1CLAHR2  ‐  the first NB columns of A complex general n‐BY‐(n‐k+1)
2matrix A so that elements below the  k‐th  subdiagonal  are  zero
3SUBROUTINE CLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
4    INTEGER K, LDA, LDT, LDY, N, NB
5    COMPLEX  A(  LDA,  * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )
6CLAHR2 reduces the first NB columns of A complex general n‐BY‐(n‐
7k+1) matrix A so that elements below the k‐th subdiagonal are ze‐
8ro. The reduction is performed by an unitary similarity transfor‐
9mation Q' * A * Q. The routine returns the matrices V and T which
10determine Q as a block reflector I ‐ V*T*V', and also the  matrix
11Y = A * V * T.
12
13This is an auxiliary routine called by CGEHRD.
14
15N        (input) INTEGER The order of the matrix A.  K       (in‐
16put) INTEGER The offset for the reduction. Elements below the  k‐
17th  subdiagonal in the first NB columns are reduced to zero.  K <
18N.  NB      (input) INTEGER The number of columns to be  reduced.
19A        (input/output)  COMPLEX  array, dimension (LDA,N‐K+1) On
20entry, the n‐by‐(n‐k+1) general matrix A.  On exit, the  elements
21on  and  above  the  k‐th subdiagonal in the first NB columns are
22overwritten with the corresponding elements of  the  reduced  ma‐
23trix;  the  elements  below  the k‐th subdiagonal, with the array
24TAU, represent the matrix Q as a product  of  elementary  reflec‐
25tors.  The other columns of A are unchanged. See Further Details.
26LDA     (input) INTEGER The leading dimension  of  the  array  A.
27LDA  >= max(1,N).  TAU     (output) COMPLEX array, dimension (NB)
28The scalar factors of the elementary reflectors. See Further  De‐
29tails.   T        (output)  COMPLEX array, dimension (LDT,NB) The
30upper triangular matrix T.  LDT     (input) INTEGER  The  leading
31dimension  of  the array T.  LDT >= NB.  Y       (output) COMPLEX
32array, dimension (LDY,NB) The n‐by‐nb matrix Y.  LDY      (input)
33INTEGER  The leading dimension of the array Y. LDY >= N.  The ma‐
34trix Q is represented as a product of nb elementary reflectors
35
36   Q = H(1) H(2) . . . H(nb).
37
38Each H(i) has the form
39
40   H(i) = I ‐ tau * v * v'
41
42where tau is a complex scalar, and v is  a  complex  vector  with
43v(1:i+k‐1)  =  0,  v(i+k)  =  1;  v(i+k+1:n) is stored on exit in
44A(i+k+1:n,i), and tau in TAU(i).
45
46The elements of the vectors v together form the (n‐k+1)‐by‐nb ma‐
47trix V which is needed, with T and Y, to apply the transformation
48to the unreduced part of the matrix, using an update of the form:
49A := (I ‐ V*T*V') * (A ‐ Y*V').
50
51The  contents of A on exit are illustrated by the following exam‐
52ple with n = 7, k = 3 and nb = 2:
53
54   ( a   a   a   a   a )
55   ( a   a   a   a   a )
56   ( a   a   a   a   a )
57   ( h   h   a   a   a )
58   ( v1  h   a   a   a )
59   ( v1  v2  a   a   a )
60   ( v1  v2  a   a   a )
61
62where a denotes an element of the original matrix A, h denotes  a
63modified element of the upper Hessenberg matrix H, and vi denotes
64an element of the vector defining H(i).
65
66This file is a slight modification of LAPACK‐3.0's CLAHRD  incor‐
67porating improvements proposed by Quintana‐Orti and Van de Gejin.
68Note that the entries of A(1:K,2:NB) differ from  those  returned
69by  the  original  LAPACK  routine. This function is not backward
70compatible with LAPACK3.0.
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