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
|
*> \brief \b SSYT22
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
* V, LDV, TAU, WORK, RESULT )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SSYT22 generally checks a decomposition of the form
*>
*> A U = U S
*>
*> where A is symmetric, the columns of U are orthonormal, and S
*> is diagonal (if KBAND=0) or symmetric tridiagonal (if
*> KBAND=1). If ITYPE=1, then U is represented as a dense matrix,
*> otherwise the U is expressed as a product of Householder
*> transformations, whose vectors are stored in the array "V" and
*> whose scaling constants are in "TAU"; we shall use the letter
*> "V" to refer to the product of Householder transformations
*> (which should be equal to U).
*>
*> Specifically, if ITYPE=1, then:
*>
*> RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC> RESULT(2) = | I - U'U | / ( m ulp )
*> \endverbatim
*
* Arguments:
* ==========
*
*> \verbatim
*> ITYPE INTEGER
*> Specifies the type of tests to be performed.
*> 1: U expressed as a dense orthogonal matrix:
*> RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp )
*>
*> UPLO CHARACTER
*> If UPLO='U', the upper triangle of A will be used and the
*> (strictly) lower triangle will not be referenced. If
*> UPLO='L', the lower triangle of A will be used and the
*> (strictly) upper triangle will not be referenced.
*> Not modified.
*>
*> N INTEGER
*> The size of the matrix. If it is zero, SSYT22 does nothing.
*> It must be at least zero.
*> Not modified.
*>
*> M INTEGER
*> The number of columns of U. If it is zero, SSYT22 does
*> nothing. It must be at least zero.
*> Not modified.
*>
*> KBAND INTEGER
*> The bandwidth of the matrix. It may only be zero or one.
*> If zero, then S is diagonal, and E is not referenced. If
*> one, then S is symmetric tri-diagonal.
*> Not modified.
*>
*> A REAL array, dimension (LDA , N)
*> The original (unfactored) matrix. It is assumed to be
*> symmetric, and only the upper (UPLO='U') or only the lower
*> (UPLO='L') will be referenced.
*> Not modified.
*>
*> LDA INTEGER
*> The leading dimension of A. It must be at least 1
*> and at least N.
*> Not modified.
*>
*> D REAL array, dimension (N)
*> The diagonal of the (symmetric tri-) diagonal matrix.
*> Not modified.
*>
*> E REAL array, dimension (N)
*> The off-diagonal of the (symmetric tri-) diagonal matrix.
*> E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
*> Not referenced if KBAND=0.
*> Not modified.
*>
*> U REAL array, dimension (LDU, N)
*> If ITYPE=1 or 3, this contains the orthogonal matrix in
*> the decomposition, expressed as a dense matrix. If ITYPE=2,
*> then it is not referenced.
*> Not modified.
*>
*> LDU INTEGER
*> The leading dimension of U. LDU must be at least N and
*> at least 1.
*> Not modified.
*>
*> V REAL array, dimension (LDV, N)
*> If ITYPE=2 or 3, the lower triangle of this array contains
*> the Householder vectors used to describe the orthogonal
*> matrix in the decomposition. If ITYPE=1, then it is not
*> referenced.
*> Not modified.
*>
*> LDV INTEGER
*> The leading dimension of V. LDV must be at least N and
*> at least 1.
*> Not modified.
*>
*> TAU REAL array, dimension (N)
*> If ITYPE >= 2, then TAU(j) is the scalar factor of
*> v(j) v(j)' in the Householder transformation H(j) of
*> the product U = H(1)...H(n-2)
*> If ITYPE < 2, then TAU is not referenced.
*> Not modified.
*>
*> WORK REAL array, dimension (2*N**2)
*> Workspace.
*> Modified.
*>
*> RESULT REAL array, dimension (2)
*> The values computed by the two tests described above. The
*> values are currently limited to 1/ulp, to avoid overflow.
*> RESULT(1) is always modified. RESULT(2) is modified only
*> if LDU is at least N.
*> Modified.
*> \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 SSYT22( ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU,
$ V, LDV, TAU, WORK, RESULT )
*
* -- LAPACK test routine (version 3.1) --
* -- 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 ..
CHARACTER UPLO
INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
INTEGER J, JJ, JJ1, JJ2, NN, NNP1
REAL ANORM, ULP, UNFL, WNORM
* ..
* .. External Functions ..
REAL SLAMCH, SLANSY
EXTERNAL SLAMCH, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SGEMM, SSYMM
* ..
* .. Intrinsic Functions ..
INTRINSIC 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( 'Precision' )
*
* Do Test 1
*
* Norm of A:
*
ANORM = MAX( SLANSY( '1', UPLO, N, A, LDA, WORK ), UNFL )
*
* Compute error matrix:
*
* ITYPE=1: error = U' A U - S
*
CALL SSYMM( 'L', UPLO, N, M, ONE, A, LDA, U, LDU, ZERO, WORK, N )
NN = N*N
NNP1 = NN + 1
CALL SGEMM( 'T', 'N', M, M, N, ONE, U, LDU, WORK, N, ZERO,
$ WORK( NNP1 ), N )
DO 10 J = 1, M
JJ = NN + ( J-1 )*N + J
WORK( JJ ) = WORK( JJ ) - D( J )
10 CONTINUE
IF( KBAND.EQ.1 .AND. N.GT.1 ) THEN
DO 20 J = 2, M
JJ1 = NN + ( J-1 )*N + J - 1
JJ2 = NN + ( J-2 )*N + J
WORK( JJ1 ) = WORK( JJ1 ) - E( J-1 )
WORK( JJ2 ) = WORK( JJ2 ) - E( J-1 )
20 CONTINUE
END IF
WNORM = SLANSY( '1', UPLO, M, WORK( NNP1 ), N, WORK( 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
*
IF( ITYPE.EQ.1 )
$ CALL SORT01( 'Columns', N, M, U, LDU, WORK, 2*N*N,
$ RESULT( 2 ) )
*
RETURN
*
* End of SSYT22
*
END
|
