
SGEHRD(l) ) SGEHRD(l)
NAME
SGEHRD  reduce a real general matrix A to upper Hessenberg form H by an orthogonal simi
larity 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 simi
larity 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:ILO1 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 NbyN 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 (N1)
The scalar factors of the elementary reflectors (see Further Details). Elements
1:ILO1 and IHI:N1 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 ith argument had an illegal value.
FURTHER DETAILS
The matrix Q is represented as a product of (ihiilo) elementary reflectors
Q = H(ilo) H(ilo+1) . . . H(ihi1).
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).
LAPACK version 3.0 15 June 2000 SGEHRD(l) 
