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
|
.TH CLATRD l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
CLATRD - reduce NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A
.SH SYNOPSIS
.TP 19
SUBROUTINE CLATRD(
UPLO, N, NB, A, LDA, E, TAU, W, LDW )
.TP 19
.ti +4
CHARACTER
UPLO
.TP 19
.ti +4
INTEGER
LDA, LDW, N, NB
.TP 19
.ti +4
REAL
E( * )
.TP 19
.ti +4
COMPLEX
A( LDA, * ), TAU( * ), W( LDW, * )
.SH PURPOSE
CLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q' * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.
If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
.br
if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
.br
This is an auxiliary routine called by CHETRD.
.br
.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
.br
= 'U': Upper triangular
.br
= 'L': Lower triangular
.TP 8
N (input) INTEGER
The order of the matrix A.
.TP 8
NB (input) INTEGER
The number of rows and columns to be reduced.
.TP 8
A (input/output) COMPLEX array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit:
if UPLO = 'U', the last NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements above the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors;
if UPLO = 'L', the first NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements below the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors.
See Further Details.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
.TP 8
E (output) REAL array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
elements of the last NB columns of the reduced matrix;
if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
the first NB columns of the reduced matrix.
.TP 8
TAU (output) COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors, stored in
TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
See Further Details.
W (output) COMPLEX array, dimension (LDW,NB)
The n-by-nb matrix W required to update the unreduced part
of A.
.TP 8
LDW (input) INTEGER
The leading dimension of the array W. LDW >= max(1,N).
.SH FURTHER DETAILS
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
.br
Q = H(n) H(n-1) . . . H(n-nb+1).
.br
Each H(i) has the form
.br
H(i) = I - tau * v * v'
.br
where tau is a complex scalar, and v is a complex vector with
v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
and tau in TAU(i-1).
.br
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
.br
Q = H(1) H(2) . . . H(nb).
.br
Each H(i) has the form
.br
H(i) = I - tau * v * v'
.br
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and tau in TAU(i).
.br
The elements of the vectors v together form the n-by-nb matrix V
which is needed, with W, to apply the transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W' - W*V'.
.br
The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:
.br
if UPLO = 'U': if UPLO = 'L':
.br
( a a a v4 v5 ) ( d )
( a a v4 v5 ) ( 1 d )
( a 1 v5 ) ( v1 1 a )
( d 1 ) ( v1 v2 a a )
( d ) ( v1 v2 a a a )
where d denotes a diagonal element of the reduced matrix, a denotes
an element of the original matrix that is unchanged, and vi denotes
an element of the vector defining H(i).
.br
|
