## table of contents

complex16_blas_level3(3) | LAPACK | complex16_blas_level3(3) |

# NAME¶

complex16_blas_level3# SYNOPSIS¶

## Functions¶

subroutine

**zgemm**(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)

**ZGEMM**subroutine

**zhemm**(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)

**ZHEMM**subroutine

**zher2k**(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)

**ZHER2K**subroutine

**zherk**(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)

**ZHERK**subroutine

**zsymm**(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)

**ZSYMM**subroutine

**zsyr2k**(UPLO, TRANS, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)

**ZSYR2K**subroutine

**zsyrk**(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)

**ZSYRK**subroutine

**ztrmm**(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)

**ZTRMM**subroutine

**ztrsm**(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)

**ZTRSM**

# Detailed Description¶

This is the group of complex16 LEVEL 3 BLAS routines.# Function Documentation¶

## 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)¶

**ZGEMM**

**Purpose: **

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.

**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*16 On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*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.

*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*16 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*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 ).

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine 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)¶

**ZHEMM**

**Purpose: **

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.

**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*16 On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*16 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*16 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*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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine 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)¶

**ZHER2K**

**Purpose: **

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.

**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*16 . On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*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.

*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 ). Unchanged on exit.

*BETA*

BETA is DOUBLE PRECISION . On entry, BETA specifies the scalar beta.

*C*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**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 DBLE( C(J,J) ) when BETA = 1. Ed Anderson, Cray Research Inc.

## 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)¶

**ZHERK**

**Purpose: **

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.

**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 DOUBLE PRECISION . On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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 DOUBLE PRECISION. On entry, BETA specifies the scalar beta.

*C*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**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 DBLE( C(J,J) ) when BETA = 1. Ed Anderson, Cray Research Inc.

## 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)¶

**ZSYMM**

**Purpose: **

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.

**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*16 On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*16 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*16 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*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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

Level 3 Blas routine. -- Written on 8-February-1989. Jack Dongarra, Argonne National Laboratory. Iain Duff, AERE Harwell. Jeremy Du Croz, Numerical Algorithms Group Ltd. Sven Hammarling, Numerical Algorithms Group Ltd.

## subroutine 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)¶

**ZSYR2K**

**Purpose: **

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.

**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*16 On entry, ALPHA specifies the scalar alpha.

*A*

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.

*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*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.

*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*16 On entry, BETA specifies the scalar beta.

*C*

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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

## 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)¶

**ZSYRK**

**Purpose: **

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.

**Parameters:**

*UPLO*

*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*16 On entry, ALPHA specifies the scalar alpha.

*A*

*LDA*

*BETA*

BETA is COMPLEX*16 On entry, BETA specifies the scalar beta.

*C*

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.

*LDC*

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

## 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)¶

**ZTRMM**

**Purpose: **

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.

**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*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.

*A*

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.

*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*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.

*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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

## 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)¶

**ZTRSM**

**Purpose: **

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.

**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*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.

*A*

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.

*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*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.

*LDB*

**Author:**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**

**Further Details: **

# Author¶

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