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
|
SUBROUTINE DLALN2( LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B,
$ LDB, WR, WI, X, LDX, SCALE, XNORM, INFO )
*
* -- LAPACK auxiliary routine (version 2.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL LTRANS
INTEGER INFO, LDA, LDB, LDX, NA, NW
DOUBLE PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* DLALN2 solves a system of the form (ca A - w D ) X = s B
* or (ca A' - w D) X = s B with possible scaling ("s") and
* perturbation of A. (A' means A-transpose.)
*
* A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
* real diagonal matrix, w is a real or complex value, and X and B are
* NA x 1 matrices -- real if w is real, complex if w is complex. NA
* may be 1 or 2.
*
* If w is complex, X and B are represented as NA x 2 matrices,
* the first column of each being the real part and the second
* being the imaginary part.
*
* "s" is a scaling factor (.LE. 1), computed by DLALN2, which is
* so chosen that X can be computed without overflow. X is further
* scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
* than overflow.
*
* If both singular values of (ca A - w D) are less than SMIN,
* SMIN*identity will be used instead of (ca A - w D). If only one
* singular value is less than SMIN, one element of (ca A - w D) will be
* perturbed enough to make the smallest singular value roughly SMIN.
* If both singular values are at least SMIN, (ca A - w D) will not be
* perturbed. In any case, the perturbation will be at most some small
* multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
* are computed by infinity-norm approximations, and thus will only be
* correct to a factor of 2 or so.
*
* Note: all input quantities are assumed to be smaller than overflow
* by a reasonable factor. (See BIGNUM.)
*
* Arguments
* ==========
*
* LTRANS (input) LOGICAL
* =.TRUE.: A-transpose will be used.
* =.FALSE.: A will be used (not transposed.)
*
* NA (input) INTEGER
* The size of the matrix A. It may (only) be 1 or 2.
*
* NW (input) INTEGER
* 1 if "w" is real, 2 if "w" is complex. It may only be 1
* or 2.
*
* SMIN (input) DOUBLE PRECISION
* The desired lower bound on the singular values of A. This
* should be a safe distance away from underflow or overflow,
* say, between (underflow/machine precision) and (machine
* precision * overflow ). (See BIGNUM and ULP.)
*
* CA (input) DOUBLE PRECISION
* The coefficient c, which A is multiplied by.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,NA)
* The NA x NA matrix A.
*
* LDA (input) INTEGER
* The leading dimension of A. It must be at least NA.
*
* D1 (input) DOUBLE PRECISION
* The 1,1 element in the diagonal matrix D.
*
* D2 (input) DOUBLE PRECISION
* The 2,2 element in the diagonal matrix D. Not used if NW=1.
*
* B (input) DOUBLE PRECISION array, dimension (LDB,NW)
* The NA x NW matrix B (right-hand side). If NW=2 ("w" is
* complex), column 1 contains the real part of B and column 2
* contains the imaginary part.
*
* LDB (input) INTEGER
* The leading dimension of B. It must be at least NA.
*
* WR (input) DOUBLE PRECISION
* The real part of the scalar "w".
*
* WI (input) DOUBLE PRECISION
* The imaginary part of the scalar "w". Not used if NW=1.
*
* X (output) DOUBLE PRECISION array, dimension (LDX,NW)
* The NA x NW matrix X (unknowns), as computed by DLALN2.
* If NW=2 ("w" is complex), on exit, column 1 will contain
* the real part of X and column 2 will contain the imaginary
* part.
*
* LDX (input) INTEGER
* The leading dimension of X. It must be at least NA.
*
* SCALE (output) DOUBLE PRECISION
* The scale factor that B must be multiplied by to insure
* that overflow does not occur when computing X. Thus,
* (ca A - w D) X will be SCALE*B, not B (ignoring
* perturbations of A.) It will be at most 1.
*
* XNORM (output) DOUBLE PRECISION
* The infinity-norm of X, when X is regarded as an NA x NW
* real matrix.
*
* INFO (output) INTEGER
* An error flag. It will be set to zero if no error occurs,
* a negative number if an argument is in error, or a positive
* number if ca A - w D had to be perturbed.
* The possible values are:
* = 0: No error occurred, and (ca A - w D) did not have to be
* perturbed.
* = 1: (ca A - w D) had to be perturbed to make its smallest
* (or only) singular value greater than SMIN.
* NOTE: In the interests of speed, this routine does not
* check the inputs for errors.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
* ..
