Man page - gehrd(3)

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Manual

gehrd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgehrd (integer n, integer ilo, integer ihi, complex, dimension(lda, * ) a, integer lda, complex, dimension( * ) tau, complex,dimension( * ) work, integer lwork, integer info)
subroutine dgehrd (integer n, integer ilo, integer ihi, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( * )tau, double precision, dimension( * ) work, integer lwork, integerinfo)
subroutine sgehrd (integer n, integer ilo, integer ihi, real, dimension(lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * )work, integer lwork, integer info)
subroutine zgehrd (integer n, integer ilo, integer ihi, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( * ) tau,complex*16, dimension( * ) work, integer lwork, integer info)
Author

NAME

gehrd - gehrd: reduction to Hessenberg

SYNOPSIS

Functions

subroutine cgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
CGEHRD
subroutine dgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
DGEHRD
subroutine sgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
SGEHRD
subroutine zgehrd (n, ilo, ihi, a, lda, tau, work, lwork, info)
ZGEHRD

Detailed Description

Function Documentation

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

CGEHRD

Purpose:

CGEHRD reduces a complex general matrix A to upper Hessenberg form H by
an 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.

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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

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).

This file is a slight modification of LAPACK-3.0’s CGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See CLAHR2.)

subroutine dgehrd (integer n, integer ilo, integer ihi, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( * )tau, double precision, dimension( * ) work, integer lwork, integerinfo)

DGEHRD

Purpose:

DGEHRD 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.

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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.

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).

This file is a slight modification of LAPACK-3.0’s DGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See DLAHR2.)

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

SGEHRD

Purpose:

SGEHRD 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.

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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.

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).

This file is a slight modification of LAPACK-3.0’s SGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See SLAHR2.)

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

ZGEHRD

Purpose:

ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
an 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 <= N, if N > 0; ILO=1 and IHI=0, if N=0.

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). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK. LWORK >= max(1,N).
For good performance, LWORK should generally be larger.

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).

This file is a slight modification of LAPACK-3.0’s ZGEHRD
subroutine incorporating improvements proposed by Quintana-Orti and
Van de Geijn (2006). (See ZLAHR2.)

Author

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