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
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
|
SUBROUTINE MD03AD( XINIT, ALG, STOR, UPLO, FCN, JPJ, M, N, ITMAX,
$ NPRINT, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEV, NJEV, TOL, CGTOL, DWORK,
$ LDWORK, IWARN, INFO )
C
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To minimize the sum of the squares of m nonlinear functions, e, in
C n variables, x, by a modification of the Levenberg-Marquardt
C algorithm, using either a Cholesky-based or a conjugate gradients
C solver. The user must provide a subroutine FCN which calculates
C the functions and the Jacobian J (possibly by finite differences),
C and another subroutine JPJ, which computes either J'*J + par*I
C (if ALG = 'D'), or (J'*J + par*I)*x (if ALG = 'I'), where par is
C the Levenberg factor, exploiting the possible structure of the
C Jacobian matrix. Template implementations of these routines are
C included in the SLICOT Library.
C
C ARGUMENTS
C
C Mode Parameters
C
C XINIT CHARACTER*1
C Specifies how the variables x are initialized, as follows:
C = 'R' : the array X is initialized to random values; the
C entries DWORK(1:4) are used to initialize the
C random number generator: the first three values
C are converted to integers between 0 and 4095, and
C the last one is converted to an odd integer
C between 1 and 4095;
C = 'G' : the given entries of X are used as initial values
C of variables.
C
C ALG CHARACTER*1
C Specifies the algorithm used for solving the linear
C systems involving a Jacobian matrix J, as follows:
C = 'D' : a direct algorithm, which computes the Cholesky
C factor of the matrix J'*J + par*I is used;
C = 'I' : an iterative Conjugate Gradients algorithm, which
C only needs the matrix J, is used.
C In both cases, matrix J is stored in a compressed form.
C
C STOR CHARACTER*1
C If ALG = 'D', specifies the storage scheme for the
C symmetric matrix J'*J, as follows:
C = 'F' : full storage is used;
C = 'P' : packed storage is used.
C The option STOR = 'F' usually ensures a faster execution.
C This parameter is not relevant if ALG = 'I'.
C
C UPLO CHARACTER*1
C If ALG = 'D', specifies which part of the matrix J'*J
C is stored, as follows:
C = 'U' : the upper triagular part is stored;
C = 'L' : the lower triagular part is stored.
C The option UPLO = 'U' usually ensures a faster execution.
C This parameter is not relevant if ALG = 'I'.
C
C Function Parameters
C
C FCN EXTERNAL
C Subroutine which evaluates the functions and the Jacobian.
C FCN must be declared in an external statement in the user
C calling program, and must have the following interface:
C
C SUBROUTINE FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1,
C $ DPAR2, LDPAR2, X, NFEVL, E, J, LDJ, JTE,
C $ DWORK, LDWORK, INFO )
C
C where
C
C IFLAG (input/output) INTEGER
C On entry, this parameter must contain a value
C defining the computations to be performed:
C = 0 : Optionally, print the current iterate X,
C function values E, and Jacobian matrix J,
C or other results defined in terms of these
C values. See the argument NPRINT of MD03AD.
C Do not alter E and J.
C = 1 : Calculate the functions at X and return
C this vector in E. Do not alter J.
C = 2 : Calculate the Jacobian at X and return
C this matrix in J. Also return J'*e in JTE
C and NFEVL (see below). Do not alter E.
C = 3 : Do not compute neither the functions nor
C the Jacobian, but return in LDJ and
C IPAR/DPAR1,DPAR2 (some of) the integer/real
C parameters needed.
C On exit, the value of this parameter should not be
C changed by FCN unless the user wants to terminate
C execution of MD03AD, in which case IFLAG must be
C set to a negative integer.
C
C M (input) INTEGER
C The number of functions. M >= 0.
C
C N (input) INTEGER
C The number of variables. M >= N >= 0.
C
C IPAR (input/output) INTEGER array, dimension (LIPAR)
C The integer parameters describing the structure of
C the Jacobian matrix or needed for problem solving.
C IPAR is an input parameter, except for IFLAG = 3
C on entry, when it is also an output parameter.
C On exit, if IFLAG = 3, IPAR(1) contains the length
C of the array J, for storing the Jacobian matrix,
C and the entries IPAR(2:5) contain the workspace
C required by FCN for IFLAG = 1, FCN for IFLAG = 2,
C JPJ for ALG = 'D', and JPJ for ALG = 'I',
C respectively.