* .. Local Scalars ..
INTEGER ICMAX, J
DOUBLE PRECISION BBND, BI1, BI2, BIGNUM, BNORM, BR1, BR2, CI21,
$ CI22, CMAX, CNORM, CR21, CR22, CSI, CSR, LI21,
$ LR21, SMINI, SMLNUM, TEMP, U22ABS, UI11, UI11R,
$ UI12, UI12S, UI22, UR11, UR11R, UR12, UR12S,
$ UR22, XI1, XI2, XR1, XR2
* ..
* .. Local Arrays ..
LOGICAL RSWAP( 4 ), ZSWAP( 4 )
INTEGER IPIVOT( 4, 4 )
DOUBLE PRECISION CI( 2, 2 ), CIV( 4 ), CR( 2, 2 ), CRV( 4 )
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DLADIV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Equivalences ..
EQUIVALENCE ( CI( 1, 1 ), CIV( 1 ) ),
$ ( CR( 1, 1 ), CRV( 1 ) )
* ..
* .. Data statements ..
DATA ZSWAP / .FALSE., .FALSE., .TRUE., .TRUE. /
DATA RSWAP / .FALSE., .TRUE., .FALSE., .TRUE. /
DATA IPIVOT / 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, 4,
$ 3, 2, 1 /
* ..
* .. Executable Statements ..
*
* Compute BIGNUM
*
SMLNUM = TWO*DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
SMINI = MAX( SMIN, SMLNUM )
*
* Don't check for input errors
*
INFO = 0
*
* Standard Initializations
*
SCALE = ONE
*
IF( NA.EQ.1 ) THEN
*
* 1 x 1 (i.e., scalar) system C X = B
*
IF( NW.EQ.1 ) THEN
*
* Real 1x1 system.
*
* C = ca A - w D
*
CSR = CA*A( 1, 1 ) - WR*D1
CNORM = ABS( CSR )
*
* If | C | < SMINI, use C = SMINI
*
IF( CNORM.LT.SMINI ) THEN
CSR = SMINI
CNORM = SMINI
INFO = 1
END IF
*
* Check scaling for X = B / C
*
BNORM = ABS( B( 1, 1 ) )
IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*CNORM )
$ SCALE = ONE / BNORM
END IF
*
* Compute X
*
X( 1, 1 ) = ( B( 1, 1 )*SCALE ) / CSR
XNORM = ABS( X( 1, 1 ) )
ELSE
*
* Complex 1x1 system (w is complex)
*
* C = ca A - w D
*
CSR = CA*A( 1, 1 ) - WR*D1
CSI = -WI*D1
CNORM = ABS( CSR ) + ABS( CSI )
*
* If | C | < SMINI, use C = SMINI
*
IF( CNORM.LT.SMINI ) THEN
CSR = SMINI
CSI = ZERO
CNORM = SMINI
INFO = 1
END IF
*
* Check scaling for X = B / C
*
BNORM = ABS( B( 1, 1 ) ) + ABS( B( 1, 2 ) )
IF( CNORM.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*CNORM )
$ SCALE = ONE / BNORM
END IF
*
* Compute X
*
CALL DLADIV( SCALE*B( 1, 1 ), SCALE*B( 1, 2 ), CSR, CSI,
$ X( 1, 1 ), X( 1, 2 ) )
XNORM = ABS( X( 1, 1 ) ) + ABS( X( 1, 2 ) )
END IF
*
ELSE
*
* 2x2 System
*
* Compute the real part of C = ca A - w D (or ca A' - w D )
*
CR( 1, 1 ) = CA*A( 1, 1 ) - WR*D1
CR( 2, 2 ) = CA*A( 2, 2 ) - WR*D2
IF( LTRANS ) THEN
CR( 1, 2 ) = CA*A( 2, 1 )
CR( 2, 1 ) = CA*A( 1, 2 )
ELSE
CR( 2, 1 ) = CA*A( 2, 1 )
CR( 1, 2 ) = CA*A( 1, 2 )
END IF
*
IF( NW.EQ.1 ) THEN
*
* Real 2x2 system (w is real)
*
* Find the largest element in C
*
CMAX = ZERO
ICMAX = 0
*
DO 10 J = 1, 4
IF( ABS( CRV( J ) ).GT.CMAX ) THEN
CMAX = ABS( CRV( J ) )
ICMAX = J
END IF
10 CONTINUE
*
* If norm(C) < SMINI, use SMINI*identity.
