1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
|
---
:name: zlahr2
:md5sum: 9ac30357bebcece7da5d61625afa8bb7
:category: :subroutine
:arguments:
- n:
:type: integer
:intent: input
- k:
:type: integer
:intent: input
- nb:
:type: integer
:intent: input
- a:
:type: doublecomplex
:intent: input/output
:dims:
- lda
- n-k+1
- lda:
:type: integer
:intent: input
- tau:
:type: doublecomplex
:intent: output
:dims:
- MAX(1,nb)
- t:
:type: doublecomplex
:intent: output
:dims:
- ldt
- MAX(1,nb)
- ldt:
:type: integer
:intent: input
- y:
:type: doublecomplex
:intent: output
:dims:
- ldy
- MAX(1,nb)
- ldy:
:type: integer
:intent: input
:substitutions:
ldy: n
ldt: nb
:fortran_help: " SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )\n\n\
* Purpose\n\
* =======\n\
*\n\
* ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)\n\
* matrix A so that elements below the k-th subdiagonal are zero. The\n\
* reduction is performed by an unitary similarity transformation\n\
* Q' * A * Q. The routine returns the matrices V and T which determine\n\
* Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.\n\
*\n\
* This is an auxiliary routine called by ZGEHRD.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A.\n\
*\n\
* K (input) INTEGER\n\
* The offset for the reduction. Elements below the k-th\n\
* subdiagonal in the first NB columns are reduced to zero.\n\
* K < N.\n\
*\n\
* NB (input) INTEGER\n\
* The number of columns to be reduced.\n\
*\n\
* A (input/output) COMPLEX*16 array, dimension (LDA,N-K+1)\n\
* On entry, the n-by-(n-k+1) general matrix A.\n\
* On exit, the elements on and above the k-th subdiagonal in\n\
* the first NB columns are overwritten with the corresponding\n\
* elements of the reduced matrix; the elements below the k-th\n\
* subdiagonal, with the array TAU, represent the matrix Q as a\n\
* product of elementary reflectors. The other columns of A are\n\
* unchanged. See Further Details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,N).\n\
*\n\
* TAU (output) COMPLEX*16 array, dimension (NB)\n\
* The scalar factors of the elementary reflectors. See Further\n\
* Details.\n\
*\n\
* T (output) COMPLEX*16 array, dimension (LDT,NB)\n\
* The upper triangular matrix T.\n\
*\n\
* LDT (input) INTEGER\n\
* The leading dimension of the array T. LDT >= NB.\n\
*\n\
* Y (output) COMPLEX*16 array, dimension (LDY,NB)\n\
* The n-by-nb matrix Y.\n\
*\n\
* LDY (input) INTEGER\n\
* The leading dimension of the array Y. LDY >= N.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* The matrix Q is represented as a product of nb elementary reflectors\n\
*\n\
* Q = H(1) H(2) . . . H(nb).\n\
*\n\
* Each H(i) has the form\n\
*\n\
* H(i) = I - tau * v * v'\n\
*\n\
* where tau is a complex scalar, and v is a complex vector with\n\
* v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in\n\
* A(i+k+1:n,i), and tau in TAU(i).\n\
*\n\
* The elements of the vectors v together form the (n-k+1)-by-nb matrix\n\
* V which is needed, with T and Y, to apply the transformation to the\n\
* unreduced part of the matrix, using an update of the form:\n\
* A := (I - V*T*V') * (A - Y*V').\n\
*\n\
* The contents of A on exit are illustrated by the following example\n\
* with n = 7, k = 3 and nb = 2:\n\
*\n\
* ( a a a a a )\n\
* ( a a a a a )\n\
* ( a a a a a )\n\
* ( h h a a a )\n\
* ( v1 h a a a )\n\
* ( v1 v2 a a a )\n\
* ( v1 v2 a a a )\n\
*\n\
* where a denotes an element of the original matrix A, h denotes a\n\
* modified element of the upper Hessenberg matrix H, and vi denotes an\n\
* element of the vector defining H(i).\n\
*\n\
* This subroutine is a slight modification of LAPACK-3.0's DLAHRD\n\
* incorporating improvements proposed by Quintana-Orti and Van de\n\
* Gejin. Note that the entries of A(1:K,2:NB) differ from those\n\
* returned by the original LAPACK-3.0's DLAHRD routine. (This\n\
* subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)\n\
*\n\
* References\n\
* ==========\n\
*\n\
* Gregorio Quintana-Orti and Robert van de Geijn, \"Improving the\n\
* performance of reduction to Hessenberg form,\" ACM Transactions on\n\
* Mathematical Software, 32(2):180-194, June 2006.\n\
*\n\
* =====================================================================\n\
*\n"
|
