DGEHRD - reduce a real general matrix A to upper Hessen
berg form H by an orthogonal similarity transformation
SYNOPSIS
SUBROUTINE DGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK,
INFO )
INTEGER IHI, ILO, INFO, LDA, LWORK, N
DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK(
* )
PURPOSE
DGEHRD reduces a real general matrix A to upper Hessenberg
form H by an orthogonal 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 DGEBAL; otherwise they
should be set to 1 and N respectively. See Further
Details.
A (input/output) DOUBLE PRECISION array, dimension
(LDA,N)
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)
INTEGER The leading dimension of the array A. LDA
>= max(1,N).
TAU (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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).
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 ille
gal value.
FURTHER DETAILS
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 exam
ple, 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).
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