---
name: "hpmv"
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/hpmv"
---

# hpmv(3)

{hp,sp}mv: Hermitian/symmetric matrix-vector multiply

## NAME

hpmv - {hp,sp}mv: Hermitian/symmetric matrix-vector multiply

## SYNOPSIS

### Functions

subroutine **chpmv** (uplo, n, alpha, ap, x, incx, beta, y, incy)\
 **CHPMV**  subroutine **dspmv** (uplo, n, alpha, ap, x, incx, beta, y, incy)\
 **DSPMV**  subroutine **sspmv** (uplo, n, alpha, ap, x, incx, beta, y, incy)\
 **SSPMV**  subroutine **zhpmv** (uplo, n, alpha, ap, x, incx, beta, y, incy)\
 **ZHPMV**  subroutine **cspmv** (uplo, n, alpha, ap, x, incx, beta, y, incy)\
 **CSPMV** computes a matrix-vector product for complex vectors using a complex symmetric packed matrix  subroutine **zspmv** (uplo, n, alpha, ap, x, incx, beta, y, incy)\
 **ZSPMV** computes a matrix-vector product for complex vectors using a complex symmetric packed matrix

## Detailed Description

## Function Documentation

### subroutine chpmv (character uplo, integer n, complex alpha, complex, dimension(\*) ap, complex, dimension(\*) x, integer incx, complex beta, complex, dimension(\*) y, integer incy)

**CHPMV**

**Purpose:**

```
!>
!> CHPMV  performs the matrix-vector operation
!>
!>    y := alpha*A*x + beta*y,
!>
!> where alpha and beta are scalars, x and y are n element vectors and
!> A is an n by n hermitian matrix, supplied in packed form.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the upper or lower
!>           triangular part of the matrix A is supplied in the packed
!>           array AP as follows:
!>
!>              UPLO = 'U' or 'u'   The upper triangular part of A is
!>                                  supplied in AP.
!>
!>              UPLO = 'L' or 'l'   The lower triangular part of A is
!>                                  supplied in AP.
!>
```

*N*

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

*ALPHA*

```
!>          ALPHA is COMPLEX
!>           On entry, ALPHA specifies the scalar alpha.
!>
```

*AP*

```
!>          AP is COMPLEX array, dimension at least
!>           ( ( n*( n + 1 ) )/2 ).
!>           Before entry with UPLO = 'U' or 'u', the array AP must
!>           contain the upper triangular part of the hermitian matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
!>           and a( 2, 2 ) respectively, and so on.
!>           Before entry with UPLO = 'L' or 'l', the array AP must
!>           contain the lower triangular part of the hermitian matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
!>           and a( 3, 1 ) respectively, and so on.
!>           Note that the imaginary parts of the diagonal elements need
!>           not be set and are assumed to be zero.
!>
```

*X*

```
!>          X is COMPLEX array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the n
!>           element vector x.
!>
```

*INCX*

```
!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX must not be zero.
!>
```

*BETA*

```
!>          BETA is COMPLEX
!>           On entry, BETA specifies the scalar beta. When BETA is
!>           supplied as zero then Y need not be set on input.
!>
```

*Y*

```
!>          Y is COMPLEX array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCY ) ).
!>           Before entry, the incremented array Y must contain the n
!>           element vector y. On exit, Y is overwritten by the updated
!>           vector y.
!>
```

*INCY*

```
!>          INCY is INTEGER
!>           On entry, INCY specifies the increment for the elements of
!>           Y. INCY must not be zero.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

```
!>
!>  Level 2 Blas routine.
!>  The vector and matrix arguments are not referenced when N = 0, or M = 0
!>
!>  -- Written on 22-October-1986.
!>     Jack Dongarra, Argonne National Lab.
!>     Jeremy Du Croz, Nag Central Office.
!>     Sven Hammarling, Nag Central Office.
!>     Richard Hanson, Sandia National Labs.
!>
```

Definition at line **148** of file **chpmv.f**.

### subroutine cspmv (character uplo, integer n, complex alpha, complex, dimension( \* ) ap, complex, dimension( \* ) x, integer incx, complex beta, complex, dimension( \* ) y, integer incy)

**CSPMV** computes a matrix-vector product for complex vectors using a complex symmetric packed matrix

**Purpose:**

