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
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
|
*> \brief \b SGET53
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SGET53( A, LDA, B, LDB, SCALE, WR, WI, RESULT, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB
* REAL RESULT, SCALE, WI, WR
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), B( LDB, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGET53 checks the generalized eigenvalues computed by SLAG2.
*>
*> The basic test for an eigenvalue is:
*>
*> | det( s A - w B ) |
*> RESULT = ---------------------------------------------------
*> ulp max( s norm(A), |w| norm(B) )*norm( s A - w B )
*>
*> Two "safety checks" are performed:
*>
*> (1) ulp*max( s*norm(A), |w|*norm(B) ) must be at least
*> safe_minimum. This insures that the test performed is
*> not essentially det(0*A + 0*B)=0.
*>
*> (2) s*norm(A) + |w|*norm(B) must be less than 1/safe_minimum.
*> This insures that s*A - w*B will not overflow.
*>
*> If these tests are not passed, then s and w are scaled and
*> tested anyway, if this is possible.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (LDA, 2)
*> The 2x2 matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. It must be at least 2.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is REAL array, dimension (LDB, N)
*> The 2x2 upper-triangular matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. It must be at least 2.
*> \endverbatim
*>
*> \param[in] SCALE
*> \verbatim
*> SCALE is REAL
*> The "scale factor" s in the formula s A - w B . It is
*> assumed to be non-negative.
*> \endverbatim
*>
*> \param[in] WR
*> \verbatim
*> WR is REAL
*> The real part of the eigenvalue w in the formula
*> s A - w B .
*> \endverbatim
*>
*> \param[in] WI
*> \verbatim
*> WI is REAL
*> The imaginary part of the eigenvalue w in the formula
*> s A - w B .
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is REAL
*> If INFO is 2 or less, the value computed by the test
*> described above.
*> If INFO=3, this will just be 1/ulp.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> =0: The input data pass the "safety checks".
*> =1: s*norm(A) + |w|*norm(B) > 1/safe_minimum.
*> =2: ulp*max( s*norm(A), |w|*norm(B) ) < safe_minimum
*> =3: same as INFO=2, but s and w could not be scaled so
*> as to compute the test.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup single_eig
*
* =====================================================================
SUBROUTINE SGET53( A, LDA, B, LDB, SCALE, WR, WI, RESULT, INFO )
*
* -- LAPACK test 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 ..
INTEGER INFO, LDA, LDB
REAL RESULT, SCALE, WI, WR
* ..
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0, ONE = 1.0 )
* ..
* .. Local Scalars ..
REAL ABSW, ANORM, BNORM, CI11, CI12, CI22, CNORM,
$ CR11, CR12, CR21, CR22, CSCALE, DETI, DETR, S1,
$ SAFMIN, SCALES, SIGMIN, TEMP, ULP, WIS, WRS
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Initialize
*
INFO = 0
RESULT = ZERO
SCALES = SCALE
WRS = WR
WIS = WI
*
* Machine constants and norms
*
SAFMIN = SLAMCH( 'Safe minimum' )
ULP = SLAMCH( 'Epsilon' )*SLAMCH( 'Base' )
ABSW = ABS( WRS ) + ABS( WIS )
ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
$ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 1, 2 ) )+ABS( B( 2, 2 ) ),
$ SAFMIN )
*
* Check for possible overflow.
*
TEMP = ( SAFMIN*BNORM )*ABSW + ( SAFMIN*ANORM )*SCALES
IF( TEMP.GE.ONE ) THEN
*
* Scale down to avoid overflow
*
INFO = 1
TEMP = ONE / TEMP
SCALES = SCALES*TEMP
WRS = WRS*TEMP
WIS = WIS*TEMP
ABSW = ABS( WRS ) + ABS( WIS )
END IF
S1 = MAX( ULP*MAX( SCALES*ANORM, ABSW*BNORM ),
$ SAFMIN*MAX( SCALES, ABSW ) )
*
* Check for W and SCALE essentially zero.
*
IF( S1.LT.SAFMIN ) THEN
INFO = 2
IF( SCALES.LT.SAFMIN .AND. ABSW.LT.SAFMIN ) THEN
INFO = 3
RESULT = ONE / ULP
RETURN
END IF
*
* Scale up to avoid underflow
*
TEMP = ONE / MAX( SCALES*ANORM+ABSW*BNORM, SAFMIN )
SCALES = SCALES*TEMP
WRS = WRS*TEMP
WIS = WIS*TEMP
ABSW = ABS( WRS ) + ABS( WIS )
S1 = MAX( ULP*MAX( SCALES*ANORM, ABSW*BNORM ),
$ SAFMIN*MAX( SCALES, ABSW ) )
IF( S1.LT.SAFMIN ) THEN
INFO = 3
RESULT = ONE / ULP
RETURN
END IF
END IF
*
* Compute C = s A - w B
*
CR11 = SCALES*A( 1, 1 ) - WRS*B( 1, 1 )
CI11 = -WIS*B( 1, 1 )
CR21 = SCALES*A( 2, 1 )
CR12 = SCALES*A( 1, 2 ) - WRS*B( 1, 2 )
CI12 = -WIS*B( 1, 2 )
CR22 = SCALES*A( 2, 2 ) - WRS*B( 2, 2 )
CI22 = -WIS*B( 2, 2 )
*
* Compute the smallest singular value of s A - w B:
*
* |det( s A - w B )|
* sigma_min = ------------------
* norm( s A - w B )
*
CNORM = MAX( ABS( CR11 )+ABS( CI11 )+ABS( CR21 ),
$ ABS( CR12 )+ABS( CI12 )+ABS( CR22 )+ABS( CI22 ), SAFMIN )
CSCALE = ONE / SQRT( CNORM )
DETR = ( CSCALE*CR11 )*( CSCALE*CR22 ) -
$ ( CSCALE*CI11 )*( CSCALE*CI22 ) -
$ ( CSCALE*CR12 )*( CSCALE*CR21 )
DETI = ( CSCALE*CR11 )*( CSCALE*CI22 ) +
$ ( CSCALE*CI11 )*( CSCALE*CR22 ) -
$ ( CSCALE*CI12 )*( CSCALE*CR21 )
SIGMIN = ABS( DETR ) + ABS( DETI )
RESULT = SIGMIN / S1
RETURN
*
* End of SGET53
*
END
|
