summaryrefslogtreecommitdiff
path: root/src/fortran/lapack/dsytf2.f
blob: d52346257abd25da68079b95d0fcb0e74aaf299e (plain)
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
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
      SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO )
*
*  -- LAPACK routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          UPLO
      INTEGER            INFO, LDA, N
*     ..
*     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
*     ..
*
*  Purpose
*  =======
*
*  DSYTF2 computes the factorization of a real symmetric matrix A using
*  the Bunch-Kaufman diagonal pivoting method:
*
*     A = U*D*U'  or  A = L*D*L'
*
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, U' is the transpose of U, and D is symmetric and
*  block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
*
*  This is the unblocked version of the algorithm, calling Level 2 BLAS.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the upper or lower triangular part of the
*          symmetric matrix A is stored:
*          = 'U':  Upper triangular
*          = 'L':  Lower triangular
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          n-by-n upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading n-by-n lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*
*          On exit, the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L (see below for further details).
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D.
*          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*          interchanged and D(k,k) is a 1-by-1 diagonal block.
*          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -k, the k-th argument had an illegal value
*          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
*               has been completed, but the block diagonal matrix D is
*               exactly singular, and division by zero will occur if it
*               is used to solve a system of equations.
*
*  Further Details
*  ===============
*
*  09-29-06 - patch from
*    Bobby Cheng, MathWorks
*
*    Replace l.204 and l.372
*         IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
*    by
*         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*
*  01-01-96 - Based on modifications by
*    J. Lewis, Boeing Computer Services Company
*    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*  1-96 - Based on modifications by J. Lewis, Boeing Computer Services
*         Company
*
*  If UPLO = 'U', then A = U*D*U', where
*     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    v    0   )   k-s
*     U(k) =  (   0    I    0   )   s
*             (   0    0    I   )   n-k
*                k-s   s   n-k
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*
*  If UPLO = 'L', then A = L*D*L', where
*     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    0     0   )  k-1
*     L(k) =  (   0    I     0   )  s
*             (   0    v     I   )  n-k-s+1
*                k-1   s  n-k-s+1
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
      DOUBLE PRECISION   EIGHT, SEVTEN
      PARAMETER          ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IMAX, J, JMAX, K, KK, KP, KSTEP
      DOUBLE PRECISION   ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22, R1,
     $                   ROWMAX, T, WK, WKM1, WKP1
*     ..
*     .. External Functions ..
      LOGICAL            LSAME, DISNAN
      INTEGER            IDAMAX
      EXTERNAL           LSAME, IDAMAX, DISNAN
*     ..
*     .. External Subroutines ..
      EXTERNAL           DSCAL, DSWAP, DSYR, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. 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
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -4
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DSYTF2', -INFO )
         RETURN
      END IF
*
*     Initialize ALPHA for use in choosing pivot block size.
*
      ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
*
      IF( UPPER ) THEN
*
*        Factorize A as U*D*U' using the upper triangle of A
*
*        K is the main loop index, decreasing from N to 1 in steps of
*        1 or 2
*
         K = N
   10    CONTINUE
*
*        If K < 1, exit from loop
*
         IF( K.LT.1 )
     $      GO TO 70
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( A( K, K ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value
*
         IF( K.GT.1 ) THEN
            IMAX = IDAMAX( K-1, A( 1, K ), 1 )
            COLMAX = ABS( A( IMAX, K ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*
*           Column K is zero or contains a NaN: set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA )
               ROWMAX = ABS( A( IMAX, JMAX ) )
               IF( IMAX.GT.1 ) THEN
                  JMAX = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
               ELSE
*
*                 interchange rows and columns K-1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
            KK = K - KSTEP + 1
            IF( KP.NE.KK ) THEN
*
*              Interchange rows and columns KK and KP in the leading
*              submatrix A(1:k,1:k)
*
               CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
               CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
     $                     LDA )
               T = A( KK, KK )
               A( KK, KK ) = A( KP, KP )
               A( KP, KP ) = T
               IF( KSTEP.EQ.2 ) THEN
                  T = A( K-1, K )
                  A( K-1, K ) = A( KP, K )
                  A( KP, K ) = T
               END IF
            END IF
*
*           Update the leading submatrix
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = U(k)*D(k)
*
*              where U(k) is the k-th column of U
*
*              Perform a rank-1 update of A(1:k-1,1:k-1) as
*
*              A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)'
*
               R1 = ONE / A( K, K )
               CALL DSYR( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA )
*
*              Store U(k) in column k
*
               CALL DSCAL( K-1, R1, A( 1, K ), 1 )
            ELSE
*
*              2-by-2 pivot block D(k): columns k and k-1 now hold
*
*              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
