DGEHRD - reduce a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation

SUBROUTINE DGEHRD(

N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )

INTEGER IHI, ILO, INFO, LDA, LWORK, N

DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )

DGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal 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 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 subdiagonal of A are overwritten 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 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). For optimum performance LWORK >= N*NB, where NB is 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 illegal value.

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 modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).