dgehrd(3P) Sun Performance Library dgehrd(3P)NAMEdgehrd - reduce a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation
SYNOPSIS
SUBROUTINE DGEHRD(N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
INTEGER N, ILO, IHI, LDA, LWORKIN, INFO
DOUBLE PRECISION A(LDA,*), TAU(*), WORKIN(*)
SUBROUTINE DGEHRD_64(N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
INTEGER*8 N, ILO, IHI, LDA, LWORKIN, INFO
DOUBLE PRECISION A(LDA,*), TAU(*), WORKIN(*)
F95 INTERFACE
SUBROUTINE GEHRD([N], ILO, IHI, A, [LDA], TAU, [WORKIN], [LWORKIN],
[INFO])
INTEGER :: N, ILO, IHI, LDA, LWORKIN, INFO
REAL(8), DIMENSION(:) :: TAU, WORKIN
REAL(8), DIMENSION(:,:) :: A
SUBROUTINE GEHRD_64([N], ILO, IHI, A, [LDA], TAU, [WORKIN], [LWORKIN],
[INFO])
INTEGER(8) :: N, ILO, IHI, LDA, LWORKIN, INFO
REAL(8), DIMENSION(:) :: TAU, WORKIN
REAL(8), DIMENSION(:,:) :: A
C INTERFACE
#include <sunperf.h>
void dgehrd(int n, int ilo, int ihi, double *a, int lda, double *tau,
int *info);
void dgehrd_64(long n, long ilo, long ihi, double *a, long lda, double
*tau, long *info);
PURPOSEdgehrd reduces a real general matrix A to upper Hessenberg form H by an
orthogonal 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 DGEBAL; 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 orthogonal matrix Q as a product of elementary reflec‐
tors. See Further Details.
LDA (input)
The leading dimension of the array A. LDA >= max(1,N).
TAU (output) 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.
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 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 modi‐
fied element of the upper Hessenberg matrix H, and vi denotes an ele‐
ment of the vector defining H(i).
6 Mar 2009 dgehrd(3P)