension(lda,*) A, integer LDA, complex*16, dimension(ldb,*) B, integer LDB)"
.PP
\fBZTRSM\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTRSM 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\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIUPLO\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fITRANSA\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIDIAG\fP
.PP
.nf
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\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
On entry, M specifies the number of rows of B\&. M must be at
least zero\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
On entry, N specifies the number of columns of B\&. N must be
at least zero\&.
.fi
.PP
.br
\fIALPHA\fP
.PP
.nf
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha\&. When alpha is
zero then A is not referenced and B need not be set before
entry\&.
.fi
.PP
.br
\fIA\fP
.PP
.nf
A is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDA\fP
.PP
.nf
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 )\&.
.fi
.PP
.br
\fIB\fP
.PP
.nf
B is COMPLEX*16 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\&.
.fi
.PP
.br
\fILDB\fP
.PP
.nf
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 )\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
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\&.
.fi
.PP
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.