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
|
SUBROUTINE ZHPTRI( UPLO, N, AP, IPIV, WORK, INFO )
*
* -- LAPACK routine (version 3.2) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 AP( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* ZHPTRI computes the inverse of a complex Hermitian indefinite matrix
* A in packed storage using the factorization A = U*D*U**H or
* A = L*D*L**H computed by ZHPTRF.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the details of the factorization are stored
* as an upper or lower triangular matrix.
* = 'U': Upper triangular, form is A = U*D*U**H;
* = 'L': Lower triangular, form is A = L*D*L**H.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
* On entry, the block diagonal matrix D and the multipliers
* used to obtain the factor U or L as computed by ZHPTRF,
* stored as a packed triangular matrix.
*
* On exit, if INFO = 0, the (Hermitian) inverse of the original
* matrix, stored as a packed triangular matrix. The j-th column
* of inv(A) is stored in the array AP as follows:
* if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
* if UPLO = 'L',
* AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.
*
* IPIV (input) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D
* as determined by ZHPTRF.
*
* WORK (workspace) COMPLEX*16 array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
* > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
* inverse could not be computed.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
COMPLEX*16 CONE, ZERO
PARAMETER ( ONE = 1.0D+0, CONE = ( 1.0D+0, 0.0D+0 ),
$ ZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, K, KC, KCNEXT, KP, KPC, KSTEP, KX, NPP
DOUBLE PRECISION AK, AKP1, D, T
COMPLEX*16 AKKP1, TEMP
* ..
* .. External Functions ..
LOGICAL LSAME
COMPLEX*16 ZDOTC
EXTERNAL LSAME, ZDOTC
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZCOPY, ZHPMV, ZSWAP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCONJG
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHPTRI', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Check that the diagonal matrix D is nonsingular.
*
IF( UPPER ) THEN
*
* Upper triangular storage: examine D from bottom to top
*
KP = N*( N+1 ) / 2
DO 10 INFO = N, 1, -1
IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO )
$ RETURN
KP = KP - INFO
10 CONTINUE
ELSE
*
* Lower triangular storage: examine D from top to bottom.
*
KP = 1
DO 20 INFO = 1, N
IF( IPIV( INFO ).GT.0 .AND. AP( KP ).EQ.ZERO )
$ RETURN
KP = KP + N - INFO + 1
20 CONTINUE
END IF
INFO = 0
*
IF( UPPER ) THEN
*
* Compute inv(A) from the factorization A = U*D*U'.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
KC = 1
30 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 50
*
KCNEXT = KC + K
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
AP( KC+K-1 ) = ONE / DBLE( AP( KC+K-1 ) )
*
* Compute column K of the inverse.
*
IF( K.GT.1 ) THEN
CALL ZCOPY( K-1, AP( KC ), 1, WORK, 1 )
CALL ZHPMV( UPLO, K-1, -CONE, AP, WORK, 1, ZERO,
$ AP( KC ), 1 )
AP( KC+K-1 ) = AP( KC+K-1 ) -
$ DBLE( ZDOTC( K-1, WORK, 1, AP( KC ), 1 ) )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( AP( KCNEXT+K-1 ) )
AK = DBLE( AP( KC+K-1 ) ) / T
AKP1 = DBLE( AP( KCNEXT+K ) ) / T
AKKP1 = AP( KCNEXT+K-1 ) / T
D = T*( AK*AKP1-ONE )
AP( KC+K-1 ) = AKP1 / D
AP( KCNEXT+K ) = AK / D
AP( KCNEXT+K-1 ) = -AKKP1 / D
*
* Compute columns K and K+1 of the inverse.
*
IF( K.GT.1 ) THEN
CALL ZCOPY( K-1, AP( KC ), 1, WORK, 1 )
CALL ZHPMV( UPLO, K-1, -CONE, AP, WORK, 1, ZERO,
$ AP( KC ), 1 )
AP( KC+K-1 ) = AP( KC+K-1 ) -
$ DBLE( ZDOTC( K-1, WORK, 1, AP( KC ), 1 ) )
AP( KCNEXT+K-1 ) = AP( KCNEXT+K-1 ) -
$ ZDOTC( K-1, AP( KC ), 1, AP( KCNEXT ),
$ 1 )
CALL ZCOPY( K-1, AP( KCNEXT ), 1, WORK, 1 )
CALL ZHPMV( UPLO, K-1, -CONE, AP, WORK, 1, ZERO,
$ AP( KCNEXT ), 1 )
AP( KCNEXT+K ) = AP( KCNEXT+K ) -
$ DBLE( ZDOTC( K-1, WORK, 1, AP( KCNEXT ),
$ 1 ) )
END IF
KSTEP = 2
KCNEXT = KCNEXT + K + 1
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the leading
* submatrix A(1:k+1,1:k+1)
*
KPC = ( KP-1 )*KP / 2 + 1
CALL ZSWAP( KP-1, AP( KC ), 1, AP( KPC ), 1 )
KX = KPC + KP - 1
DO 40 J = KP + 1, K - 1
KX = KX + J - 1
TEMP = DCONJG( AP( KC+J-1 ) )
AP( KC+J-1 ) = DCONJG( AP( KX ) )
AP( KX ) = TEMP
40 CONTINUE
AP( KC+KP-1 ) = DCONJG( AP( KC+KP-1 ) )
TEMP = AP( KC+K-1 )
AP( KC+K-1 ) = AP( KPC+KP-1 )
AP( KPC+KP-1 ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = AP( KC+K+K-1 )
AP( KC+K+K-1 ) = AP( KC+K+KP-1 )
AP( KC+K+KP-1 ) = TEMP
END IF
END IF
*
K = K + KSTEP
KC = KCNEXT
GO TO 30
50 CONTINUE
*
ELSE
*
* Compute inv(A) from the factorization A = L*D*L'.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
NPP = N*( N+1 ) / 2
K = N
KC = NPP
60 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 80
*
KCNEXT = KC - ( N-K+2 )
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Invert the diagonal block.
