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Manual

gehd2

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgehd2 (integer n, integer ilo, integer ihi, complex, dimension(lda, * ) a, integer lda, complex, dimension( * ) tau, complex,dimension( * ) work, integer info)
subroutine dgehd2 (integer n, integer ilo, integer ihi, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( * )tau, double precision, dimension( * ) work, integer info)
subroutine sgehd2 (integer n, integer ilo, integer ihi, real, dimension(lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * )work, integer info)
subroutine zgehd2 (integer n, integer ilo, integer ihi, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau,complex*16, dimension( * ) work, integer info)
Author

NAME

gehd2 - gehd2: reduction to Hessenberg, level 2

SYNOPSIS

Functions

subroutine cgehd2 (n, ilo, ihi, a, lda, tau, work, info)
CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine dgehd2 (n, ilo, ihi, a, lda, tau, work, info)
DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine sgehd2 (n, ilo, ihi, a, lda, tau, work, info)
SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.
subroutine zgehd2 (n, ilo, ihi, a, lda, tau, work, info)
ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Detailed Description

Function Documentation

subroutine cgehd2 (integer n, integer ilo, integer ihi, complex, dimension(lda, * ) a, integer lda, complex, dimension( * ) tau, complex,dimension( * ) work, integer info)

CGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

CGEHD2 reduces a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation: Q**H * A * Q = H .

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to CGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).

A is COMPLEX array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the unitary matrix Q as a product of elementary
reflectors. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TAU

TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is COMPLEX array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

subroutine dgehd2 (integer n, integer ilo, integer ihi, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( * )tau, double precision, dimension( * ) work, integer info)

DGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H .

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to DGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is DOUBLE PRECISION array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

subroutine sgehd2 (integer n, integer ilo, integer ihi, real, dimension(lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * )work, integer info)

SGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q**T * A * Q = H .

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to SGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).

A is REAL array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TAU

TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is REAL array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

subroutine zgehd2 (integer n, integer ilo, integer ihi, complex16,dimension( lda, ) a, integer lda, complex16, dimension( ) tau,complex16, dimension( ) work, integer info)

ZGEHD2 reduces a general square matrix to upper Hessenberg form using an unblocked algorithm.

Purpose:

ZGEHD2 reduces a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation: Q**H * A * Q = H .

Parameters

N is INTEGER
The order of the matrix A. N >= 0.

ILO

ILO is INTEGER

IHI

IHI is INTEGER

It is assumed that A is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
set by a previous call to ZGEBAL; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ILO <= IHI <= max(1,N).

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the n by n general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array TAU,
represent the unitary matrix Q as a product of elementary
reflectors. See Further Details.

LDA

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).

TAU

TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

WORK

WORK is COMPLEX*16 array, dimension (N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors

Q = H(ilo) H(ilo+1) . . . H(ihi-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in TAU(i).

The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:

on entry, on exit,

where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).

Author

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