```
!>
!> CSPMV  performs the matrix-vector operation
!>
!>    y := alpha*A*x + beta*y,
!>
!> where alpha and beta are scalars, x and y are n element vectors and
!> A is an n by n symmetric matrix, supplied in packed form.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the upper or lower
!>           triangular part of the matrix A is supplied in the packed
!>           array AP as follows:
!>
!>              UPLO = 'U' or 'u'   The upper triangular part of A is
!>                                  supplied in AP.
!>
!>              UPLO = 'L' or 'l'   The lower triangular part of A is
!>                                  supplied in AP.
!>
!>           Unchanged on exit.
!>
```

*N*

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

*ALPHA*

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

*AP*

```
!>          AP is COMPLEX array, dimension at least
!>           ( ( N*( N + 1 ) )/2 ).
!>           Before entry, with UPLO = 'U' or 'u', the array AP must
!>           contain the upper triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
!>           and a( 2, 2 ) respectively, and so on.
!>           Before entry, with UPLO = 'L' or 'l', the array AP must
!>           contain the lower triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
!>           and a( 3, 1 ) respectively, and so on.
!>           Unchanged on exit.
!>
```

*X*

```
!>          X is COMPLEX array, dimension at least
!>           ( 1 + ( N - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the N-
!>           element vector x.
!>           Unchanged on exit.
!>
```

*INCX*

```
!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX must not be zero.
!>           Unchanged on exit.
!>
```

*BETA*

```
!>          BETA is COMPLEX
!>           On entry, BETA specifies the scalar beta. When BETA is
!>           supplied as zero then Y need not be set on input.
!>           Unchanged on exit.
!>
```

*Y*

```
!>          Y is COMPLEX array, dimension at least
!>           ( 1 + ( N - 1 )*abs( INCY ) ).
!>           Before entry, the incremented array Y must contain the n
!>           element vector y. On exit, Y is overwritten by the updated
!>           vector y.
!>
```

*INCY*

```
!>          INCY is INTEGER
!>           On entry, INCY specifies the increment for the elements of
!>           Y. INCY must not be zero.
!>           Unchanged on exit.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **150** of file **cspmv.f**.

### subroutine dspmv (character uplo, integer n, double precision alpha, double precision, dimension(\*) ap, double precision, dimension(\*) x, integer incx, double precision beta, double precision, dimension(\*) y, integer incy)

**DSPMV**

**Purpose:**

```
!>
!> DSPMV  performs the matrix-vector operation
!>
!>    y := alpha*A*x + beta*y,
!>
!> where alpha and beta are scalars, x and y are n element vectors and
!> A is an n by n symmetric matrix, supplied in packed form.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the upper or lower
!>           triangular part of the matrix A is supplied in the packed
!>           array AP as follows:
!>
!>              UPLO = 'U' or 'u'   The upper triangular part of A is
!>                                  supplied in AP.
!>
!>              UPLO = 'L' or 'l'   The lower triangular part of A is
!>                                  supplied in AP.
!>
```

*N*

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

*ALPHA*

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

*AP*

```
!>          AP is DOUBLE PRECISION array, dimension at least
!>           ( ( n*( n + 1 ) )/2 ).
!>           Before entry with UPLO = 'U' or 'u', the array AP must
!>           contain the upper triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
!>           and a( 2, 2 ) respectively, and so on.
!>           Before entry with UPLO = 'L' or 'l', the array AP must
!>           contain the lower triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
!>           and a( 3, 1 ) respectively, and so on.
!>
```

*X*

```
!>          X is DOUBLE PRECISION array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the n
!>           element vector x.
!>
```

*INCX*

```
!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX must not be zero.
!>
```

*BETA*

```
!>          BETA is DOUBLE PRECISION.
!>           On entry, BETA specifies the scalar beta. When BETA is
!>           supplied as zero then Y need not be set on input.
!>
```

*Y*

```
!>          Y is DOUBLE PRECISION array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCY ) ).
!>           Before entry, the incremented array Y must contain the n
!>           element vector y. On exit, Y is overwritten by the updated
!>           vector y.
!>
```

*INCY*

```
!>          INCY is INTEGER
!>           On entry, INCY specifies the increment for the elements of
!>           Y. INCY must not be zero.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