C
C LIPAR (input) INTEGER
C The length of the array IPAR. LIPAR >= 5.
C
C DPAR1 (input/output) DOUBLE PRECISION array, dimension
C (LDPAR1,*) or (LDPAR1)
C A first set of real parameters needed for
C describing or solving the problem.
C DPAR1 can also be used as an additional array for
C intermediate results when computing the functions
C or the Jacobian. For control problems, DPAR1 could
C store the input trajectory of a system.
C
C LDPAR1 (input) INTEGER
C The leading dimension or the length of the array
C DPAR1, as convenient. LDPAR1 >= 0. (LDPAR1 >= 1,
C if leading dimension.)
C
C DPAR2 (input/output) DOUBLE PRECISION array, dimension
C (LDPAR2,*) or (LDPAR2)
C A second set of real parameters needed for
C describing or solving the problem.
C DPAR2 can also be used as an additional array for
C intermediate results when computing the functions
C or the Jacobian. For control problems, DPAR2 could
C store the output trajectory of a system.
C
C LDPAR2 (input) INTEGER
C The leading dimension or the length of the array
C DPAR2, as convenient. LDPAR2 >= 0. (LDPAR2 >= 1,
C if leading dimension.)
C
C X (input) DOUBLE PRECISION array, dimension (N)
C This array must contain the value of the
C variables x where the functions or the Jacobian
C must be evaluated.
C
C NFEVL (input/output) INTEGER
C The number of function evaluations needed to
C compute the Jacobian by a finite difference
C approximation.
C NFEVL is an input parameter if IFLAG = 0, or an
C output parameter if IFLAG = 2. If the Jacobian is
C computed analytically, NFEVL should be set to a
C non-positive value.
C
C E (input/output) DOUBLE PRECISION array,
C dimension (M)
C This array contains the value of the (error)
C functions e evaluated at X.
C E is an input parameter if IFLAG = 0 or 2, or an
C output parameter if IFLAG = 1.
C
C J (input/output) DOUBLE PRECISION array, dimension
C (LDJ,NC), where NC is the number of columns
C needed.
C This array contains a possibly compressed
C representation of the Jacobian matrix evaluated
C at X. If full Jacobian is stored, then NC = N.
C J is an input parameter if IFLAG = 0, or an output
C parameter if IFLAG = 2.
C
C LDJ (input/output) INTEGER
C The leading dimension of array J. LDJ >= 1.
C LDJ is essentially used inside the routines FCN
C and JPJ.
C LDJ is an input parameter, except for IFLAG = 3
C on entry, when it is an output parameter.
C It is assumed in MD03AD that LDJ is not larger
C than needed.
C
C JTE (output) DOUBLE PRECISION array, dimension (N)
C If IFLAG = 2, the matrix-vector product J'*e.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C The workspace array for subroutine FCN.
C On exit, if INFO = 0, DWORK(1) returns the optimal
C value of LDWORK.
C
C LDWORK (input) INTEGER
C The size of the array DWORK (as large as needed
C in the subroutine FCN). LDWORK >= 1.
C
C INFO INTEGER
C Error indicator, set to a negative value if an
C input (scalar) argument is erroneous, and to
C positive values for other possible errors in the
C subroutine FCN. The LAPACK Library routine XERBLA
C should be used in conjunction with negative INFO.
C INFO must be zero if the subroutine finished
C successfully.
C
C Parameters marked with "(input)" must not be changed.
C
C JPJ EXTERNAL
C Subroutine which computes J'*J + par*I, if ALG = 'D', and
C J'*J*x + par*x, if ALG = 'I', where J is the Jacobian as
C described above.
C
C JPJ must have the following interface:
C
C SUBROUTINE JPJ( STOR, UPLO, N, IPAR, LIPAR, DPAR, LDPAR,
C $ J, LDJ, JTJ, LDJTJ, DWORK, LDWORK, INFO )
C
C if ALG = 'D', and
C
C SUBROUTINE JPJ( N, IPAR, LIPAR, DPAR, LDPAR, J, LDJ, X,
C $ INCX, DWORK, LDWORK, INFO )
C
C if ALG = 'I', where
C
C STOR (input) CHARACTER*1
C Specifies the storage scheme for the symmetric
C matrix J'*J, as follows:
C = 'F' : full storage is used;
C = 'P' : packed storage is used.
C
C UPLO (input) CHARACTER*1
C Specifies which part of the matrix J'*J is stored,
C as follows:
C = 'U' : the upper triagular part is stored;
C = 'L' : the lower triagular part is stored.
