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
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
|
SUBROUTINE ZHPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
$ TAU, WORK, RWORK, RESULT )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER ITYPE, KBAND, LDU, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * )
COMPLEX*16 AP( * ), TAU( * ), U( LDU, * ), VP( * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* ZHPT21 generally checks a decomposition of the form
*
* A = U S U*
*
* where * means conjugate transpose, A is hermitian, U is
* unitary, and S is diagonal (if KBAND=0) or (real) 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) = | A - U S U* | / ( |A| n ulp ) *and*
* RESULT(2) = | I - UU* | / ( n ulp )
*
* If ITYPE=2, then:
*
* RESULT(1) = | A - V S V* | / ( |A| n ulp )
*
* If ITYPE=3, then:
*
* RESULT(1) = | I - UV* | / ( n ulp )
*
* Packed storage means that, for example, if UPLO='U', then the columns
* of the upper triangle of A are stored one after another, so that
* A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if
* UPLO='L', then the columns of the lower triangle of A are stored one
* after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
* in the array AP. This means that A(i,j) is stored in:
*
* AP( i + j*(j-1)/2 ) if UPLO='U'
*
* AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L'
*
* The array VP bears the same relation to the matrix V that A does to
* AP.
*
* For ITYPE > 1, the transformation U is expressed as a product
* of Householder transformations:
*
* If UPLO='U', then V = H(n-1)...H(1), where
*
* H(j) = I - tau(j) v(j) v(j)*
*
* and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
* (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
* the j-th element is 1, and the last n-j elements are 0.
*
* If UPLO='L', then V = H(1)...H(n-1), where
*
* H(j) = I - tau(j) v(j) v(j)*
*
* and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
* (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
* in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)
*
* Arguments
* =========
*
* ITYPE (input) INTEGER
* Specifies the type of tests to be performed.
* 1: U expressed as a dense unitary matrix:
* RESULT(1) = | A - U S U* | / ( |A| n ulp ) *and*
* RESULT(2) = | I - UU* | / ( n ulp )
*
* 2: U expressed as a product V of Housholder transformations:
* RESULT(1) = | A - V S V* | / ( |A| n ulp )
*
* 3: U expressed both as a dense unitary matrix and
* as a product of Housholder transformations:
* RESULT(1) = | I - UV* | / ( n ulp )
*
* UPLO (input) CHARACTER
* If UPLO='U', the upper triangle of A and V will be used and
* the (strictly) lower triangle will not be referenced.
* If UPLO='L', the lower triangle of A and V will be used and
* the (strictly) upper triangle will not be referenced.
*
* N (input) INTEGER
* The size of the matrix. If it is zero, ZHPT21 does nothing.
* It must be at least zero.
*
* KBAND (input) 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.
*
* AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
* The original (unfactored) matrix. It is assumed to be
* hermitian, and contains the columns of just the upper
* triangle (UPLO='U') or only the lower triangle (UPLO='L'),
* packed one after another.
*
* D (input) DOUBLE PRECISION array, dimension (N)
* The diagonal of the (symmetric tri-) diagonal matrix.
*
* E (input) DOUBLE PRECISION array, dimension (N)
* The off-diagonal of the (symmetric tri-) diagonal matrix.
* E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
* (3,2) element, etc.
* Not referenced if KBAND=0.
*
* U (input) COMPLEX*16 array, dimension (LDU, N)
* If ITYPE=1 or 3, this contains the unitary matrix in
* the decomposition, expressed as a dense matrix. If ITYPE=2,
* then it is not referenced.
*
* LDU (input) INTEGER
* The leading dimension of U. LDU must be at least N and
* at least 1.
*
* VP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
* If ITYPE=2 or 3, the columns of this array contain the
* Householder vectors used to describe the unitary matrix
* in the decomposition, as described in purpose.
* *NOTE* If ITYPE=2 or 3, V is modified and restored. The
* subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
* is set to one, and later reset to its original value, during
* the course of the calculation.
* If ITYPE=1, then it is neither referenced nor modified.
*
* TAU (input) COMPLEX*16 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.
*
* WORK (workspace) COMPLEX*16 array, dimension (N**2)
* Workspace.
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
* Workspace.
*
* RESULT (output) DOUBLE PRECISION 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 ITYPE=1.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, TEN
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 10.0D+0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 1.0D+0 / 2.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LOWER
CHARACTER CUPLO
INTEGER IINFO, J, JP, JP1, JR, LAP
DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
COMPLEX*16 TEMP, VSAVE
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHP
COMPLEX*16 ZDOTC
EXTERNAL LSAME, DLAMCH, ZLANGE, ZLANHP, ZDOTC
* ..
* .. External Subroutines ..
EXTERNAL ZAXPY, ZCOPY, ZGEMM, ZHPMV, ZHPR, ZHPR2,
$ ZLACPY, ZLASET, ZUPMTR
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE, DCMPLX, MAX, MIN
* ..
* .. Executable Statements ..
*
* Constants
*
RESULT( 1 ) = ZERO
IF( ITYPE.EQ.1 )
$ RESULT( 2 ) = ZERO
IF( N.LE.0 )
$ RETURN
*
LAP = ( N*( N+1 ) ) / 2
*
IF( LSAME( UPLO, 'U' ) ) THEN
LOWER = .FALSE.
CUPLO = 'U'
ELSE
LOWER = .TRUE.
