1 SUBROUTINE ZLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
2 $ SCALE, CNORM, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER DIAG, NORMIN, TRANS, UPLO
11 INTEGER INFO, KD, LDAB, N
12 DOUBLE PRECISION SCALE
13 * ..
14 * .. Array Arguments ..
15 DOUBLE PRECISION CNORM( * )
16 COMPLEX*16 AB( LDAB, * ), X( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZLATBS solves one of the triangular systems
23 *
24 * A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
25 *
26 * with scaling to prevent overflow, where A is an upper or lower
27 * triangular band matrix. Here A**T denotes the transpose of A, x and b
28 * are n-element vectors, and s is a scaling factor, usually less than
29 * or equal to 1, chosen so that the components of x will be less than
30 * the overflow threshold. If the unscaled problem will not cause
31 * overflow, the Level 2 BLAS routine ZTBSV is called. If the matrix A
32 * is singular (A(j,j) = 0 for some j), then s is set to 0 and a
33 * non-trivial solution to A*x = 0 is returned.
34 *
35 * Arguments
36 * =========
37 *
38 * UPLO (input) CHARACTER*1
39 * Specifies whether the matrix A is upper or lower triangular.
40 * = 'U': Upper triangular
41 * = 'L': Lower triangular
42 *
43 * TRANS (input) CHARACTER*1
44 * Specifies the operation applied to A.
45 * = 'N': Solve A * x = s*b (No transpose)
46 * = 'T': Solve A**T * x = s*b (Transpose)
47 * = 'C': Solve A**H * x = s*b (Conjugate transpose)
48 *
49 * DIAG (input) CHARACTER*1
50 * Specifies whether or not the matrix A is unit triangular.
51 * = 'N': Non-unit triangular
52 * = 'U': Unit triangular
53 *
54 * NORMIN (input) CHARACTER*1
55 * Specifies whether CNORM has been set or not.
56 * = 'Y': CNORM contains the column norms on entry
57 * = 'N': CNORM is not set on entry. On exit, the norms will
58 * be computed and stored in CNORM.
59 *
60 * N (input) INTEGER
61 * The order of the matrix A. N >= 0.
62 *
63 * KD (input) INTEGER
64 * The number of subdiagonals or superdiagonals in the
65 * triangular matrix A. KD >= 0.
66 *
67 * AB (input) COMPLEX*16 array, dimension (LDAB,N)
68 * The upper or lower triangular band matrix A, stored in the
69 * first KD+1 rows of the array. The j-th column of A is stored
70 * in the j-th column of the array AB as follows:
71 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
72 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
73 *
74 * LDAB (input) INTEGER
75 * The leading dimension of the array AB. LDAB >= KD+1.
76 *
77 * X (input/output) COMPLEX*16 array, dimension (N)
78 * On entry, the right hand side b of the triangular system.
79 * On exit, X is overwritten by the solution vector x.
80 *
81 * SCALE (output) DOUBLE PRECISION
82 * The scaling factor s for the triangular system
83 * A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
84 * If SCALE = 0, the matrix A is singular or badly scaled, and
85 * the vector x is an exact or approximate solution to A*x = 0.
86 *
87 * CNORM (input or output) DOUBLE PRECISION array, dimension (N)
88 *
89 * If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
90 * contains the norm of the off-diagonal part of the j-th column
91 * of A. If TRANS = 'N', CNORM(j) must be greater than or equal
92 * to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
93 * must be greater than or equal to the 1-norm.
94 *
95 * If NORMIN = 'N', CNORM is an output argument and CNORM(j)
96 * returns the 1-norm of the offdiagonal part of the j-th column
97 * of A.
98 *
99 * INFO (output) INTEGER
100 * = 0: successful exit
101 * < 0: if INFO = -k, the k-th argument had an illegal value
102 *
103 * Further Details
104 * ======= =======
105 *
106 * A rough bound on x is computed; if that is less than overflow, ZTBSV
107 * is called, otherwise, specific code is used which checks for possible
108 * overflow or divide-by-zero at every operation.
109 *
110 * A columnwise scheme is used for solving A*x = b. The basic algorithm
111 * if A is lower triangular is
112 *
113 * x[1:n] := b[1:n]
114 * for j = 1, ..., n
115 * x(j) := x(j) / A(j,j)
116 * x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
117 * end
118 *
119 * Define bounds on the components of x after j iterations of the loop:
120 * M(j) = bound on x[1:j]
121 * G(j) = bound on x[j+1:n]
122 * Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
123 *
124 * Then for iteration j+1 we have
125 * M(j+1) <= G(j) / | A(j+1,j+1) |
126 * G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
127 * <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
128 *
129 * where CNORM(j+1) is greater than or equal to the infinity-norm of
130 * column j+1 of A, not counting the diagonal. Hence
131 *
132 * G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
133 * 1<=i<=j
134 * and
135 *
136 * |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
137 * 1<=i< j
138 *
139 * Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTBSV if the
140 * reciprocal of the largest M(j), j=1,..,n, is larger than
141 * max(underflow, 1/overflow).
