# herk - Man Page

{he,sy}rk: Hermitian/symmetric rank-k update

## Synopsis

### Functions

subroutine **cherk** (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)**CHERK**

subroutine **csyrk** (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)**CSYRK**

subroutine **dsyrk** (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)**DSYRK**

subroutine **ssyrk** (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)**SSYRK**

subroutine **zherk** (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)**ZHERK**

subroutine **zsyrk** (uplo, trans, n, k, alpha, a, lda, beta, c, ldc)**ZSYRK**

## Detailed Description

## Function Documentation

### subroutine cherk (character uplo, character trans, integer n, integer k, real alpha, complex, dimension(lda,*) a, integer lda, real beta, complex, dimension(ldc,*) c, integer ldc)

**CHERK**

**Purpose:**

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

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

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

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

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

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

*A*A is COMPLEX array, dimension ( LDA, ka ), where ka is k when TRANS = 'N' or 'n', and is n otherwise. Before entry with TRANS = 'N' or 'n', the leading n by k part of the array A must contain the matrix A, otherwise the leading k by n part of the array A must contain the matrix A.

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

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

*C*C is COMPLEX array, dimension ( LDC, N ) Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array C must contain the upper triangular part of the hermitian matrix and the strictly lower triangular part of C is not referenced. On exit, the upper triangular part of the array C is overwritten by the upper triangular part of the updated matrix. Before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array C must contain the lower triangular part of the hermitian matrix and the strictly upper triangular part of C is not referenced. On exit, the lower triangular part of the array C is overwritten by the lower triangular part of the updated matrix. Note that the imaginary parts of the diagonal elements need not be set, they are assumed to be zero, and on exit they are set to zero.

*LDC*LDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDC must be at least max( 1, n ).

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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**.

### subroutine csyrk (character uplo, character trans, integer n, integer k, complex alpha, complex, dimension(lda,*) a, integer lda, complex beta, complex, dimension(ldc,*) c, integer ldc)

**CSYRK**

**Purpose:**

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

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

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

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

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

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

*A*A is COMPLEX array, dimension ( LDA, ka ), where ka is k when TRANS = 'N' or 'n', and is n otherwise. Before entry with TRANS = 'N' or 'n', the leading n by k part of the array A must contain the matrix A, otherwise the leading k by n part of the array A must contain the matrix A.

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

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

*C*C is COMPLEX array, dimension ( LDC, N ) Before entry with UPLO = 'U' or 'u', the leading n by n upper triangular part of the array C must contain the upper triangular part of the symmetric matrix and the strictly lower triangular part of C is not referenced. On exit, the upper triangular part of the array C is overwritten by the upper triangular part of the updated matrix. Before entry with UPLO = 'L' or 'l', the leading n by n lower triangular part of the array C must contain the lower triangular part of the symmetric matrix and the strictly upper triangular part of C is not referenced. On exit, the lower triangular part of the array C is overwritten by the lower triangular part of the updated matrix.

*LDC*LDC is INTEGER On entry, LDC specifies the first dimension of C as declared in the calling (sub) program. LDC must be at least max( 1, n ).

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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**.

### subroutine dsyrk (character uplo, character trans, integer n, integer k, double precision alpha, double precision, dimension(lda,*) a, integer lda, double precision beta, double precision, dimension(ldc,*) c, integer ldc)

**DSYRK**

**Purpose:**

DSYRK 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. TRANS = 'C' or 'c' 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' or '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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 **168** of file **dsyrk.f**.

### subroutine ssyrk (character uplo, character trans, integer n, integer k, real alpha, real, dimension(lda,*) a, integer lda, real beta, real, dimension(ldc,*) c, integer ldc)

**SSYRK**

**Purpose:**

SSYRK 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. TRANS = 'C' or 'c' 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' or '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 REAL 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**BETA*BETA is REAL On entry, BETA specifies the scalar beta.

*C*C is REAL 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 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 **168** of file **ssyrk.f**.

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

Definition at line **172** of file **zherk.f**.

### 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*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**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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

Definition at line **166** of file **zsyrk.f**.

## Author

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