ZGEHRD(1) LAPACK routine (version 3.2) ZGEHRD(1)NAMEZGEHRD - reduces a complex general matrix A to upper Hessenberg form H
by an unitary similarity transformation
SYNOPSIS
SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
INTEGER IHI, ILO, INFO, LDA, LWORK, N
COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
PURPOSEZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
an unitary 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 ZGEBAL; otherwise they
should be set to 1 and N respectively. See Further Details.
A (input/output) 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 overwrit‐
ten with the upper Hessenberg matrix H, and the elements below
the first subdiagonal, with the array TAU, represent the uni‐
tary matrix Q as a product of elementary reflectors. See Fur‐
ther Details. LDA (input) INTEGER The leading dimension of
the array A. LDA >= max(1,N).
TAU (output) 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 (workspace/output) COMPLEX*16 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 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 ZGEHRD subroutine
incorporating improvements proposed by Quintana-Orti and Van de Geijn
(2005).
LAPACK routine (version 3.2) November 2008 ZGEHRD(1)