1DLAHRD(1)           LAPACK auxiliary routine (version 3.1)           DLAHRD(1)
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NAME

6       DLAHRD  -  the first NB columns of a real general n-by-(n-k+1) matrix A
7       so that elements below the k-th subdiagonal are zero
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SYNOPSIS

10       SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
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12           INTEGER        K, LDA, LDT, LDY, N, NB
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14           DOUBLE         PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB  ),  Y(
15                          LDY, NB )
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PURPOSE

18       DLAHRD  reduces  the  first  NB  columns of a real general n-by-(n-k+1)
19       matrix A so that elements below the  k-th  subdiagonal  are  zero.  The
20       reduction  is performed by an orthogonal similarity transformation Q' *
21       A * Q. The routine returns the matrices V and T which determine Q as  a
22       block reflector I - V*T*V', and also the matrix Y = A * V * T.
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24       This is an OBSOLETE auxiliary routine.
25       This routine will be 'deprecated' in a  future release.
26       Please use the new routine DLAHR2 instead.
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ARGUMENTS

30       N       (input) INTEGER
31               The order of the matrix A.
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33       K       (input) INTEGER
34               The offset for the reduction. Elements below the k-th subdiago‐
35               nal in the first NB columns are reduced to zero.
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37       NB      (input) INTEGER
38               The number of columns to be reduced.
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40       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
41               On entry, the n-by-(n-k+1) general matrix A.  On exit, the ele‐
42               ments on and above the k-th subdiagonal in the first NB columns
43               are overwritten with the corresponding elements of the  reduced
44               matrix; the elements below the k-th subdiagonal, with the array
45               TAU, represent the matrix Q as a product of elementary  reflec‐
46               tors.  The  other  columns  of  A  are  unchanged.  See Further
47               Details.  LDA     (input) INTEGER The leading dimension of  the
48               array A.  LDA >= max(1,N).
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50       TAU     (output) DOUBLE PRECISION array, dimension (NB)
51               The  scalar  factors  of the elementary reflectors. See Further
52               Details.
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54       T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
55               The upper triangular matrix T.
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57       LDT     (input) INTEGER
58               The leading dimension of the array T.  LDT >= NB.
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60       Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
61               The n-by-nb matrix Y.
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63       LDY     (input) INTEGER
64               The leading dimension of the array Y. LDY >= N.
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FURTHER DETAILS

67       The matrix Q is represented as a product of nb elementary reflectors
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69          Q = H(1) H(2) . . . H(nb).
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71       Each H(i) has the form
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73          H(i) = I - tau * v * v'
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75       where tau is a real scalar, and v is a real vector with
76       v(1:i+k-1)  =  0,  v(i+k)  =  1;  v(i+k+1:n)  is  stored  on  exit   in
77       A(i+k+1:n,i), and tau in TAU(i).
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79       The  elements of the vectors v together form the (n-k+1)-by-nb matrix V
80       which is needed, with T and Y, to apply the transformation to the unre‐
81       duced  part  of  the  matrix,  using  an  update of the form: A := (I -
82       V*T*V') * (A - Y*V').
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84       The contents of A on exit are illustrated by the following example with
85       n = 7, k = 3 and nb = 2:
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87          ( a   h   a   a   a )
88          ( a   h   a   a   a )
89          ( a   h   a   a   a )
90          ( h   h   a   a   a )
91          ( v1  h   a   a   a )
92          ( v1  v2  a   a   a )
93          ( v1  v2  a   a   a )
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95       where  a denotes an element of the original matrix A, h denotes a modi‐
96       fied element of the upper Hessenberg matrix H, and vi denotes  an  ele‐
97       ment of the vector defining H(i).
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102 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       DLAHRD(1)