C
C N (input) INTEGER
C The number of columns of the matrix J. N >= 0.
C
C IPAR (input) INTEGER array, dimension (LIPAR)
C The integer parameters describing the structure of
C the Jacobian matrix.
C
C LIPAR (input) INTEGER
C The length of the array IPAR. LIPAR >= 0.
C
C DPAR (input) DOUBLE PRECISION array, dimension (LDPAR)
C DPAR(1) must contain an initial estimate of the
C Levenberg-Marquardt parameter, par. DPAR(1) >= 0.
C
C LDPAR (input) INTEGER
C The length of the array DPAR. LDPAR >= 1.
C
C J (input) DOUBLE PRECISION array, dimension
C (LDJ, NC), where NC is the number of columns.
C The leading NR-by-NC part of this array must
C contain the (compressed) representation of the
C Jacobian matrix J, where NR is the number of rows
C of J (function of IPAR entries).
C
C LDJ (input) INTEGER
C The leading dimension of array J.
C LDJ >= MAX(1,NR).
C
C JTJ (output) DOUBLE PRECISION array,
C dimension (LDJTJ,N), if STOR = 'F',
C dimension (N*(N+1)/2), if STOR = 'P'.
C The leading N-by-N (if STOR = 'F'), or N*(N+1)/2
C (if STOR = 'P') part of this array contains the
C upper or lower triangle of the matrix J'*J+par*I,
C depending on UPLO = 'U', or UPLO = 'L',
C respectively, stored either as a two-dimensional,
C or one-dimensional array, depending on STOR.
C
C LDJTJ (input) INTEGER
C The leading dimension of the array JTJ.
C LDJTJ >= MAX(1,N), if STOR = 'F'.
C LDJTJ >= 1, if STOR = 'P'.
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C The workspace array for subroutine JPJ.
C
C LDWORK (input) INTEGER
C The size of the array DWORK (as large as needed
C in the subroutine JPJ).
C
C INFO INTEGER
C Error indicator, set to a negative value if an
C input (scalar) argument is erroneous, and to
C positive values for other possible errors in the
C subroutine JPJ. The LAPACK Library routine XERBLA
C should be used in conjunction with negative INFO
C values. INFO must be zero if the subroutine
C finished successfully.
C
C If ALG = 'I', the parameters in common with those for
C ALG = 'D', have the same meaning, and the additional
C parameters are:
C
C X (input/output) DOUBLE PRECISION array, dimension
C (1+(N-1)*INCX)
C On entry, this incremented array must contain the
C vector x.
C On exit, this incremented array contains the value
C of the matrix-vector product (J'*J + par)*x.
C
C INCX (input) INTEGER
C The increment for the elements of X. INCX > 0.
C
C Parameters marked with "(input)" must not be changed.
C
C Input/Output Parameters
C
C M (input) INTEGER
C The number of functions. M >= 0.
C
C N (input) INTEGER
C The number of variables. M >= N >= 0.
C
C ITMAX (input) INTEGER
C The maximum number of iterations. ITMAX >= 0.
C
C NPRINT (input) INTEGER
C This parameter enables controlled printing of iterates if
C it is positive. In this case, FCN is called with IFLAG = 0
C at the beginning of the first iteration and every NPRINT
C iterations thereafter and immediately prior to return,
C with X, E, and J available for printing. If NPRINT is not
C positive, no special calls of FCN with IFLAG = 0 are made.
C
C IPAR (input) INTEGER array, dimension (LIPAR)
C The integer parameters needed, for instance, for
C describing the structure of the Jacobian matrix, which
C are handed over to the routines FCN and JPJ.
C The first five entries of this array are modified
C internally by a call to FCN (with IFLAG = 3), but are
C restored on exit.
C
C LIPAR (input) INTEGER
C The length of the array IPAR. LIPAR >= 5.
C
C DPAR1 (input/output) DOUBLE PRECISION array, dimension
C (LDPAR1,*) or (LDPAR1)
C A first set of real parameters needed for describing or
C solving the problem. This argument is not used by MD03AD
C routine, but it is passed to the routine FCN.
C
C LDPAR1 (input) INTEGER
C The leading dimension or the length of the array DPAR1, as
C convenient. LDPAR1 >= 0. (LDPAR1 >= 1, if leading
C dimension.)
