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
173
174
175
176
177
178
179
180
181
182
183
184
185
186
|
// Copyright ©2017 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package gonum
import "math"
// Dlags2 computes 2-by-2 orthogonal matrices U, V and Q with the
// triangles of A and B specified by upper.
//
// If upper is true
//
// Uᵀ*A*Q = Uᵀ*[ a1 a2 ]*Q = [ x 0 ]
// [ 0 a3 ] [ x x ]
//
// and
//
// Vᵀ*B*Q = Vᵀ*[ b1 b2 ]*Q = [ x 0 ]
// [ 0 b3 ] [ x x ]
//
// otherwise
//
// Uᵀ*A*Q = Uᵀ*[ a1 0 ]*Q = [ x x ]
// [ a2 a3 ] [ 0 x ]
//
// and
//
// Vᵀ*B*Q = Vᵀ*[ b1 0 ]*Q = [ x x ]
// [ b2 b3 ] [ 0 x ].
//
// The rows of the transformed A and B are parallel, where
//
// U = [ csu snu ], V = [ csv snv ], Q = [ csq snq ]
// [ -snu csu ] [ -snv csv ] [ -snq csq ]
//
// Dlags2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlags2(upper bool, a1, a2, a3, b1, b2, b3 float64) (csu, snu, csv, snv, csq, snq float64) {
if upper {
// Input matrices A and B are upper triangular matrices.
//
// Form matrix C = A*adj(B) = [ a b ]
// [ 0 d ]
a := a1 * b3
d := a3 * b1
b := a2*b1 - a1*b2
// The SVD of real 2-by-2 triangular C.
//
// [ csl -snl ]*[ a b ]*[ csr snr ] = [ r 0 ]
// [ snl csl ] [ 0 d ] [ -snr csr ] [ 0 t ]
_, _, snr, csr, snl, csl := impl.Dlasv2(a, b, d)
if math.Abs(csl) >= math.Abs(snl) || math.Abs(csr) >= math.Abs(snr) {
// Compute the [0, 0] and [0, 1] elements of Uᵀ*A and Vᵀ*B,
// and [0, 1] element of |U|ᵀ*|A| and |V|ᵀ*|B|.
ua11r := csl * a1
ua12 := csl*a2 + snl*a3
vb11r := csr * b1
vb12 := csr*b2 + snr*b3
aua12 := math.Abs(csl)*math.Abs(a2) + math.Abs(snl)*math.Abs(a3)
avb12 := math.Abs(csr)*math.Abs(b2) + math.Abs(snr)*math.Abs(b3)
// Zero [0, 1] elements of Uᵀ*A and Vᵀ*B.
if math.Abs(ua11r)+math.Abs(ua12) != 0 {
if aua12/(math.Abs(ua11r)+math.Abs(ua12)) <= avb12/(math.Abs(vb11r)+math.Abs(vb12)) {
csq, snq, _ = impl.Dlartg(-ua11r, ua12)
} else {
csq, snq, _ = impl.Dlartg(-vb11r, vb12)
}
} else {
csq, snq, _ = impl.Dlartg(-vb11r, vb12)
}
csu = csl
snu = -snl
csv = csr
snv = -snr
} else {
// Compute the [1, 0] and [1, 1] elements of Uᵀ*A and Vᵀ*B,
// and [1, 1] element of |U|ᵀ*|A| and |V|ᵀ*|B|.
ua21 := -snl * a1
ua22 := -snl*a2 + csl*a3
vb21 := -snr * b1
vb22 := -snr*b2 + csr*b3
aua22 := math.Abs(snl)*math.Abs(a2) + math.Abs(csl)*math.Abs(a3)
avb22 := math.Abs(snr)*math.Abs(b2) + math.Abs(csr)*math.Abs(b3)
// Zero [1, 1] elements of Uᵀ*A and Vᵀ*B, and then swap.
if math.Abs(ua21)+math.Abs(ua22) != 0 {
if aua22/(math.Abs(ua21)+math.Abs(ua22)) <= avb22/(math.Abs(vb21)+math.Abs(vb22)) {
csq, snq, _ = impl.Dlartg(-ua21, ua22)
} else {
csq, snq, _ = impl.Dlartg(-vb21, vb22)
}
} else {
csq, snq, _ = impl.Dlartg(-vb21, vb22)
}
csu = snl
snu = csl
csv = snr
snv = csr
}
} else {
// Input matrices A and B are lower triangular matrices
//
// Form matrix C = A*adj(B) = [ a 0 ]
// [ c d ]
a := a1 * b3
d := a3 * b1
c := a2*b3 - a3*b2
// The SVD of real 2-by-2 triangular C
//
// [ csl -snl ]*[ a 0 ]*[ csr snr ] = [ r 0 ]
// [ snl csl ] [ c d ] [ -snr csr ] [ 0 t ]
_, _, snr, csr, snl, csl := impl.Dlasv2(a, c, d)
if math.Abs(csr) >= math.Abs(snr) || math.Abs(csl) >= math.Abs(snl) {
// Compute the [1, 0] and [1, 1] elements of Uᵀ*A and Vᵀ*B,
// and [1, 0] element of |U|ᵀ*|A| and |V|ᵀ*|B|.
ua21 := -snr*a1 + csr*a2
ua22r := csr * a3
vb21 := -snl*b1 + csl*b2
vb22r := csl * b3
aua21 := math.Abs(snr)*math.Abs(a1) + math.Abs(csr)*math.Abs(a2)
avb21 := math.Abs(snl)*math.Abs(b1) + math.Abs(csl)*math.Abs(b2)
// Zero [1, 0] elements of Uᵀ*A and Vᵀ*B.
if (math.Abs(ua21) + math.Abs(ua22r)) != 0 {
if aua21/(math.Abs(ua21)+math.Abs(ua22r)) <= avb21/(math.Abs(vb21)+math.Abs(vb22r)) {
csq, snq, _ = impl.Dlartg(ua22r, ua21)
} else {
csq, snq, _ = impl.Dlartg(vb22r, vb21)
}
} else {
csq, snq, _ = impl.Dlartg(vb22r, vb21)
}
csu = csr
snu = -snr
csv = csl
snv = -snl
} else {
// Compute the [0, 0] and [0, 1] elements of Uᵀ *A and Vᵀ *B,
// and [0, 0] element of |U|ᵀ*|A| and |V|ᵀ*|B|.
ua11 := csr*a1 + snr*a2
ua12 := snr * a3
vb11 := csl*b1 + snl*b2
vb12 := snl * b3
aua11 := math.Abs(csr)*math.Abs(a1) + math.Abs(snr)*math.Abs(a2)
avb11 := math.Abs(csl)*math.Abs(b1) + math.Abs(snl)*math.Abs(b2)
// Zero [0, 0] elements of Uᵀ*A and Vᵀ*B, and then swap.
if (math.Abs(ua11) + math.Abs(ua12)) != 0 {
if aua11/(math.Abs(ua11)+math.Abs(ua12)) <= avb11/(math.Abs(vb11)+math.Abs(vb12)) {
csq, snq, _ = impl.Dlartg(ua12, ua11)
} else {
csq, snq, _ = impl.Dlartg(vb12, vb11)
}
} else {
csq, snq, _ = impl.Dlartg(vb12, vb11)
}
csu = snr
snu = csr
csv = snl
snv = csl
}
}
return csu, snu, csv, snv, csq, snq
}
|