142 *
143 * The bound on x(j) is also used to determine when a step in the
144 * columnwise method can be performed without fear of overflow. If
145 * the computed bound is greater than a large constant, x is scaled to
146 * prevent overflow, but if the bound overflows, x is set to 0, x(j) to
147 * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
148 *
149 * Similarly, a row-wise scheme is used to solve A**T *x = b or
150 * A**H *x = b. The basic algorithm for A upper triangular is
151 *
152 * for j = 1, ..., n
153 * x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
154 * end
155 *
156 * We simultaneously compute two bounds
157 * G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
158 * M(j) = bound on x(i), 1<=i<=j
159 *
160 * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
161 * add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
162 * Then the bound on x(j) is
163 *
164 * M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
165 *
166 * <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
167 * 1<=i<=j
168 *
169 * and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both greater
170 * than max(underflow, 1/overflow).
171 *
172 * =====================================================================
173 *
174 * .. Parameters ..
175 DOUBLE PRECISION ZERO, HALF, ONE, TWO
176 PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
177 $ TWO = 2.0D+0 )
178 * ..
179 * .. Local Scalars ..
180 LOGICAL NOTRAN, NOUNIT, UPPER
181 INTEGER I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
182 DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
183 $ XBND, XJ, XMAX
184 COMPLEX*16 CSUMJ, TJJS, USCAL, ZDUM
185 * ..
186 * .. External Functions ..
187 LOGICAL LSAME
188 INTEGER IDAMAX, IZAMAX
189 DOUBLE PRECISION DLAMCH, DZASUM
190 COMPLEX*16 ZDOTC, ZDOTU, ZLADIV
191 EXTERNAL LSAME, IDAMAX, IZAMAX, DLAMCH, DZASUM, ZDOTC,
192 $ ZDOTU, ZLADIV
193 * ..
194 * .. External Subroutines ..
195 EXTERNAL DSCAL, XERBLA, ZAXPY, ZDSCAL, ZTBSV
196 * ..
197 * .. Intrinsic Functions ..
198 INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
199 * ..
200 * .. Statement Functions ..
201 DOUBLE PRECISION CABS1, CABS2
202 * ..
203 * .. Statement Function definitions ..
204 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
205 CABS2( ZDUM ) = ABS( DBLE( ZDUM ) / 2.D0 ) +
206 $ ABS( DIMAG( ZDUM ) / 2.D0 )
207 * ..
208 * .. Executable Statements ..
209 *
210 INFO = 0
211 UPPER = LSAME( UPLO, 'U' )
212 NOTRAN = LSAME( TRANS, 'N' )
213 NOUNIT = LSAME( DIAG, 'N' )
214 *
215 * Test the input parameters.
216 *
217 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
218 INFO = -1
219 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
220 $ LSAME( TRANS, 'C' ) ) THEN
221 INFO = -2
222 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
223 INFO = -3
224 ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
225 $ LSAME( NORMIN, 'N' ) ) THEN
226 INFO = -4
227 ELSE IF( N.LT.0 ) THEN
228 INFO = -5
229 ELSE IF( KD.LT.0 ) THEN
230 INFO = -6
231 ELSE IF( LDAB.LT.KD+1 ) THEN
232 INFO = -8
233 END IF
234 IF( INFO.NE.0 ) THEN
235 CALL XERBLA( 'ZLATBS', -INFO )
236 RETURN
237 END IF
238 *
239 * Quick return if possible
240 *
241 IF( N.EQ.0 )
242 $ RETURN
243 *
244 * Determine machine dependent parameters to control overflow.
245 *
246 SMLNUM = DLAMCH( 'Safe minimum' )
247 BIGNUM = ONE / SMLNUM
248 CALL DLABAD( SMLNUM, BIGNUM )
249 SMLNUM = SMLNUM / DLAMCH( 'Precision' )
250 BIGNUM = ONE / SMLNUM
251 SCALE = ONE
252 *
253 IF( LSAME( NORMIN, 'N' ) ) THEN
254 *
255 * Compute the 1-norm of each column, not including the diagonal.
256 *
257 IF( UPPER ) THEN
258 *
259 * A is upper triangular.
260 *
261 DO 10 J = 1, N
262 JLEN = MIN( KD, J-1 )
263 CNORM( J ) = DZASUM( JLEN, AB( KD+1-JLEN, J ), 1 )
264 10 CONTINUE
265 ELSE
266 *
267 * A is lower triangular.