C
C DPAR2 (input/output) DOUBLE PRECISION array, dimension
C (LDPAR2,*) or (LDPAR2)
C A second set of real parameters needed for describing or
C solving the problem. This argument is not used by MD03AD
C routine, but it is passed to the routine FCN.
C
C LDPAR2 (input) INTEGER
C The leading dimension or the length of the array DPAR2, as
C convenient. LDPAR2 >= 0. (LDPAR2 >= 1, if leading
C dimension.)
C
C X (input/output) DOUBLE PRECISION array, dimension (N)
C On entry, if XINIT = 'G', this array must contain the
C vector of initial variables x to be optimized.
C If XINIT = 'R', this array need not be set before entry,
C and random values will be used to initialize x.
C On exit, if INFO = 0, this array contains the vector of
C values that (approximately) minimize the sum of squares of
C error functions. The values returned in IWARN and
C DWORK(1:5) give details on the iterative process.
C
C NFEV (output) INTEGER
C The number of calls to FCN with IFLAG = 1. If FCN is
C properly implemented, this includes the function
C evaluations needed for finite difference approximation
C of the Jacobian.
C
C NJEV (output) INTEGER
C The number of calls to FCN with IFLAG = 2.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C If TOL >= 0, the tolerance which measures the relative
C error desired in the sum of squares. Termination occurs
C when the actual relative reduction in the sum of squares
C is at most TOL. If the user sets TOL < 0, then SQRT(EPS)
C is used instead TOL, where EPS is the machine precision
C (see LAPACK Library routine DLAMCH).
C
C CGTOL DOUBLE PRECISION
C If ALG = 'I' and CGTOL > 0, the tolerance which measures
C the relative residual of the solutions computed by the
C conjugate gradients (CG) algorithm. Termination of a
C CG process occurs when the relative residual is at
C most CGTOL. If the user sets CGTOL <= 0, then SQRT(EPS)
C is used instead CGTOL.
C
C Workspace
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK, DWORK(2) returns the residual error norm (the
C sum of squares), DWORK(3) returns the number of iterations
C performed, DWORK(4) returns the total number of conjugate
C gradients iterations performed (zero, if ALG = 'D'), and
C DWORK(5) returns the final Levenberg factor.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= max( 5, M + 2*N + size(J) +
C max( DW( FCN|IFLAG = 1 ) + N,
C DW( FCN|IFLAG = 2 ),
C DW( sol ) ) ),
C where size(J) is the size of the Jacobian (provided by FCN
C in IPAR(1), for IFLAG = 3), DW( f ) is the workspace
C needed by the routine f, where f is FCN or JPJ (provided
C by FCN in IPAR(2:5), for IFLAG = 3), and DW( sol ) is the
C workspace needed for solving linear systems,
C DW( sol ) = N*N + DW( JPJ ), if ALG = 'D', STOR = 'F';
C DW( sol ) = N*(N+1)/2 + DW( JPJ ),
C if ALG = 'D', STOR = 'P';
C DW( sol ) = 3*N + DW( JPJ ), if ALG = 'I'.
C
C Warning Indicator
C
C IWARN INTEGER
C < 0: the user set IFLAG = IWARN in the subroutine FCN;
C = 0: no warning;
C = 1: if the iterative process did not converge in ITMAX
C iterations with tolerance TOL;
C = 2: if ALG = 'I', and in one or more iterations of the
C Levenberg-Marquardt algorithm, the conjugate
C gradient algorithm did not finish after 3*N
C iterations, with the accuracy required in the
C call;
C = 3: the cosine of the angle between e and any column of
C the Jacobian is at most FACTOR*EPS in absolute
C value, where FACTOR = 100 is defined in a PARAMETER
C statement;
C = 4: TOL is too small: no further reduction in the sum
C of squares is possible.
C In all these cases, DWORK(1:5) are set as described
C above.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value;
C = 1: user-defined routine FCN returned with INFO <> 0
C for IFLAG = 1;
C = 2: user-defined routine FCN returned with INFO <> 0
C for IFLAG = 2;
C = 3: SLICOT Library routine MB02XD, if ALG = 'D', or
C SLICOT Library routine MB02WD, if ALG = 'I' (or
C user-defined routine JPJ), returned with INFO <> 0.
C
C METHOD
C
C If XINIT = 'R', the initial value for X is set to a vector of
C pseudo-random values uniformly distributed in [-1,1].