```
!>
!>  Level 2 Blas routine.
!>  The vector and matrix arguments are not referenced when N = 0, or M = 0
!>
!>  -- Written on 22-October-1986.
!>     Jack Dongarra, Argonne National Lab.
!>     Jeremy Du Croz, Nag Central Office.
!>     Sven Hammarling, Nag Central Office.
!>     Richard Hanson, Sandia National Labs.
!>
```

Definition at line **146** of file **dspmv.f**.

### subroutine sspmv (character uplo, integer n, real alpha, real, dimension(\*) ap, real, dimension(\*) x, integer incx, real beta, real, dimension(\*) y, integer incy)

**SSPMV**

**Purpose:**

```
!>
!> SSPMV  performs the matrix-vector operation
!>
!>    y := alpha*A*x + beta*y,
!>
!> where alpha and beta are scalars, x and y are n element vectors and
!> A is an n by n symmetric matrix, supplied in packed form.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the upper or lower
!>           triangular part of the matrix A is supplied in the packed
!>           array AP as follows:
!>
!>              UPLO = 'U' or 'u'   The upper triangular part of A is
!>                                  supplied in AP.
!>
!>              UPLO = 'L' or 'l'   The lower triangular part of A is
!>                                  supplied in AP.
!>
```

*N*

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

*ALPHA*

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

*AP*

```
!>          AP is REAL array, dimension at least
!>           ( ( n*( n + 1 ) )/2 ).
!>           Before entry with UPLO = 'U' or 'u', the array AP must
!>           contain the upper triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
!>           and a( 2, 2 ) respectively, and so on.
!>           Before entry with UPLO = 'L' or 'l', the array AP must
!>           contain the lower triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
!>           and a( 3, 1 ) respectively, and so on.
!>
```

*X*

```
!>          X is REAL array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the n
!>           element vector x.
!>
```

*INCX*

```
!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX must not be zero.
!>
```

*BETA*

```
!>          BETA is REAL
!>           On entry, BETA specifies the scalar beta. When BETA is
!>           supplied as zero then Y need not be set on input.
!>
```

*Y*

```
!>          Y is REAL array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCY ) ).
!>           Before entry, the incremented array Y must contain the n
!>           element vector y. On exit, Y is overwritten by the updated
!>           vector y.
!>
```

*INCY*

```
!>          INCY is INTEGER
!>           On entry, INCY specifies the increment for the elements of
!>           Y. INCY must not be zero.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

```
!>
!>  Level 2 Blas routine.
!>  The vector and matrix arguments are not referenced when N = 0, or M = 0
!>
!>  -- Written on 22-October-1986.
!>     Jack Dongarra, Argonne National Lab.
!>     Jeremy Du Croz, Nag Central Office.
!>     Sven Hammarling, Nag Central Office.
!>     Richard Hanson, Sandia National Labs.
!>
```

Definition at line **146** of file **sspmv.f**.

### subroutine zhpmv (character uplo, integer n, complex\*16 alpha, complex\*16, dimension(\*) ap, complex\*16, dimension(\*) x, integer incx, complex\*16 beta, complex\*16, dimension(\*) y, integer incy)

**ZHPMV**

**Purpose:**

```
!>
!> ZHPMV  performs the matrix-vector operation
!>
!>    y := alpha*A*x + beta*y,
!>
!> where alpha and beta are scalars, x and y are n element vectors and
!> A is an n by n hermitian matrix, supplied in packed form.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the upper or lower
!>           triangular part of the matrix A is supplied in the packed
!>           array AP as follows:
!>
!>              UPLO = 'U' or 'u'   The upper triangular part of A is
!>                                  supplied in AP.
!>
!>              UPLO = 'L' or 'l'   The lower triangular part of A is
!>                                  supplied in AP.
!>
```

*N*

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

*ALPHA*

```
!>          ALPHA is COMPLEX*16
!>           On entry, ALPHA specifies the scalar alpha.
!>
```

*AP*

```
!>          AP is COMPLEX*16 array, dimension at least
!>           ( ( n*( n + 1 ) )/2 ).
!>           Before entry with UPLO = 'U' or 'u', the array AP must
!>           contain the upper triangular part of the hermitian matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
!>           and a( 2, 2 ) respectively, and so on.
!>           Before entry with UPLO = 'L' or 'l', the array AP must
!>           contain the lower triangular part of the hermitian matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
!>           and a( 3, 1 ) respectively, and so on.
!>           Note that the imaginary parts of the diagonal elements need
!>           not be set and are assumed to be zero.
!>
```

