SGEHRD(3f)

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

SYNOPSIS

SUBROUTINE SGEHRD(
    N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )

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

REAL
    A( LDA, * ), TAU( * ), WORK( * )

PURPOSE
SGEHRD 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 SGEBAL; otherwise they should be set to 1 and N respectively. See Further Details.

A       (input/output) REAL 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) 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.

WORK    (workspace/output) REAL 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.

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 modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).
This file is a slight modification of LAPACK-3.0's SGEHRD subroutine incorporating improvements proposed by Quintana-Orti and Van de Geijn (2005).