C
C The Levenberg-Marquardt algorithm (described in [1]) is used for
C optimizing the parameters. This algorithm needs the Jacobian
C matrix J, which is provided by the subroutine FCN. The algorithm
C tries to update x by the formula
C
C x = x - p,
C
C using the solution of the system of linear equations
C
C (J'*J + PAR*I)*p = J'*e,
C
C where I is the identity matrix, and e the error function vector.
C The Levenberg factor PAR is decreased after each successfull step
C and increased in the other case.
C
C If ALG = 'D', a direct method, which evaluates the matrix product
C J'*J + par*I and then factors it using Cholesky algorithm,
C implemented in the SLICOT Libray routine MB02XD, is used for
C solving the linear system above.
C
C If ALG = 'I', the Conjugate Gradients method, described in [2],
C and implemented in the SLICOT Libray routine MB02WD, is used for
C solving the linear system above. The main advantage of this method
C is that in most cases the solution of the system can be computed
C in less time than the time needed to compute the matrix J'*J
C This is, however, problem dependent.
C
C REFERENCES
C
C [1] Kelley, C.T.
C Iterative Methods for Optimization.
C Society for Industrial and Applied Mathematics (SIAM),
C Philadelphia (Pa.), 1999.
C
C [2] Golub, G.H. and van Loan, C.F.
C Matrix Computations. Third Edition.
C M. D. Johns Hopkins University Press, Baltimore, pp. 520-528,
C 1996.
C
C [3] More, J.J.
C The Levenberg-Marquardt algorithm: implementation and theory.
C In Watson, G.A. (Ed.), Numerical Analysis, Lecture Notes in
C Mathematics, vol. 630, Springer-Verlag, Berlin, Heidelberg
C and New York, pp. 105-116, 1978.
C
C NUMERICAL ASPECTS
C
C The Levenberg-Marquardt algorithm described in [3] is scaling
C invariant and globally convergent to (maybe local) minima.
C According to [1], the convergence rate near a local minimum is
C quadratic, if the Jacobian is computed analytically, and linear,
C if the Jacobian is computed numerically.
C
C Whether or not the direct algorithm is faster than the iterative
C Conjugate Gradients algorithm for solving the linear systems
C involved depends on several factors, including the conditioning
C of the Jacobian matrix, and the ratio between its dimensions.
C
C CONTRIBUTORS
C
C A. Riedel, R. Schneider, Chemnitz University of Technology,
C Oct. 2000.
C V. Sima, Research Institute for Informatics, Bucharest, Dec. 2001.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, Mar. 2001,
C Mar. 2002.
C
C KEYWORDS
C
C Conjugate gradients, least-squares approximation,
C Levenberg-Marquardt algorithm, matrix operations, optimization.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE, FOUR, FIVE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, FOUR = 4.0D0,
$ FIVE = 5.0D0 )
DOUBLE PRECISION FACTOR, MARQF, MINIMP, PARMAX
PARAMETER ( FACTOR = 10.0D0**2, MARQF = 2.0D0**2,
$ MINIMP = 2.0D0**(-3), PARMAX = 1.0D20 )
C .. Scalar Arguments ..
CHARACTER ALG, STOR, UPLO, XINIT
INTEGER INFO, ITMAX, IWARN, LDPAR1, LDPAR2, LDWORK,
$ LIPAR, M, N, NFEV, NJEV, NPRINT
DOUBLE PRECISION CGTOL, TOL
C .. Array Arguments ..
DOUBLE PRECISION DPAR1(LDPAR1,*), DPAR2(LDPAR2,*), DWORK(*), X(*)
INTEGER IPAR(*)
C .. Local Scalars ..
LOGICAL CHOL, FULL, INIT, UPPER
INTEGER DWJTJ, E, I, IFLAG, INFOL, ITER, ITERCG, IW1,
$ IW2, IWARNL, JAC, JTE, JW1, JW2, JWORK, LDJ,
$ LDW, LFCN1, LFCN2, LJTJ, LJTJD, LJTJI, NFEVL,
$ SIZEJ, WRKOPT
DOUBLE PRECISION ACTRED, BIGNUM, CGTDEF, EPSMCH, FNORM, FNORM1,
$ GNORM, GSMIN, PAR, SMLNUM, SQREPS, TOLDEF
C .. Local Arrays ..
INTEGER SEED(4)
C .. External Functions ..
DOUBLE PRECISION DDOT, DLAMCH, DNRM2
LOGICAL LSAME
EXTERNAL DDOT, DLAMCH, DNRM2, LSAME
C .. External Subroutines ..