*
IF( CMAX.LT.SMINI ) THEN
BNORM = MAX( ABS( B( 1, 1 ) ), ABS( B( 2, 1 ) ) )
IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*SMINI )
$ SCALE = ONE / BNORM
END IF
TEMP = SCALE / SMINI
X( 1, 1 ) = TEMP*B( 1, 1 )
X( 2, 1 ) = TEMP*B( 2, 1 )
XNORM = TEMP*BNORM
INFO = 1
RETURN
END IF
*
* Gaussian elimination with complete pivoting.
*
UR11 = CRV( ICMAX )
CR21 = CRV( IPIVOT( 2, ICMAX ) )
UR12 = CRV( IPIVOT( 3, ICMAX ) )
CR22 = CRV( IPIVOT( 4, ICMAX ) )
UR11R = ONE / UR11
LR21 = UR11R*CR21
UR22 = CR22 - UR12*LR21
*
* If smaller pivot < SMINI, use SMINI
*
IF( ABS( UR22 ).LT.SMINI ) THEN
UR22 = SMINI
INFO = 1
END IF
IF( RSWAP( ICMAX ) ) THEN
BR1 = B( 2, 1 )
BR2 = B( 1, 1 )
ELSE
BR1 = B( 1, 1 )
BR2 = B( 2, 1 )
END IF
BR2 = BR2 - LR21*BR1
BBND = MAX( ABS( BR1*( UR22*UR11R ) ), ABS( BR2 ) )
IF( BBND.GT.ONE .AND. ABS( UR22 ).LT.ONE ) THEN
IF( BBND.GE.BIGNUM*ABS( UR22 ) )
$ SCALE = ONE / BBND
END IF
*
XR2 = ( BR2*SCALE ) / UR22
XR1 = ( SCALE*BR1 )*UR11R - XR2*( UR11R*UR12 )
IF( ZSWAP( ICMAX ) ) THEN
X( 1, 1 ) = XR2
X( 2, 1 ) = XR1
ELSE
X( 1, 1 ) = XR1
X( 2, 1 ) = XR2
END IF
XNORM = MAX( ABS( XR1 ), ABS( XR2 ) )
*
* Further scaling if norm(A) norm(X) > overflow
*
IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
IF( XNORM.GT.BIGNUM / CMAX ) THEN
TEMP = CMAX / BIGNUM
X( 1, 1 ) = TEMP*X( 1, 1 )
X( 2, 1 ) = TEMP*X( 2, 1 )
XNORM = TEMP*XNORM
SCALE = TEMP*SCALE
END IF
END IF
ELSE
*
* Complex 2x2 system (w is complex)
*
* Find the largest element in C
*
CI( 1, 1 ) = -WI*D1
CI( 2, 1 ) = ZERO
CI( 1, 2 ) = ZERO
CI( 2, 2 ) = -WI*D2
CMAX = ZERO
ICMAX = 0
*
DO 20 J = 1, 4
IF( ABS( CRV( J ) )+ABS( CIV( J ) ).GT.CMAX ) THEN
CMAX = ABS( CRV( J ) ) + ABS( CIV( J ) )
ICMAX = J
END IF
20 CONTINUE
*
* If norm(C) < SMINI, use SMINI*identity.
*
IF( CMAX.LT.SMINI ) THEN
BNORM = MAX( ABS( B( 1, 1 ) )+ABS( B( 1, 2 ) ),
$ ABS( B( 2, 1 ) )+ABS( B( 2, 2 ) ) )
IF( SMINI.LT.ONE .AND. BNORM.GT.ONE ) THEN
IF( BNORM.GT.BIGNUM*SMINI )
$ SCALE = ONE / BNORM
END IF
TEMP = SCALE / SMINI
X( 1, 1 ) = TEMP*B( 1, 1 )
X( 2, 1 ) = TEMP*B( 2, 1 )
X( 1, 2 ) = TEMP*B( 1, 2 )
X( 2, 2 ) = TEMP*B( 2, 2 )
XNORM = TEMP*BNORM
INFO = 1
RETURN
END IF
*
* Gaussian elimination with complete pivoting.
