hpr(3)

NAME
hpr - {hp,sp}r: Hermitian/symmetric rank-1 update

SYNOPSIS

  Functions

subroutine chpr (uplo, n, alpha, x, incx, ap)
CHPR
subroutine dspr (uplo, n, alpha, x, incx, ap)
DSPR
subroutine sspr (uplo, n, alpha, x, incx, ap)
SSPR
subroutine zhpr (uplo, n, alpha, x, incx, ap)
ZHPR
subroutine cspr (uplo, n, alpha, x, incx, ap)
CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.
subroutine zspr (uplo, n, alpha, x, incx, ap)
ZSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.

Detailed Description

Function Documentation

  subroutine chpr (character uplo, integer n, real alpha, complex, dimension(*) x, integer incx, complex, dimension(*) ap)

CHPR

Purpose:

    !>
    !> CHPR    performs the hermitian rank 1 operation
    !>
    !>    A := alpha*x*x**H + A,
    !>
    !> where alpha is a real scalar, x is an n element vector 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 REAL
    !>           On entry, ALPHA specifies the scalar alpha.
    !>

    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.
    !>

    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. On exit, the array
    !>           AP is overwritten by the upper triangular part of the
    !>           updated matrix.
    !>           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. On exit, the array
    !>           AP 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Level 2 Blas routine.
    !>
    !>  -- 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.
    !>

  subroutine cspr (character uplo, integer n, complex alpha, complex, dimension( * ) x, integer incx, complex, dimension( * ) ap)

CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.

Purpose:

    !>
    !> CSPR    performs the symmetric rank 1 operation
    !>
    !>    A := alpha*x*x**H + A,
    !>
    !> where alpha is a complex scalar, x is an n element vector 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.
    !>

    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.
    !>

    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. On exit, the array
    !>           AP is overwritten by the upper triangular part of the
    !>           updated matrix.
    !>           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. On exit, the array
    !>           AP 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dspr (character uplo, integer n, double precision alpha, double precision, dimension(*) x, integer incx, double precision, dimension(*) ap)

DSPR

Purpose:

    !>
    !> DSPR    performs the symmetric rank 1 operation
    !>
    !>    A := alpha*x*x**T + A,
    !>
    !> where alpha is a real scalar, x is an n element vector 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.
    !>

    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.
    !>

    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. On exit, the array
    !>           AP is overwritten by the upper triangular part of the
    !>           updated matrix.
    !>           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. On exit, the array
    !>           AP is overwritten by the lower triangular part of the
    !>           updated matrix.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Level 2 Blas routine.
    !>
    !>  -- 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.
    !>

  subroutine sspr (character uplo, integer n, real alpha, real, dimension(*) x, integer incx, real, dimension(*) ap)

SSPR

Purpose:

    !>
    !> SSPR    performs the symmetric rank 1 operation
    !>
    !>    A := alpha*x*x**T + A,
    !>
    !> where alpha is a real scalar, x is an n element vector 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.
    !>

    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.
    !>

    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. On exit, the array
    !>           AP is overwritten by the upper triangular part of the
    !>           updated matrix.
    !>           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. On exit, the array
    !>           AP is overwritten by the lower triangular part of the
    !>           updated matrix.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Level 2 Blas routine.
    !>
    !>  -- 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.
    !>

  subroutine zhpr (character uplo, integer n, double precision alpha, complex*16, dimension(*) x, integer incx, complex*16, dimension(*) ap)

ZHPR

Purpose:

    !>
    !> ZHPR    performs the hermitian rank 1 operation
    !>
    !>    A := alpha*x*x**H + A,
    !>
    !> where alpha is a real scalar, x is an n element vector 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 DOUBLE PRECISION.
    !>           On entry, ALPHA specifies the scalar alpha.
    !>

    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.
    !>

    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. On exit, the array
    !>           AP is overwritten by the upper triangular part of the
    !>           updated matrix.
    !>           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. On exit, the array
    !>           AP 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  Level 2 Blas routine.
    !>
    !>  -- 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.
    !>

  subroutine zspr (character uplo, integer n, complex*16 alpha, complex*16, dimension( * ) x, integer incx, complex*16, dimension( * ) ap)

ZSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.

Purpose:

    !>
    !> ZSPR    performs the symmetric rank 1 operation
    !>
    !>    A := alpha*x*x**H + A,
    !>
    !> where alpha is a complex scalar, x is an n element vector 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.
    !>

    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.
    !>

    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. On exit, the array
    !>           AP is overwritten by the upper triangular part of the
    !>           updated matrix.
    !>           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. On exit, the array
    !>           AP 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.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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