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 SSTT22
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SSTT22( N, M, KBAND, AD, AE, SD, SE, U, LDU, WORK,
* LDWORK, RESULT )
*
* .. Scalar Arguments ..
* INTEGER KBAND, LDU, LDWORK, M, N
* ..
* .. Array Arguments ..
* REAL AD( * ), AE( * ), RESULT( 2 ), SD( * ),
* $ SE( * ), U( LDU, * ), WORK( LDWORK, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SSTT22 checks a set of M eigenvalues and eigenvectors,
*>
*> A U = U S
*>
*> where A is symmetric tridiagonal, the columns of U are orthogonal,
*> and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
*> Two tests are performed:
*>
*> RESULT(1) = | U' A U - S | / ( |A| m ulp )
*>
*> RESULT(2) = | I - U'U | / ( m ulp )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The size of the matrix. If it is zero, SSTT22 does nothing.
*> It must be at least zero.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of eigenpairs to check. If it is zero, SSTT22
*> does nothing. It must be at least zero.
*> \endverbatim
*>
*> \param[in] KBAND
*> \verbatim
*> KBAND is INTEGER
*> The bandwidth of the matrix S. It may only be zero or one.
*> If zero, then S is diagonal, and SE is not referenced. If
*> one, then S is symmetric tri-diagonal.
*> \endverbatim
*>
*> \param[in] AD
*> \verbatim
*> AD is REAL array, dimension (N)
*> The diagonal of the original (unfactored) matrix A. A is
*> assumed to be symmetric tridiagonal.
*> \endverbatim
*>
*> \param[in] AE
*> \verbatim
*> AE is REAL array, dimension (N)
*> The off-diagonal of the original (unfactored) matrix A. A
*> is assumed to be symmetric tridiagonal. AE(1) is ignored,
*> AE(2) is the (1,2) and (2,1) element, etc.
*> \endverbatim
*>
*> \param[in] SD
*> \verbatim
*> SD is REAL array, dimension (N)
*> The diagonal of the (symmetric tri-) diagonal matrix S.
*> \endverbatim
*>
*> \param[in] SE
*> \verbatim
*> SE is REAL array, dimension (N)
*> The off-diagonal of the (symmetric tri-) diagonal matrix S.
*> Not referenced if KBSND=0. If KBAND=1, then AE(1) is
*> ignored, SE(2) is the (1,2) and (2,1) element, etc.
*> \endverbatim
*>
*> \param[in] U
*> \verbatim
*> U is REAL array, dimension (LDU, N)
*> The orthogonal matrix in the decomposition.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of U. LDU must be at least N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (LDWORK, M+1)
*> \endverbatim
*>
*> \param[in] LDWORK
*> \verbatim
*> LDWORK is INTEGER
*> The leading dimension of WORK. LDWORK must be at least
*> max(1,M).
*> \endverbatim
*>
*> \param[out] RESULT
*> \verbatim
*> RESULT is REAL array, dimension (2)
*> The values computed by the two tests described above. The
*> values are currently limited to 1/ulp, to avoid overflow.
*> \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 SSTT22( N, M, KBAND, AD, AE, SD, SE, U, LDU, WORK,
$ LDWORK, RESULT )
*
* -- 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 KBAND, LDU, LDWORK, M, N
* ..
* .. Array Arguments ..
REAL AD( * ), AE( * ), RESULT( 2 ), SD( * ),
$ SE( * ), U( LDU, * ), WORK( LDWORK, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
INTEGER I, J, K
REAL ANORM, AUKJ, ULP, UNFL, WNORM
* ..
* .. External Functions ..
REAL SLAMCH, SLANGE, SLANSY
EXTERNAL SLAMCH, SLANGE, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SGEMM
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
RESULT( 1 ) = ZERO
RESULT( 2 ) = ZERO
IF( N.LE.0 .OR. M.LE.0 )
$ RETURN
*
UNFL = SLAMCH( 'Safe minimum' )
ULP = SLAMCH( 'Epsilon' )
*
* Do Test 1
*
* Compute the 1-norm of A.
*
IF( N.GT.1 ) THEN
ANORM = ABS( AD( 1 ) ) + ABS( AE( 1 ) )
DO 10 J = 2, N - 1
ANORM = MAX( ANORM, ABS( AD( J ) )+ABS( AE( J ) )+
$ ABS( AE( J-1 ) ) )
10 CONTINUE
ANORM = MAX( ANORM, ABS( AD( N ) )+ABS( AE( N-1 ) ) )
ELSE
ANORM = ABS( AD( 1 ) )
END IF
ANORM = MAX( ANORM, UNFL )
*
* Norm of U'AU - S
*
DO 40 I = 1, M
DO 30 J = 1, M
WORK( I, J ) = ZERO
DO 20 K = 1, N
AUKJ = AD( K )*U( K, J )
IF( K.NE.N )
$ AUKJ = AUKJ + AE( K )*U( K+1, J )
IF( K.NE.1 )
$ AUKJ = AUKJ + AE( K-1 )*U( K-1, J )
WORK( I, J ) = WORK( I, J ) + U( K, I )*AUKJ
20 CONTINUE
30 CONTINUE
WORK( I, I ) = WORK( I, I ) - SD( I )
IF( KBAND.EQ.1 ) THEN
IF( I.NE.1 )
$ WORK( I, I-1 ) = WORK( I, I-1 ) - SE( I-1 )
IF( I.NE.N )
$ WORK( I, I+1 ) = WORK( I, I+1 ) - SE( I )
END IF
40 CONTINUE
*
WNORM = SLANSY( '1', 'L', M, WORK, M, WORK( 1, M+1 ) )
*
IF( ANORM.GT.WNORM ) THEN
RESULT( 1 ) = ( WNORM / ANORM ) / ( M*ULP )
ELSE
IF( ANORM.LT.ONE ) THEN
RESULT( 1 ) = ( MIN( WNORM, M*ANORM ) / ANORM ) / ( M*ULP )
ELSE
RESULT( 1 ) = MIN( WNORM / ANORM, REAL( M ) ) / ( M*ULP )
END IF
END IF
*
* Do Test 2
*
* Compute U'U - I
*
CALL SGEMM( 'T', 'N', M, M, N, ONE, U, LDU, U, LDU, ZERO, WORK,
$ M )
*
DO 50 J = 1, M
WORK( J, J ) = WORK( J, J ) - ONE
50 CONTINUE
*
RESULT( 2 ) = MIN( REAL( M ), SLANGE( '1', M, M, WORK, M, WORK( 1,
$ M+1 ) ) ) / ( M*ULP )
*
RETURN
*
* End of SSTT22
*
END
|