EXTERNAL DCOPY, DLABAD, DLARNV, FCN, JPJ, MB02WD, MB02XD,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, MOD, SQRT
C ..
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
INIT = LSAME( XINIT, 'R' )
CHOL = LSAME( ALG, 'D' )
FULL = LSAME( STOR, 'F' )
UPPER = LSAME( UPLO, 'U' )
C
C Check the scalar input parameters.
C
IWARN = 0
INFO = 0
IF( .NOT.( INIT .OR. LSAME( XINIT, 'G' ) ) ) THEN
INFO = -1
ELSEIF ( .NOT.( CHOL .OR. LSAME( ALG, 'I' ) ) ) THEN
INFO = -2
ELSEIF ( CHOL .AND. .NOT.( FULL .OR. LSAME( STOR, 'P' ) ) ) THEN
INFO = -3
ELSEIF ( CHOL .AND. .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
INFO = -4
ELSEIF ( M.LT.0 ) THEN
INFO = -7
ELSEIF ( N.LT.0 .OR. N.GT.M ) THEN
INFO = -8
ELSEIF ( ITMAX.LT.0 ) THEN
INFO = -9
ELSEIF ( LIPAR.LT.5 ) THEN
INFO = -12
ELSEIF( LDPAR1.LT.0 ) THEN
INFO = -14
ELSEIF( LDPAR2.LT.0 ) THEN
INFO = -16
ELSEIF ( LDWORK.LT.5 ) THEN
INFO = -23
ENDIF
C
C Return if there are illegal arguments.
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MD03AD', -INFO )
RETURN
ENDIF
C
C Quick return if possible.
C
NFEV = 0
NJEV = 0
IF ( MIN( N, ITMAX ).EQ.0 ) THEN
DWORK(1) = FIVE
DWORK(2) = ZERO
DWORK(3) = ZERO
DWORK(4) = ZERO
DWORK(5) = ZERO
RETURN
ENDIF
C
C Call FCN to get the size of the array J, for storing the Jacobian
C matrix, the leading dimension LDJ and the workspace required
C by FCN for IFLAG = 1 and IFLAG = 2, and JPJ. The entries
C DWORK(1:4) should not be modified by the special call of FCN
C below, if XINIT = 'R' and the values in DWORK(1:4) are explicitly
C desired for initialization of the random number generator.
C
IFLAG = 3
IW1 = IPAR(1)
IW2 = IPAR(2)
JW1 = IPAR(3)
JW2 = IPAR(4)
LJTJ = IPAR(5)
C
CALL FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2, LDPAR2,
$ X, NFEVL, DWORK, DWORK, LDJ, DWORK, DWORK, LDWORK,
$ INFOL )
C
SIZEJ = IPAR(1)
LFCN1 = IPAR(2)
LFCN2 = IPAR(3)
LJTJD = IPAR(4)
LJTJI = IPAR(5)
C
IPAR(1) = IW1
IPAR(2) = IW2
IPAR(3) = JW1
IPAR(4) = JW2
IPAR(5) = LJTJ
C
C Define pointers to the array variables stored in DWORK.
C
JAC = 1
E = JAC + SIZEJ
JTE = E + M
IW1 = JTE + N
IW2 = IW1 + N
JW1 = IW2
JW2 = IW2 + N
C
C Check the workspace length.
C
JWORK = JW1
IF ( CHOL ) THEN
IF ( FULL ) THEN
LDW = N*N
ELSE
LDW = ( N*( N + 1 ) ) / 2
ENDIF
DWJTJ = JWORK
JWORK = DWJTJ + LDW
LJTJ = LJTJD
ELSE
LDW = 3*N
LJTJ = LJTJI
ENDIF
IF ( LDWORK.LT.MAX( 5, SIZEJ + M + 2*N +
$ MAX( LFCN1 + N, LFCN2, LDW + LJTJ ) ) )
$ THEN
INFO = -23
ENDIF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'MD03AD', -INFO )
RETURN
ENDIF
C
C Set default tolerances. SQREPS is the square root of the machine
C precision, and GSMIN is used in the tests of the gradient norm.
C
EPSMCH = DLAMCH( 'Epsilon' )
SQREPS = SQRT( EPSMCH )
TOLDEF = TOL
IF ( TOLDEF.LT.ZERO )
$ TOLDEF = SQREPS
CGTDEF = CGTOL
IF ( CGTDEF.LE.ZERO )
$ CGTDEF = SQREPS
GSMIN = FACTOR*EPSMCH
WRKOPT = 5
C
SMLNUM = DLAMCH( 'Safe minimum' ) / DLAMCH( 'Precision' )
BIGNUM = ONE / SMLNUM
CALL DLABAD( SMLNUM, BIGNUM )
C
C Initialization.
