1SGEHD2(1)                LAPACK routine (version 3.1)                SGEHD2(1)
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NAME

6       SGEHD2  -  a  real  general  matrix  A to upper Hessenberg form H by an
7       orthogonal similarity transformation
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SYNOPSIS

10       SUBROUTINE SGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
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12           INTEGER        IHI, ILO, INFO, LDA, N
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14           REAL           A( LDA, * ), TAU( * ), WORK( * )
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PURPOSE

17       SGEHD2 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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ARGUMENTS

22       N       (input) INTEGER
23               The order of the matrix A.  N >= 0.
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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 SGEBAL; otherwise they
29               should be set to 1 and N respectively. See Further Details.
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31       A       (input/output) REAL 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).
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40       TAU     (output) REAL array, dimension (N-1)
41               The scalar factors of the elementary  reflectors  (see  Further
42               Details).
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44       WORK    (workspace) REAL array, dimension (N)
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46       INFO    (output) INTEGER
47               = 0:  successful exit.
48               < 0:  if INFO = -i, the i-th argument had an illegal value.
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FURTHER DETAILS

51       The  matrix  Q  is  represented  as  a  product of (ihi-ilo) elementary
52       reflectors
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54          Q = H(ilo) H(ilo+1) . . . H(ihi-1).
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56       Each H(i) has the form
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58          H(i) = I - tau * v * v'
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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).
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64       The contents of A are illustrated by the following example, with n = 7,
65       ilo = 2 and ihi = 6:
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67       on entry,                        on exit,
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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 )
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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).
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84 LAPACK routine (version 3.1)    November 2006                       SGEHD2(1)