Contents
sgehrd - reduce a real general matrix A to upper Hessenberg
form H by an orthogonal similarity transformation
SUBROUTINE SGEHRD(N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
INTEGER N, ILO, IHI, LDA, LWORKIN, INFO
REAL A(LDA,*), TAU(*), WORKIN(*)
SUBROUTINE SGEHRD_64(N, ILO, IHI, A, LDA, TAU, WORKIN, LWORKIN, INFO)
INTEGER*8 N, ILO, IHI, LDA, LWORKIN, INFO
REAL 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, DIMENSION(:) :: TAU, WORKIN
REAL, DIMENSION(:,:) :: A
SUBROUTINE GEHRD_64([N], ILO, IHI, A, [LDA], TAU, [WORKIN], [LWORKIN],
[INFO])
INTEGER(8) :: N, ILO, IHI, LDA, LWORKIN, INFO
REAL, DIMENSION(:) :: TAU, WORKIN
REAL, DIMENSION(:,:) :: A
C INTERFACE
#include <sunperf.h>
void sgehrd(int n, int ilo, int ihi, float *a, int lda,
float *tau, int *info);
void sgehrd_64(long n, long ilo, long ihi, float *a, long
lda, float *tau, long *info);
sgehrd reduces a real general matrix A to upper Hessenberg
form H by an orthogonal similarity transformation: Q' * A *
Q = H .
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 SGEBAL;
otherwise they should be set to 1 and N respec-
tively. 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 subdiag-
onal of A are overwritten with the upper Hessen-
berg 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)
The leading dimension of the array A. LDA >=
max(1,N).
TAU (output)
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 routine 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 ille-
gal value.
The matrix Q is represented as a product of (ihi-ilo) ele-
mentary 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).