C
IF ( INIT ) THEN
C
C SEED is the initial state of the random number generator.
C SEED(4) must be odd.
C
SEED(1) = MOD( INT( DWORK(1) ), 4096 )
SEED(2) = MOD( INT( DWORK(2) ), 4096 )
SEED(3) = MOD( INT( DWORK(3) ), 4096 )
SEED(4) = MOD( 2*INT( DWORK(4) ) + 1, 4096 )
CALL DLARNV( 2, SEED, N, X )
ENDIF
C
C Evaluate the function at the starting point and calculate
C its norm.
C Workspace: need: SIZEJ + M + 2*N + LFCN1;
C prefer: larger.
C
IFLAG = 1
CALL FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2, LDPAR2,
$ X, NFEVL, DWORK(E), DWORK(JAC), LDJ, DWORK(JTE),
$ DWORK(JW1), LDWORK-JW1+1, INFOL )
C
IF ( INFOL.NE.0 ) THEN
INFO = 1
RETURN
END IF
WRKOPT = MAX( WRKOPT, INT( DWORK(JW1) ) + JW1 - 1 )
NFEV = 1
FNORM = DNRM2( M, DWORK(E), 1 )
ACTRED = ZERO
ITERCG = 0
ITER = 0
IWARNL = 0
PAR = ZERO
IF ( IFLAG.LT.0 .OR. FNORM.EQ.ZERO )
$ GO TO 40
C
C Set the initial vector for the conjugate gradients algorithm.
C
DWORK(IW1) = ZERO
CALL DCOPY( N, DWORK(IW1), 0, DWORK(IW1), 1 )
C
C WHILE ( nonconvergence and ITER < ITMAX ) DO
C
C Beginning of the outer loop.
C
10 CONTINUE
C
C Calculate the Jacobian matrix.
C Workspace: need: SIZEJ + M + 2*N + LFCN2;
C prefer: larger.
C
ITER = ITER + 1
IFLAG = 2
CALL FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEVL, DWORK(E), DWORK(JAC), LDJ,
$ DWORK(JTE), DWORK(JW1), LDWORK-JW1+1, INFOL )
C
IF ( INFOL.NE.0 ) THEN
INFO = 2
RETURN
END IF
C
C Compute the gradient norm.
C
GNORM = DNRM2( N, DWORK(JTE), 1 )
IF ( NFEVL.GT.0 )
$ NFEV = NFEV + NFEVL
NJEV = NJEV + 1
IF ( GNORM.LE.GSMIN )
$ IWARN = 3
IF ( IWARN.NE.0 )
$ GO TO 40
IF ( ITER.EQ.1 ) THEN
WRKOPT = MAX( WRKOPT, INT( DWORK(JW1) ) + JW1 - 1 )
PAR = MIN( GNORM, SQRT( PARMAX ) )
END IF
IF ( IFLAG.LT.0 )
$ GO TO 40
C
C If requested, call FCN to enable printing of iterates.
C
IF ( NPRINT.GT.0 ) THEN
IFLAG = 0
IF ( MOD( ITER-1, NPRINT ).EQ.0 ) THEN
CALL FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEV, DWORK(E), DWORK(JAC), LDJ,
$ DWORK(JTE), DWORK(JW1), LDWORK-JW1+1, INFOL )
C
IF ( IFLAG.LT.0 )
$ GO TO 40
END IF
END IF
C
C Beginning of the inner loop.
C
20 CONTINUE
C
C Store the Levenberg factor in DWORK(E) (which is no longer
C needed), to pass it to JPJ routine.
C
DWORK(E) = PAR
C
C Solve (J'*J + PAR*I)*x = J'*e, and store x in DWORK(IW1).
C Additional workspace:
C N*N + DW(JPJ), if ALG = 'D', STOR = 'F';
C N*( N + 1)/2 + DW(JPJ), if ALG = 'D', STOR = 'P';
C 3*N + DW(JPJ), if ALG = 'I'.
