CGEHD2(1) LAPACK routine (version 3.2) CGEHD2(1)NAME
CGEHD2 - reduces a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation
SUBROUTINE CGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
INTEGER IHI, ILO, INFO, LDA, N
COMPLEX A( LDA, * ), TAU( * ), WORK( * )
CGEHD2 reduces a complex general matrix A to upper Hessenberg form H by
a unitary similarity transformation: Q' * A * Q = H .
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 CGEBAL; otherwise they
should be set to 1 and N respectively. See Further Details.
A (input/output) 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 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 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
WORK (workspace) COMPLEX array, dimension (N)
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
The matrix Q is represented as a product of (ihi-ilo) elementary
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).
LAPACK routine (version 3.2) November 2008 CGEHD2(1)