cgehrd(3P) Sun Performance Library cgehrd(3P)NAMEcgehrd - reduce a complex general matrix A to upper Hessenberg form H
by a unitary similarity transformation
SYNOPSIS
SUBROUTINE CGEHRD(N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
COMPLEX A(LDA,*), TAU(*), WORKIN(*)
INTEGER N, ILO, IHI, LDA, LWORKIN, INFO
SUBROUTINE CGEHRD_64(N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
COMPLEX A(LDA,*), TAU(*), WORKIN(*)
INTEGER*8 N, ILO, IHI, LDA, LWORKIN, INFO
F95 INTERFACE
SUBROUTINE GEHRD([N], ILO, IHI, A, [LDA], TAU, [WORKIN], [LWORKIN],
[INFO])
COMPLEX, DIMENSION(:) :: TAU, WORKIN
COMPLEX, DIMENSION(:,:) :: A
INTEGER :: N, ILO, IHI, LDA, LWORKIN, INFO
SUBROUTINE GEHRD_64([N], ILO, IHI, A, [LDA], TAU, [WORKIN], [LWORKIN],
[INFO])
COMPLEX, DIMENSION(:) :: TAU, WORKIN
COMPLEX, DIMENSION(:,:) :: A
INTEGER(8) :: N, ILO, IHI, LDA, LWORKIN, INFO
C INTERFACE
#include <sunperf.h>
void cgehrd(int n, int ilo, int ihi, complex *a, int lda, complex *tau,
int *info);
void cgehrd_64(long n, long ilo, long ihi, complex *a, long lda, com‐
plex *tau, long *info);
PURPOSEcgehrd reduces a complex general matrix A to upper Hessenberg form H by
a unitary similarity transformation: Q' * A * Q = H .
ARGUMENTS
N (input) The order of the matrix A. N >= 0.
ILO (input)
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.
IHI (input)
See the description of ILO.
A (input/output)
On entry, the N-by-N general matrix to be reduced. On exit,
the upper triangle and the first subdiagonal of A are over‐
written 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 (input)
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
Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
zero.
WORKIN (workspace)
On exit, if INFO = 0, WORKIN(1) returns the optimal LWORKIN.
LWORKIN (input)
The length of the array WORKIN. LWORKIN >= max(1,N). For
optimum performance LWORKIN >= N*NB, where NB is the optimal
blocksize.
If LWORKIN = -1, then a workspace query is assumed; the rou‐
tine only calculates the optimal size of the WORKIN array,
returns this value as the first entry of the WORKIN array,
and no error message related to LWORKIN is issued by XERBLA.
INFO (output)
= 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 modi‐
fied element of the upper Hessenberg matrix H, and vi denotes an ele‐
ment of the vector defining H(i).
6 Mar 2009 cgehrd(3P)
