DGEHRD(1) LAPACK routine (version 3.2) DGEHRD(1)NAME
DGEHRD - reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation
SYNOPSIS
SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER IHI, ILO, INFO, LDA, LWORK, N
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
DGEHRD reduces a real general matrix A to upper Hessenberg form H by an
orthogonal similarity transformation: Q' * A * Q = H .
ARGUMENTS
N (input) INTEGER
The order of the matrix A. N >= 0.
ILO (input) INTEGER
IHI (input) 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.
A (input/output) 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 overwrit‐
ten 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 (input) INTEGER The leading dimension
of the array A. LDA >= max(1,N).
TAU (output) 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 (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,N). For optimum
performance LWORK >= N*NB, where NB is the optimal blocksize.
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 (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
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'
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,
( 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 DGEHRD subroutine
incorporating improvements proposed by Quintana-Orti and Van de Geijn
(2005).
LAPACK routine (version 3.2) November 2008 DGEHRD(1)
