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

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

10       SUBROUTINE ZLAHRD( 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           COMPLEX*16     A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )
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PURPOSE

17       ZLAHRD reduces the first NB columns of a complex  general  n-by-(n-k+1)
18       matrix  A  so  that  elements  below the k-th subdiagonal are zero. The
19       reduction is performed by a unitary similarity transformation Q' * A  *
20       Q.  The  routine  returns  the  matrices V and T which determine Q as a
21       block reflector I - V*T*V', and also the matrix Y = A * V * T.
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23       This is an OBSOLETE auxiliary routine.
24       This routine will be 'deprecated' in a  future release.
25       Please use the new routine ZLAHR2 instead.
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ARGUMENTS

29       N       (input) INTEGER
30               The order of the matrix A.
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32       K       (input) INTEGER
33               The offset for the reduction. Elements below the k-th subdiago‐
34               nal in the first NB columns are reduced to zero.
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36       NB      (input) INTEGER
37               The number of columns to be reduced.
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39       A       (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)
40               On entry, the n-by-(n-k+1) general matrix A.  On exit, the ele‐
41               ments on and above the k-th subdiagonal in the first NB columns
42               are  overwritten with the corresponding elements of the reduced
43               matrix; the elements below the k-th subdiagonal, with the array
44               TAU,  represent the matrix Q as a product of elementary reflec‐
45               tors. The  other  columns  of  A  are  unchanged.  See  Further
46               Details.   LDA     (input) INTEGER The leading dimension of the
47               array A.  LDA >= max(1,N).
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49       TAU     (output) COMPLEX*16 array, dimension (NB)
50               The scalar factors of the elementary  reflectors.  See  Further
51               Details.
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53       T       (output) COMPLEX*16 array, dimension (LDT,NB)
54               The upper triangular matrix T.
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56       LDT     (input) INTEGER
57               The leading dimension of the array T.  LDT >= NB.
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59       Y       (output) COMPLEX*16 array, dimension (LDY,NB)
60               The n-by-nb matrix Y.
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62       LDY     (input) INTEGER
63               The leading dimension of the array Y. LDY >= max(1,N).
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FURTHER DETAILS

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