CUPLO = 'L'
END IF
*
UNFL = DLAMCH( 'Safe minimum' )
ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
*
* Some Error Checks
*
IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
RESULT( 1 ) = TEN / ULP
RETURN
END IF
*
* Do Test 1
*
* Norm of A:
*
IF( ITYPE.EQ.3 ) THEN
ANORM = ONE
ELSE
ANORM = MAX( ZLANHP( '1', CUPLO, N, AP, RWORK ), UNFL )
END IF
*
* Compute error matrix:
*
IF( ITYPE.EQ.1 ) THEN
*
* ITYPE=1: error = A - U S U*
*
CALL ZLASET( 'Full', N, N, CZERO, CZERO, WORK, N )
CALL ZCOPY( LAP, AP, 1, WORK, 1 )
*
DO 10 J = 1, N
CALL ZHPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
10 CONTINUE
*
IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
DO 20 J = 1, N - 1
CALL ZHPR2( CUPLO, N, -DCMPLX( E( J ) ), U( 1, J ), 1,
$ U( 1, J-1 ), 1, WORK )
20 CONTINUE
END IF
WNORM = ZLANHP( '1', CUPLO, N, WORK, RWORK )
*
ELSE IF( ITYPE.EQ.2 ) THEN
*
* ITYPE=2: error = V S V* - A
*
CALL ZLASET( 'Full', N, N, CZERO, CZERO, WORK, N )
*
IF( LOWER ) THEN
WORK( LAP ) = D( N )
DO 40 J = N - 1, 1, -1
JP = ( ( 2*N-J )*( J-1 ) ) / 2
JP1 = JP + N - J
IF( KBAND.EQ.1 ) THEN
WORK( JP+J+1 ) = ( CONE-TAU( J ) )*E( J )
DO 30 JR = J + 2, N
WORK( JP+JR ) = -TAU( J )*E( J )*VP( JP+JR )
30 CONTINUE
END IF
*
IF( TAU( J ).NE.CZERO ) THEN
VSAVE = VP( JP+J+1 )
VP( JP+J+1 ) = CONE
CALL ZHPMV( 'L', N-J, CONE, WORK( JP1+J+1 ),
$ VP( JP+J+1 ), 1, CZERO, WORK( LAP+1 ), 1 )
TEMP = -HALF*TAU( J )*ZDOTC( N-J, WORK( LAP+1 ), 1,
$ VP( JP+J+1 ), 1 )
CALL ZAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ),
$ 1 )
CALL ZHPR2( 'L', N-J, -TAU( J ), VP( JP+J+1 ), 1,
$ WORK( LAP+1 ), 1, WORK( JP1+J+1 ) )
*
VP( JP+J+1 ) = VSAVE
END IF
WORK( JP+J ) = D( J )
40 CONTINUE
ELSE
WORK( 1 ) = D( 1 )
DO 60 J = 1, N - 1
JP = ( J*( J-1 ) ) / 2
JP1 = JP + J
IF( KBAND.EQ.1 ) THEN
WORK( JP1+J ) = ( CONE-TAU( J ) )*E( J )
DO 50 JR = 1, J - 1
WORK( JP1+JR ) = -TAU( J )*E( J )*VP( JP1+JR )
50 CONTINUE
END IF
*
IF( TAU( J ).NE.CZERO ) THEN
VSAVE = VP( JP1+J )
VP( JP1+J ) = CONE
CALL ZHPMV( 'U', J, CONE, WORK, VP( JP1+1 ), 1, CZERO,
$ WORK( LAP+1 ), 1 )
TEMP = -HALF*TAU( J )*ZDOTC( J, WORK( LAP+1 ), 1,
$ VP( JP1+1 ), 1 )
CALL ZAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ),
$ 1 )
CALL ZHPR2( 'U', J, -TAU( J ), VP( JP1+1 ), 1,
$ WORK( LAP+1 ), 1, WORK )
VP( JP1+J ) = VSAVE
END IF
WORK( JP1+J+1 ) = D( J+1 )
60 CONTINUE
END IF
*
DO 70 J = 1, LAP
WORK( J ) = WORK( J ) - AP( J )
70 CONTINUE
WNORM = ZLANHP( '1', CUPLO, N, WORK, RWORK )
*
ELSE IF( ITYPE.EQ.3 ) THEN
*
* ITYPE=3: error = U V* - I
*
IF( N.LT.2 )
$ RETURN
CALL ZLACPY( ' ', N, N, U, LDU, WORK, N )
CALL ZUPMTR( 'R', CUPLO, 'C', N, N, VP, TAU, WORK, N,
$ WORK( N**2+1 ), IINFO )
IF( IINFO.NE.0 ) THEN
RESULT( 1 ) = TEN / ULP
RETURN
END IF
*
DO 80 J = 1, N
WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE
80 CONTINUE
*
WNORM = ZLANGE( '1', N, N, WORK, N, RWORK )
END IF
*
IF( ANORM.GT.WNORM ) THEN
RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
ELSE
IF( ANORM.LT.ONE ) THEN
RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
ELSE
RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
END IF
END IF
*
* Do Test 2
*
* Compute UU* - I
*
IF( ITYPE.EQ.1 ) THEN
CALL ZGEMM( 'N', 'C', N, N, N, CONE, U, LDU, U, LDU, CZERO,
$ WORK, N )
*
DO 90 J = 1, N
WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - CONE
90 CONTINUE
*
RESULT( 2 ) = MIN( ZLANGE( '1', N, N, WORK, N, RWORK ),
$ DBLE( N ) ) / ( N*ULP )
END IF
*
RETURN
*
* End of ZHPT21
*
END
|
