---
name: "hfrk"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/hfrk"
---

# hfrk(3)

hfrk: Hermitian rank-k update, RFP format

## NAME

hfrk - hfrk: Hermitian rank-k update, RFP format

## SYNOPSIS

### Functions

subroutine **chfrk** (transr, uplo, trans, n, k, alpha, a, lda, beta, c)\
 **CHFRK** performs a Hermitian rank-k operation for matrix in RFP format.  subroutine **dsfrk** (transr, uplo, trans, n, k, alpha, a, lda, beta, c)\
 **DSFRK** performs a symmetric rank-k operation for matrix in RFP format.  subroutine **ssfrk** (transr, uplo, trans, n, k, alpha, a, lda, beta, c)\
 **SSFRK** performs a symmetric rank-k operation for matrix in RFP format.  subroutine **zhfrk** (transr, uplo, trans, n, k, alpha, a, lda, beta, c)\
 **ZHFRK** performs a Hermitian rank-k operation for matrix in RFP format.

## Detailed Description

## Function Documentation

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

**CHFRK** performs a Hermitian rank-k operation for matrix in RFP format.

**Purpose:**

```
!>
!> Level 3 BLAS like routine for C in RFP Format.
!>
!> CHFRK 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**

*TRANSR*

```
!>          TRANSR is CHARACTER*1
!>          = 'N':  The Normal Form of RFP A is stored;
!>          = 'C':  The Conjugate-transpose Form of RFP A is stored.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*N*

```
!>          N is INTEGER
!>           On entry,  N specifies the order of the matrix C.  N must be
!>           at least zero.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*ALPHA*

```
!>          ALPHA is REAL
!>           On entry, ALPHA specifies the scalar alpha.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*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 ).
!>           Unchanged on exit.
!>
```

*BETA*

```
!>          BETA is REAL
!>           On entry, BETA specifies the scalar beta.
!>           Unchanged on exit.
!>
```

*C*

```
!>          C is COMPLEX array, dimension (N*(N+1)/2)
!>           On entry, the matrix A in RFP Format. RFP Format is
!>           described by TRANSR, UPLO and N. 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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

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

**DSFRK** performs a symmetric rank-k operation for matrix in RFP format.

**Purpose:**

```
!>
!> Level 3 BLAS like routine for C in RFP Format.
!>
!> DSFRK 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 real 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**

*TRANSR*

```
!>          TRANSR is CHARACTER*1
!>          = 'N':  The Normal Form of RFP A is stored;
!>          = 'T':  The Transpose Form of RFP A is stored.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*N*

```
!>          N is INTEGER
!>           On entry, N specifies the order of the matrix C. N must be
!>           at least zero.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*ALPHA*

```
!>          ALPHA is DOUBLE PRECISION
!>           On entry, ALPHA specifies the scalar alpha.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*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 ).
!>           Unchanged on exit.
!>
```

*BETA*

```
!>          BETA is DOUBLE PRECISION
!>           On entry, BETA specifies the scalar beta.
!>           Unchanged on exit.
!>
```

*C*

```
!>          C is DOUBLE PRECISION array, dimension (NT)
!>           NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
!>           Format. RFP Format is described by TRANSR, UPLO and N.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **164** of file **dsfrk.f**.

### subroutine ssfrk (character transr, character uplo, character trans, integer n, integer k, real alpha, real, dimension( lda, \* ) a, integer lda, real beta, real, dimension( \* ) c)

**SSFRK** performs a symmetric rank-k operation for matrix in RFP format.

**Purpose:**

```
!>
!> Level 3 BLAS like routine for C in RFP Format.
!>
!> SSFRK 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 real 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**

*TRANSR*

```
!>          TRANSR is CHARACTER*1
!>          = 'N':  The Normal Form of RFP A is stored;
!>          = 'T':  The Transpose Form of RFP A is stored.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*N*

```
!>          N is INTEGER
!>           On entry, N specifies the order of the matrix C. N must be
!>           at least zero.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*ALPHA*

```
!>          ALPHA is REAL
!>           On entry, ALPHA specifies the scalar alpha.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*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 ).
!>           Unchanged on exit.
!>
```

*BETA*

```
!>          BETA is REAL
!>           On entry, BETA specifies the scalar beta.
!>           Unchanged on exit.
!>
```

*C*

```
!>          C is REAL array, dimension (NT)
!>           NT = N*(N+1)/2. On entry, the symmetric matrix C in RFP
!>           Format. RFP Format is described by TRANSR, UPLO and N.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **164** of file **ssfrk.f**.

### subroutine zhfrk (character transr, character uplo, character trans, integer n, integer k, double precision alpha, complex\*16, dimension( lda, \* ) a, integer lda, double precision beta, complex\*16, dimension( \* ) c)

**ZHFRK** performs a Hermitian rank-k operation for matrix in RFP format.

**Purpose:**

```
!>
!> Level 3 BLAS like routine for C in RFP Format.
!>
!> ZHFRK 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**

*TRANSR*

```
!>          TRANSR is CHARACTER*1
!>          = 'N':  The Normal Form of RFP A is stored;
!>          = 'C':  The Conjugate-transpose Form of RFP A is stored.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*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.
!>
!>           Unchanged on exit.
!>
```

*N*

```
!>          N is INTEGER
!>           On entry,  N specifies the order of the matrix C.  N must be
!>           at least zero.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*ALPHA*

```
!>          ALPHA is DOUBLE PRECISION
!>           On entry, ALPHA specifies the scalar alpha.
!>           Unchanged on exit.
!>
```

*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.
!>           Unchanged on exit.
!>
```

*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 ).
!>           Unchanged on exit.
!>
```

*BETA*

```
!>          BETA is DOUBLE PRECISION
!>           On entry, BETA specifies the scalar beta.
!>           Unchanged on exit.
!>
```

*C*

```
!>          C is COMPLEX*16 array, dimension (N*(N+1)/2)
!>           On entry, the matrix A in RFP Format. RFP Format is
!>           described by TRANSR, UPLO and N. 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.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

## Author

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