*
UR11 = CRV( ICMAX )
UI11 = CIV( ICMAX )
CR21 = CRV( IPIVOT( 2, ICMAX ) )
CI21 = CIV( IPIVOT( 2, ICMAX ) )
UR12 = CRV( IPIVOT( 3, ICMAX ) )
UI12 = CIV( IPIVOT( 3, ICMAX ) )
CR22 = CRV( IPIVOT( 4, ICMAX ) )
CI22 = CIV( IPIVOT( 4, ICMAX ) )
IF( ICMAX.EQ.1 .OR. ICMAX.EQ.4 ) THEN
*
* Code when off-diagonals of pivoted C are real
*
IF( ABS( UR11 ).GT.ABS( UI11 ) ) THEN
TEMP = UI11 / UR11
UR11R = ONE / ( UR11*( ONE+TEMP**2 ) )
UI11R = -TEMP*UR11R
ELSE
TEMP = UR11 / UI11
UI11R = -ONE / ( UI11*( ONE+TEMP**2 ) )
UR11R = -TEMP*UI11R
END IF
LR21 = CR21*UR11R
LI21 = CR21*UI11R
UR12S = UR12*UR11R
UI12S = UR12*UI11R
UR22 = CR22 - UR12*LR21
UI22 = CI22 - UR12*LI21
ELSE
*
* Code when diagonals of pivoted C are real
*
UR11R = ONE / UR11
UI11R = ZERO
LR21 = CR21*UR11R
LI21 = CI21*UR11R
UR12S = UR12*UR11R
UI12S = UI12*UR11R
UR22 = CR22 - UR12*LR21 + UI12*LI21
UI22 = -UR12*LI21 - UI12*LR21
END IF
U22ABS = ABS( UR22 ) + ABS( UI22 )
*
* If smaller pivot < SMINI, use SMINI
*
IF( U22ABS.LT.SMINI ) THEN
UR22 = SMINI
UI22 = ZERO
INFO = 1
END IF
IF( RSWAP( ICMAX ) ) THEN
BR2 = B( 1, 1 )
BR1 = B( 2, 1 )
BI2 = B( 1, 2 )
BI1 = B( 2, 2 )
ELSE
BR1 = B( 1, 1 )
BR2 = B( 2, 1 )
BI1 = B( 1, 2 )
BI2 = B( 2, 2 )
END IF
BR2 = BR2 - LR21*BR1 + LI21*BI1
BI2 = BI2 - LI21*BR1 - LR21*BI1
BBND = MAX( ( ABS( BR1 )+ABS( BI1 ) )*
$ ( U22ABS*( ABS( UR11R )+ABS( UI11R ) ) ),
$ ABS( BR2 )+ABS( BI2 ) )
IF( BBND.GT.ONE .AND. U22ABS.LT.ONE ) THEN
IF( BBND.GE.BIGNUM*U22ABS ) THEN
SCALE = ONE / BBND
BR1 = SCALE*BR1
BI1 = SCALE*BI1
BR2 = SCALE*BR2
BI2 = SCALE*BI2
END IF
END IF
*
CALL DLADIV( BR2, BI2, UR22, UI22, XR2, XI2 )
XR1 = UR11R*BR1 - UI11R*BI1 - UR12S*XR2 + UI12S*XI2
XI1 = UI11R*BR1 + UR11R*BI1 - UI12S*XR2 - UR12S*XI2
IF( ZSWAP( ICMAX ) ) THEN
X( 1, 1 ) = XR2
X( 2, 1 ) = XR1
X( 1, 2 ) = XI2
X( 2, 2 ) = XI1
ELSE
X( 1, 1 ) = XR1
X( 2, 1 ) = XR2
X( 1, 2 ) = XI1
X( 2, 2 ) = XI2
END IF
XNORM = MAX( ABS( XR1 )+ABS( XI1 ), ABS( XR2 )+ABS( XI2 ) )
*
* Further scaling if norm(A) norm(X) > overflow
*
IF( XNORM.GT.ONE .AND. CMAX.GT.ONE ) THEN
IF( XNORM.GT.BIGNUM / CMAX ) THEN
TEMP = CMAX / BIGNUM
X( 1, 1 ) = TEMP*X( 1, 1 )
X( 2, 1 ) = TEMP*X( 2, 1 )
X( 1, 2 ) = TEMP*X( 1, 2 )
X( 2, 2 ) = TEMP*X( 2, 2 )
XNORM = TEMP*XNORM
SCALE = TEMP*SCALE
END IF
END IF
END IF
END IF
*
RETURN
*
* End of DLALN2
*
END
|
