1SGEHRD(1) LAPACK routine (version 3.2) SGEHRD(1)
2
6 SGEHRD - reduces a real general matrix A to upper Hessenberg form H by
7 an orthogonal similarity transformation
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10 SUBROUTINE SGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
11 INTEGER IHI, ILO, INFO, LDA, LWORK, N
13 REAL A( LDA, * ), TAU( * ), WORK( * )
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17 SGEHRD reduces a real general matrix A to upper Hessenberg form H by an
18 orthogonal similarity transformation: Q' * A * Q = H .
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21 N (input) INTEGER
22 The order of the matrix A. N >= 0.
23 ILO (input) INTEGER
25 IHI (input) INTEGER It is assumed that A is already upper
26 triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI
27 are normally set by a previous call to SGEBAL; otherwise they
28 should be set to 1 and N respectively. See Further Details.
29 A (input/output) REAL array, dimension (LDA,N)
31 On entry, the N-by-N general matrix to be reduced. On exit,
32 the upper triangle and the first subdiagonal of A are overwrit‐
33 ten with the upper Hessenberg matrix H, and the elements below
34 the first subdiagonal, with the array TAU, represent the
35 orthogonal matrix Q as a product of elementary reflectors. See
36 Further Details. LDA (input) INTEGER The leading dimension
37 of the array A. LDA >= max(1,N).
38 TAU (output) REAL array, dimension (N-1)
40 The scalar factors of the elementary reflectors (see Further
41 Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.
42 WORK (workspace/output) REAL array, dimension (LWORK)
44 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
45 LWORK (input) INTEGER
47 The length of the array WORK. LWORK >= max(1,N). For optimum
48 performance LWORK >= N*NB, where NB is the optimal blocksize.
49 If LWORK = -1, then a workspace query is assumed; the routine
50 only calculates the optimal size of the WORK array, returns
51 this value as the first entry of the WORK array, and no error
52 message related to LWORK is issued by XERBLA.
53 INFO (output) INTEGER
55 = 0: successful exit
56 < 0: if INFO = -i, the i-th argument had an illegal value.
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59 The matrix Q is represented as a product of (ihi-ilo) elementary
60 reflectors
61 Q = H(ilo) H(ilo+1) . . . H(ihi-1).
62 Each H(i) has the form
63 H(i) = I - tau * v * v'
64 where tau is a real scalar, and v is a real vector with
65 v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit
66 in A(i+2:ihi,i), and tau in TAU(i).
67 The contents of A are illustrated by the following example, with n = 7,
68 ilo = 2 and ihi = 6:
69 on entry, on exit,
70 ( a a a a a a a ) ( a a h h h h a ) ( a
71 a a a a a ) ( a h h h h a ) ( a a a
72 a a a ) ( h h h h h h ) ( a a a a a
73 a ) ( v2 h h h h h ) ( a a a a a a )
74 ( v2 v3 h h h h ) ( a a a a a a ) (
75 v2 v3 v4 h h h ) ( a ) (
76 a ) where a denotes an element of the original matrix A, h denotes a
77 modified element of the upper Hessenberg matrix H, and vi denotes an
78 element of the vector defining H(i).
79 This file is a slight modification of LAPACK-3.0's SGEHRD subroutine
80 incorporating improvements proposed by Quintana-Orti and Van de Geijn
81 (2005).
82 LAPACK routine (version 3.2) November 2008 SGEHRD(1)