Performs a symmetric rank-k update.

## Syntax

call ssyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)

call dsyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)

call csyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)

call zsyrk(uplo, trans, n, k, alpha, a, lda, beta, c, ldc)

call syrk(a, c [,uplo] [, trans] [,alpha][,beta])

## Include Files

• mkl.fi, blas.f90

## Description

The ?syrk routines perform a rank-k matrix-matrix operation for a symmetric matrix C using a general matrix A . The operation is defined as:

`C := alpha*A*A' + beta*C,`

or

`C := alpha*A'*A + beta*C,`

where:

alpha and beta are scalars,

C is an n-by-n symmetric matrix,

A is an n-by-k matrix in the first case and a k-by-n matrix in the second case.

## Input Parameters

uplo

CHARACTER*1. Specifies whether the upper or lower triangular part of the array c is used.

If uplo = 'U' or 'u' , then the upper triangular part of the array c is used.

If uplo = 'L' or 'l' , then the low triangular part of the array c is used.

trans

CHARACTER*1. Specifies the operation:

if trans= 'N' or 'n', then C := alpha*A*A' + beta*C;

if trans= 'T' or 't', then C := alpha*A'*A + beta*C;

if trans= 'C' or 'c', then C := alpha*A'*A + beta*C.

n

INTEGER. Specifies the order of the matrix C. The value of n must be at least zero.

k

INTEGER. On entry with trans= 'N' or 'n', k specifies the number of columns of the matrix a, and on entry with trans= 'T', 't', 'C', or 'c' , k specifies the number of rows of the matrix a.

The value of k must be at least zero.

alpha

REAL for ssyrk

DOUBLE PRECISION for dsyrk

COMPLEX for csyrk

DOUBLE COMPLEX for zsyrk

Specifies the scalar alpha.

a

REAL for ssyrk

DOUBLE PRECISION for dsyrk

COMPLEX for csyrk

DOUBLE COMPLEX for zsyrk

Array, size (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.

Array, size (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

INTEGER. Specifies the leading 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

REAL for ssyrk

DOUBLE PRECISION for dsyrk

COMPLEX for csyrk

DOUBLE COMPLEX for zsyrk

Specifies the scalar beta.

c

REAL for ssyrk

DOUBLE PRECISION for dsyrk

COMPLEX for csyrk

DOUBLE COMPLEX for zsyrk

Array, size (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.

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.

ldc

INTEGER. Specifies the leading dimension of c as declared in the calling (sub)program. The value of ldc must be at least max(1, n).

## Output Parameters

c

With uplo = 'U' or 'u', the upper triangular part of the array c is overwritten by the upper triangular part of the updated matrix.

With uplo = 'L' or 'l', the lower triangular part of the array c is overwritten by the lower triangular part of the updated matrix.

## BLAS 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see BLAS 95 Interface Conventions.

Specific details for the routine syrk interface are the following:

a

Holds the matrix A of size (ma,ka) where

ka = k if transa='N',

ka = n otherwise,

ma = n if transa='N',

ma = k otherwise.

c

Holds the matrix C of size (n,n).

uplo

Must be 'U' or 'L'. The default value is 'U'.

trans

Must be 'N', 'C', or 'T'.

The default value is 'N'.

alpha

The default value is 1.

beta

The default value is 0.

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