*
*              where U(k) and U(k-1) are the k-th and (k-1)-th columns
*              of U
*
*              Perform a rank-2 update of A(1:k-2,1:k-2) as
*
*              A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )'
*                 = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )'
*
               IF( K.GT.2 ) THEN
*
                  D12 = A( K-1, K )
                  D22 = A( K-1, K-1 ) / D12
                  D11 = A( K, K ) / D12
                  T = ONE / ( D11*D22-ONE )
                  D12 = T / D12
*
                  DO 30 J = K - 2, 1, -1
                     WKM1 = D12*( D11*A( J, K-1 )-A( J, K ) )
                     WK = D12*( D22*A( J, K )-A( J, K-1 ) )
                     DO 20 I = J, 1, -1
                        A( I, J ) = A( I, J ) - A( I, K )*WK -
     $                              A( I, K-1 )*WKM1
   20                CONTINUE
                     A( J, K ) = WK
                     A( J, K-1 ) = WKM1
   30             CONTINUE
*
               END IF
*
            END IF
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K-1 ) = -KP
         END IF
*
*        Decrease K and return to the start of the main loop
*
         K = K - KSTEP
         GO TO 10
*
      ELSE
*
*        Factorize A as L*D*L' using the lower triangle of A
*
*        K is the main loop index, increasing from 1 to N in steps of
*        1 or 2
*
         K = 1
   40    CONTINUE
*
*        If K > N, exit from loop
*
         IF( K.GT.N )
     $      GO TO 70
         KSTEP = 1
*
*        Determine rows and columns to be interchanged and whether
*        a 1-by-1 or 2-by-2 pivot block will be used
*
         ABSAKK = ABS( A( K, K ) )
*
*        IMAX is the row-index of the largest off-diagonal element in
*        column K, and COLMAX is its absolute value
*
         IF( K.LT.N ) THEN
            IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
            COLMAX = ABS( A( IMAX, K ) )
         ELSE
            COLMAX = ZERO
         END IF
*
         IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN
*
*           Column K is zero or contains a NaN: set INFO and continue
*
            IF( INFO.EQ.0 )
     $         INFO = K
            KP = K
         ELSE
            IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
*
*              no interchange, use 1-by-1 pivot block
*
               KP = K
            ELSE
*
*              JMAX is the column-index of the largest off-diagonal
*              element in row IMAX, and ROWMAX is its absolute value
*
               JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
               ROWMAX = ABS( A( IMAX, JMAX ) )
               IF( IMAX.LT.N ) THEN
                  JMAX = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ), 1 )
                  ROWMAX = MAX( ROWMAX, ABS( A( JMAX, IMAX ) ) )
               END IF
*
               IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
*
*                 no interchange, use 1-by-1 pivot block
*
                  KP = K
               ELSE IF( ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX ) THEN
*
*                 interchange rows and columns K and IMAX, use 1-by-1
*                 pivot block
*
                  KP = IMAX
               ELSE
*
*                 interchange rows and columns K+1 and IMAX, use 2-by-2
*                 pivot block
*
                  KP = IMAX
                  KSTEP = 2
               END IF
            END IF
*
            KK = K + KSTEP - 1
            IF( KP.NE.KK ) THEN
*
*              Interchange rows and columns KK and KP in the trailing
*              submatrix A(k:n,k:n)
*
               IF( KP.LT.N )
     $            CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
               CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
     $                     LDA )
               T = A( KK, KK )
               A( KK, KK ) = A( KP, KP )
               A( KP, KP ) = T
               IF( KSTEP.EQ.2 ) THEN
                  T = A( K+1, K )
                  A( K+1, K ) = A( KP, K )
                  A( KP, K ) = T
               END IF
            END IF
*
*           Update the trailing submatrix
*
            IF( KSTEP.EQ.1 ) THEN
*
*              1-by-1 pivot block D(k): column k now holds
*
*              W(k) = L(k)*D(k)
*
*              where L(k) is the k-th column of L
*
               IF( K.LT.N ) THEN
*
*                 Perform a rank-1 update of A(k+1:n,k+1:n) as
*
*                 A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)'
*
                  D11 = ONE / A( K, K )
                  CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
     $                       A( K+1, K+1 ), LDA )
*
*                 Store L(k) in column K
*
                  CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
               END IF
            ELSE
*
*              2-by-2 pivot block D(k)
*
               IF( K.LT.N-1 ) THEN
*
*                 Perform a rank-2 update of A(k+2:n,k+2:n) as
*
*                 A := A - ( (A(k) A(k+1))*D(k)**(-1) ) * (A(k) A(k+1))'
*
*                 where L(k) and L(k+1) are the k-th and (k+1)-th
*                 columns of L
*
                  D21 = A( K+1, K )
                  D11 = A( K+1, K+1 ) / D21
                  D22 = A( K, K ) / D21
                  T = ONE / ( D11*D22-ONE )
                  D21 = T / D21
*
                  DO 60 J = K + 2, N
*
                     WK = D21*( D11*A( J, K )-A( J, K+1 ) )
                     WKP1 = D21*( D22*A( J, K+1 )-A( J, K ) )
*
                     DO 50 I = J, N
                        A( I, J ) = A( I, J ) - A( I, K )*WK -
     $                              A( I, K+1 )*WKP1
   50                CONTINUE
*
                     A( J, K ) = WK
                     A( J, K+1 ) = WKP1
*
   60             CONTINUE
               END IF
            END IF
         END IF
*
*        Store details of the interchanges in IPIV
*
         IF( KSTEP.EQ.1 ) THEN
            IPIV( K ) = KP
         ELSE
            IPIV( K ) = -KP
            IPIV( K+1 ) = -KP
         END IF
*
*        Increase K and return to the start of the main loop
*
         K = K + KSTEP
         GO TO 40
*
      END IF
*
   70 CONTINUE
*
      RETURN
*
*     End of DSYTF2
*
      END