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
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
|
*> \brief \b ZLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLAQR1 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr1.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr1.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr1.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLAQR1( N, H, LDH, S1, S2, V )
*
* .. Scalar Arguments ..
* COMPLEX*16 S1, S2
* INTEGER LDH, N
* ..
* .. Array Arguments ..
* COMPLEX*16 H( LDH, * ), V( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> Given a 2-by-2 or 3-by-3 matrix H, ZLAQR1 sets v to a
*> scalar multiple of the first column of the product
*>
*> (*) K = (H - s1*I)*(H - s2*I)
*>
*> scaling to avoid overflows and most underflows.
*>
*> This is useful for starting double implicit shift bulges
*> in the QR algorithm.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is integer
*> Order of the matrix H. N must be either 2 or 3.
*> \endverbatim
*>
*> \param[in] H
*> \verbatim
*> H is COMPLEX*16 array of dimension (LDH,N)
*> The 2-by-2 or 3-by-3 matrix H in (*).
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is integer
*> The leading dimension of H as declared in
*> the calling procedure. LDH.GE.N
*> \endverbatim
*>
*> \param[in] S1
*> \verbatim
*> S1 is COMPLEX*16
*> \endverbatim
*>
*> \param[in] S2
*> \verbatim
*> S2 is COMPLEX*16
*>
*> S1 and S2 are the shifts defining K in (*) above.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is COMPLEX*16 array of dimension N
*> A scalar multiple of the first column of the
*> matrix K in (*).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup complex16OTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*>
* =====================================================================
SUBROUTINE ZLAQR1( N, H, LDH, S1, S2, V )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
COMPLEX*16 S1, S2
INTEGER LDH, N
* ..
* .. Array Arguments ..
COMPLEX*16 H( LDH, * ), V( * )
* ..
*
* ================================================================
*
* .. Parameters ..
COMPLEX*16 ZERO
PARAMETER ( ZERO = ( 0.0d0, 0.0d0 ) )
DOUBLE PRECISION RZERO
PARAMETER ( RZERO = 0.0d0 )
* ..
* .. Local Scalars ..
COMPLEX*16 CDUM, H21S, H31S
DOUBLE PRECISION S
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DIMAG
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
* ..
* .. Executable Statements ..
IF( N.EQ.2 ) THEN
S = CABS1( H( 1, 1 )-S2 ) + CABS1( H( 2, 1 ) )
IF( S.EQ.RZERO ) THEN
V( 1 ) = ZERO
V( 2 ) = ZERO
ELSE
H21S = H( 2, 1 ) / S
V( 1 ) = H21S*H( 1, 2 ) + ( H( 1, 1 )-S1 )*
$ ( ( H( 1, 1 )-S2 ) / S )
V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-S1-S2 )
END IF
ELSE
S = CABS1( H( 1, 1 )-S2 ) + CABS1( H( 2, 1 ) ) +
$ CABS1( H( 3, 1 ) )
IF( S.EQ.ZERO ) THEN
V( 1 ) = ZERO
V( 2 ) = ZERO
V( 3 ) = ZERO
ELSE
H21S = H( 2, 1 ) / S
H31S = H( 3, 1 ) / S
V( 1 ) = ( H( 1, 1 )-S1 )*( ( H( 1, 1 )-S2 ) / S ) +
$ H( 1, 2 )*H21S + H( 1, 3 )*H31S
V( 2 ) = H21S*( H( 1, 1 )+H( 2, 2 )-S1-S2 ) + H( 2, 3 )*H31S
V( 3 ) = H31S*( H( 1, 1 )+H( 3, 3 )-S1-S2 ) + H21S*H( 3, 2 )
END IF
END IF
END
|