C
IF ( CHOL ) THEN
CALL DCOPY( N, DWORK(JTE), 1, DWORK(IW1), 1 )
CALL MB02XD( 'Function', STOR, UPLO, JPJ, M, N, 1, IPAR,
$ LIPAR, DWORK(E), 1, DWORK(JAC), LDJ,
$ DWORK(IW1), N, DWORK(DWJTJ), N,
$ DWORK(JWORK), LDWORK-JWORK+1, INFOL )
ELSE
CALL MB02WD( 'Function', JPJ, N, IPAR, LIPAR, DWORK(E),
$ 1, 3*N, DWORK(JAC), LDJ, DWORK(JTE), 1,
$ DWORK(IW1), 1, CGTOL*GNORM, DWORK(JWORK),
$ LDWORK-JWORK+1, IWARN, INFOL )
ITERCG = ITERCG + INT( DWORK(JWORK) )
IWARNL = MAX( 2*IWARN, IWARNL )
ENDIF
C
IF ( INFOL.NE.0 ) THEN
INFO = 3
RETURN
ENDIF
C
C Compute updated X.
C
DO 30 I = 0, N - 1
DWORK(IW2+I) = X(I+1) - DWORK(IW1+I)
30 CONTINUE
C
C Evaluate the function at x - p and calculate its norm.
C Workspace: need: SIZEJ + M + 3*N + LFCN1;
C prefer: larger.
C
IFLAG = 1
CALL FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, DWORK(IW2), NFEVL, DWORK(E), DWORK(JAC),
$ LDJ, DWORK(JTE), DWORK(JW2), LDWORK-JW2+1, INFOL )
C
IF ( INFOL.NE.0 ) THEN
INFO = 1
RETURN
END IF
C
NFEV = NFEV + 1
IF ( IFLAG.LT.0 )
$ GO TO 40
FNORM1 = DNRM2( M, DWORK(E), 1 )
C
C Now, check whether this step was successful and update the
C Levenberg factor.
C
IF ( FNORM.LT.FNORM1 ) THEN
C
C Unsuccessful step: increase PAR.
C
ACTRED = ONE
IF ( PAR.GT.PARMAX ) THEN
IF ( PAR/MARQF.LE.BIGNUM )
$ PAR = PAR*MARQF
ELSE
PAR = PAR*MARQF
END IF
C
ELSE
C
C Successful step: update PAR, X, and FNORM.
C
ACTRED = ONE - ( FNORM1/FNORM )**2
IF ( ( FNORM - FNORM1 )*( FNORM + FNORM1 ) .LT.
$ MINIMP*DDOT( N, DWORK(IW1), 1,
$ DWORK(JTE), 1 ) ) THEN
IF ( PAR.GT.PARMAX ) THEN
IF ( PAR/MARQF.LE.BIGNUM )
$ PAR = PAR*MARQF
ELSE
PAR = PAR*MARQF
END IF
ELSE
PAR = MAX( PAR/MARQF, SMLNUM )
ENDIF
CALL DCOPY( N, DWORK(IW2), 1, X, 1 )
FNORM = FNORM1
ENDIF
C
IF ( ( ACTRED.LE.TOLDEF ) .OR. ( ITER.GT.ITMAX ) .OR.
$ ( PAR.GT.PARMAX ) )
$ GO TO 40
IF ( ACTRED.LE.EPSMCH ) THEN
IWARN = 4
GO TO 40
ENDIF
C
C End of the inner loop. Repeat if unsuccessful iteration.
C
IF ( FNORM.LT.FNORM1 )
$ GO TO 20
C
C End of the outer loop.
C
GO TO 10
C
C END WHILE 10
C
40 CONTINUE
C
C Termination, either normal or user imposed.
C
IF ( ACTRED.GT.TOLDEF )
$ IWARN = 1
IF ( IWARNL.NE.0 )
$ IWARN = 2
C
IF ( IFLAG.LT.0 )
$ IWARN = IFLAG
IF ( NPRINT.GT.0 ) THEN
IFLAG = 0
CALL FCN( IFLAG, M, N, IPAR, LIPAR, DPAR1, LDPAR1, DPAR2,
$ LDPAR2, X, NFEV, DWORK(E), DWORK(JAC), LDJ,
$ DWORK(JTE), DWORK(JW1), LDWORK-JW1+1, INFOL )
IF ( IFLAG.LT.0 )
$ IWARN = IFLAG
END IF
C
DWORK(1) = WRKOPT
DWORK(2) = FNORM
DWORK(3) = ITER
DWORK(4) = ITERCG
DWORK(5) = PAR
C
RETURN
C *** Last line of MD03AD ***
END
|