*
AP( KC ) = ONE / DBLE( AP( KC ) )
*
* Compute column K of the inverse.
*
IF( K.LT.N ) THEN
CALL ZCOPY( N-K, AP( KC+1 ), 1, WORK, 1 )
CALL ZHPMV( UPLO, N-K, -CONE, AP( KC+N-K+1 ), WORK, 1,
$ ZERO, AP( KC+1 ), 1 )
AP( KC ) = AP( KC ) - DBLE( ZDOTC( N-K, WORK, 1,
$ AP( KC+1 ), 1 ) )
END IF
KSTEP = 1
ELSE
*
* 2 x 2 diagonal block
*
* Invert the diagonal block.
*
T = ABS( AP( KCNEXT+1 ) )
AK = DBLE( AP( KCNEXT ) ) / T
AKP1 = DBLE( AP( KC ) ) / T
AKKP1 = AP( KCNEXT+1 ) / T
D = T*( AK*AKP1-ONE )
AP( KCNEXT ) = AKP1 / D
AP( KC ) = AK / D
AP( KCNEXT+1 ) = -AKKP1 / D
*
* Compute columns K-1 and K of the inverse.
*
IF( K.LT.N ) THEN
CALL ZCOPY( N-K, AP( KC+1 ), 1, WORK, 1 )
CALL ZHPMV( UPLO, N-K, -CONE, AP( KC+( N-K+1 ) ), WORK,
$ 1, ZERO, AP( KC+1 ), 1 )
AP( KC ) = AP( KC ) - DBLE( ZDOTC( N-K, WORK, 1,
$ AP( KC+1 ), 1 ) )
AP( KCNEXT+1 ) = AP( KCNEXT+1 ) -
$ ZDOTC( N-K, AP( KC+1 ), 1,
$ AP( KCNEXT+2 ), 1 )
CALL ZCOPY( N-K, AP( KCNEXT+2 ), 1, WORK, 1 )
CALL ZHPMV( UPLO, N-K, -CONE, AP( KC+( N-K+1 ) ), WORK,
$ 1, ZERO, AP( KCNEXT+2 ), 1 )
AP( KCNEXT ) = AP( KCNEXT ) -
$ DBLE( ZDOTC( N-K, WORK, 1, AP( KCNEXT+2 ),
$ 1 ) )
END IF
KSTEP = 2
KCNEXT = KCNEXT - ( N-K+3 )
END IF
*
KP = ABS( IPIV( K ) )
IF( KP.NE.K ) THEN
*
* Interchange rows and columns K and KP in the trailing
* submatrix A(k-1:n,k-1:n)
*
KPC = NPP - ( N-KP+1 )*( N-KP+2 ) / 2 + 1
IF( KP.LT.N )
$ CALL ZSWAP( N-KP, AP( KC+KP-K+1 ), 1, AP( KPC+1 ), 1 )
KX = KC + KP - K
DO 70 J = K + 1, KP - 1
KX = KX + N - J + 1
TEMP = DCONJG( AP( KC+J-K ) )
AP( KC+J-K ) = DCONJG( AP( KX ) )
AP( KX ) = TEMP
70 CONTINUE
AP( KC+KP-K ) = DCONJG( AP( KC+KP-K ) )
TEMP = AP( KC )
AP( KC ) = AP( KPC )
AP( KPC ) = TEMP
IF( KSTEP.EQ.2 ) THEN
TEMP = AP( KC-N+K-1 )
AP( KC-N+K-1 ) = AP( KC-N+KP-1 )
AP( KC-N+KP-1 ) = TEMP
END IF
END IF
*
K = K - KSTEP
KC = KCNEXT
GO TO 60
80 CONTINUE
END IF
*
RETURN
*
* End of ZHPTRI
*
END
|
