1DGEHD2(1)                LAPACK routine (version 3.1)                DGEHD2(1)
2
3
4

NAME

6       DGEHD2  -  a  real  general  matrix  A to upper Hessenberg form H by an
7       orthogonal similarity transformation
8

SYNOPSIS

10       SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
11
12           INTEGER        IHI, ILO, INFO, LDA, N
13
14           DOUBLE         PRECISION A( LDA, * ), TAU( * ), WORK( * )
15

PURPOSE

17       DGEHD2 reduces a real general matrix A to upper Hessenberg form H by an
18       orthogonal similarity transformation:  Q' * A * Q = H .
19
20

ARGUMENTS

22       N       (input) INTEGER
23               The order of the matrix A.  N >= 0.
24
25       ILO     (input) INTEGER
26               IHI      (input)  INTEGER It is assumed that A is already upper
27               triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI
28               are  normally  set by a previous call to DGEBAL; otherwise they
29               should be set to 1 and N respectively. See Further Details.
30
31       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
32               On entry, the n by n general matrix to be  reduced.   On  exit,
33               the upper triangle and the first subdiagonal of A are overwrit‐
34               ten with the upper Hessenberg matrix H, and the elements  below
35               the  first  subdiagonal,  with  the  array  TAU,  represent the
36               orthogonal matrix Q as a product of elementary reflectors.  See
37               Further Details.  LDA     (input) INTEGER The leading dimension
38               of the array A.  LDA >= max(1,N).
39
40       TAU     (output) DOUBLE PRECISION array, dimension (N-1)
41               The scalar factors of the elementary  reflectors  (see  Further
42               Details).
43
44       WORK    (workspace) DOUBLE PRECISION array, dimension (N)
45
46       INFO    (output) INTEGER
47               = 0:  successful exit.
48               < 0:  if INFO = -i, the i-th argument had an illegal value.
49

FURTHER DETAILS

51       The  matrix  Q  is  represented  as  a  product of (ihi-ilo) elementary
52       reflectors
53
54          Q = H(ilo) H(ilo+1) . . . H(ihi-1).
55
56       Each H(i) has the form
57
58          H(i) = I - tau * v * v'
59
60       where tau is a real scalar, and v is a real vector with
61       v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit
62       in A(i+2:ihi,i), and tau in TAU(i).
63
64       The contents of A are illustrated by the following example, with n = 7,
65       ilo = 2 and ihi = 6:
66
67       on entry,                        on exit,
68
69       ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a ) (     a
70       a    a    a   a   a )    (      a   h   h   h   h   a ) (     a   a   a
71       a   a   a )    (      h   h   h   h   h   h ) (     a   a   a    a    a
72       a  )     (       v2   h   h   h   h   h ) (     a   a   a   a   a   a )
73       (      v2  v3  h   h   h   h ) (     a   a    a    a    a    a  )     (
74       v2    v3    v4    h    h    h  )  (                          a  )     (
75       a )
76
77       where a denotes an element of the original matrix A, h denotes a  modi‐
78       fied  element  of the upper Hessenberg matrix H, and vi denotes an ele‐
79       ment of the vector defining H(i).
80
81
82
83
84 LAPACK routine (version 3.1)    November 2006                       DGEHD2(1)