*X*

```
!>          X is COMPLEX*16 array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the n
!>           element vector x.
!>
```

*INCX*

```
!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX must not be zero.
!>
```

*BETA*

```
!>          BETA is COMPLEX*16
!>           On entry, BETA specifies the scalar beta. When BETA is
!>           supplied as zero then Y need not be set on input.
!>
```

*Y*

```
!>          Y is COMPLEX*16 array, dimension at least
!>           ( 1 + ( n - 1 )*abs( INCY ) ).
!>           Before entry, the incremented array Y must contain the n
!>           element vector y. On exit, Y is overwritten by the updated
!>           vector y.
!>
```

*INCY*

```
!>          INCY is INTEGER
!>           On entry, INCY specifies the increment for the elements of
!>           Y. INCY must not be zero.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

```
!>
!>  Level 2 Blas routine.
!>  The vector and matrix arguments are not referenced when N = 0, or M = 0
!>
!>  -- Written on 22-October-1986.
!>     Jack Dongarra, Argonne National Lab.
!>     Jeremy Du Croz, Nag Central Office.
!>     Sven Hammarling, Nag Central Office.
!>     Richard Hanson, Sandia National Labs.
!>
```

Definition at line **148** of file **zhpmv.f**.

### subroutine zspmv (character uplo, integer n, complex\*16 alpha, complex\*16, dimension( \* ) ap, complex\*16, dimension( \* ) x, integer incx, complex\*16 beta, complex\*16, dimension( \* ) y, integer incy)

**ZSPMV** computes a matrix-vector product for complex vectors using a complex symmetric packed matrix

**Purpose:**

```
!>
!> ZSPMV  performs the matrix-vector operation
!>
!>    y := alpha*A*x + beta*y,
!>
!> where alpha and beta are scalars, x and y are n element vectors and
!> A is an n by n symmetric matrix, supplied in packed form.
!>
```

**Parameters**

*UPLO*

```
!>          UPLO is CHARACTER*1
!>           On entry, UPLO specifies whether the upper or lower
!>           triangular part of the matrix A is supplied in the packed
!>           array AP as follows:
!>
!>              UPLO = 'U' or 'u'   The upper triangular part of A is
!>                                  supplied in AP.
!>
!>              UPLO = 'L' or 'l'   The lower triangular part of A is
!>                                  supplied in AP.
!>
!>           Unchanged on exit.
!>
```

*N*

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

*ALPHA*

```
!>          ALPHA is COMPLEX*16
!>           On entry, ALPHA specifies the scalar alpha.
!>           Unchanged on exit.
!>
```

*AP*

```
!>          AP is COMPLEX*16 array, dimension at least
!>           ( ( N*( N + 1 ) )/2 ).
!>           Before entry, with UPLO = 'U' or 'u', the array AP must
!>           contain the upper triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
!>           and a( 2, 2 ) respectively, and so on.
!>           Before entry, with UPLO = 'L' or 'l', the array AP must
!>           contain the lower triangular part of the symmetric matrix
!>           packed sequentially, column by column, so that AP( 1 )
!>           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
!>           and a( 3, 1 ) respectively, and so on.
!>           Unchanged on exit.
!>
```

*X*

```
!>          X is COMPLEX*16 array, dimension at least
!>           ( 1 + ( N - 1 )*abs( INCX ) ).
!>           Before entry, the incremented array X must contain the N-
!>           element vector x.
!>           Unchanged on exit.
!>
```

*INCX*

```
!>          INCX is INTEGER
!>           On entry, INCX specifies the increment for the elements of
!>           X. INCX must not be zero.
!>           Unchanged on exit.
!>
```

*BETA*

```
!>          BETA is COMPLEX*16
!>           On entry, BETA specifies the scalar beta. When BETA is
!>           supplied as zero then Y need not be set on input.
!>           Unchanged on exit.
!>
```

*Y*

```
!>          Y is COMPLEX*16 array, dimension at least
!>           ( 1 + ( N - 1 )*abs( INCY ) ).
!>           Before entry, the incremented array Y must contain the n
!>           element vector y. On exit, Y is overwritten by the updated
!>           vector y.
!>
```

*INCY*

```
!>          INCY is INTEGER
!>           On entry, INCY specifies the increment for the elements of
!>           Y. INCY must not be zero.
!>           Unchanged on exit.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line **150** of file **zspmv.f**.

## Author

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