268 *
269 DO 20 J = 1, N
270 JLEN = MIN( KD, N-J )
271 IF( JLEN.GT.0 ) THEN
272 CNORM( J ) = DZASUM( JLEN, AB( 2, J ), 1 )
273 ELSE
274 CNORM( J ) = ZERO
275 END IF
276 20 CONTINUE
277 END IF
278 END IF
279 *
280 * Scale the column norms by TSCAL if the maximum element in CNORM is
281 * greater than BIGNUM/2.
282 *
283 IMAX = IDAMAX( N, CNORM, 1 )
284 TMAX = CNORM( IMAX )
285 IF( TMAX.LE.BIGNUM*HALF ) THEN
286 TSCAL = ONE
287 ELSE
288 TSCAL = HALF / ( SMLNUM*TMAX )
289 CALL DSCAL( N, TSCAL, CNORM, 1 )
290 END IF
291 *
292 * Compute a bound on the computed solution vector to see if the
293 * Level 2 BLAS routine ZTBSV can be used.
294 *
295 XMAX = ZERO
296 DO 30 J = 1, N
297 XMAX = MAX( XMAX, CABS2( X( J ) ) )
298 30 CONTINUE
299 XBND = XMAX
300 IF( NOTRAN ) THEN
301 *
302 * Compute the growth in A * x = b.
303 *
304 IF( UPPER ) THEN
305 JFIRST = N
306 JLAST = 1
307 JINC = -1
308 MAIND = KD + 1
309 ELSE
310 JFIRST = 1
311 JLAST = N
312 JINC = 1
313 MAIND = 1
314 END IF
315 *
316 IF( TSCAL.NE.ONE ) THEN
317 GROW = ZERO
318 GO TO 60
319 END IF
320 *
321 IF( NOUNIT ) THEN
322 *
323 * A is non-unit triangular.
324 *
325 * Compute GROW = 1/G(j) and XBND = 1/M(j).
326 * Initially, G(0) = max{x(i), i=1,...,n}.
327 *
328 GROW = HALF / MAX( XBND, SMLNUM )
329 XBND = GROW
330 DO 40 J = JFIRST, JLAST, JINC
331 *
332 * Exit the loop if the growth factor is too small.
333 *
334 IF( GROW.LE.SMLNUM )
335 $ GO TO 60
336 *
337 TJJS = AB( MAIND, J )
338 TJJ = CABS1( TJJS )
339 *
340 IF( TJJ.GE.SMLNUM ) THEN
341 *
342 * M(j) = G(j-1) / abs(A(j,j))
343 *
344 XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
345 ELSE
346 *
347 * M(j) could overflow, set XBND to 0.
348 *
349 XBND = ZERO
350 END IF
351 *
352 IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
353 *
354 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
355 *
356 GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
357 ELSE
358 *
359 * G(j) could overflow, set GROW to 0.
360 *
361 GROW = ZERO
362 END IF
363 40 CONTINUE
364 GROW = XBND
365 ELSE
366 *
367 * A is unit triangular.
368 *
369 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
370 *
371 GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
372 DO 50 J = JFIRST, JLAST, JINC
373 *
374 * Exit the loop if the growth factor is too small.
375 *
376 IF( GROW.LE.SMLNUM )
377 $ GO TO 60
378 *
379 * G(j) = G(j-1)*( 1 + CNORM(j) )
380 *
381 GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
382 50 CONTINUE
383 END IF
384 60 CONTINUE
385 *
386 ELSE
387 *
388 * Compute the growth in A**T * x = b or A**H * x = b.
389 *
390 IF( UPPER ) THEN
391 JFIRST = 1
392 JLAST = N
393 JINC = 1
394 MAIND = KD + 1
395 ELSE
396 JFIRST = N
397 JLAST = 1
398 JINC = -1
399 MAIND = 1
400 END IF
401 *
402 IF( TSCAL.NE.ONE ) THEN
403 GROW = ZERO
404 GO TO 90
405 END IF
406 *
407 IF( NOUNIT ) THEN
408 *
409 * A is non-unit triangular.
410 *
411 * Compute GROW = 1/G(j) and XBND = 1/M(j).
412 * Initially, M(0) = max{x(i), i=1,...,n}.
413 *
414 GROW = HALF / MAX( XBND, SMLNUM )
415 XBND = GROW
416 DO 70 J = JFIRST, JLAST, JINC
417 *
418 * Exit the loop if the growth factor is too small.
419 *
420 IF( GROW.LE.SMLNUM )
421 $ GO TO 90
422 *
423 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
424 *
425 XJ = ONE + CNORM( J )
426 GROW = MIN( GROW, XBND / XJ )
427 *
428 TJJS = AB( MAIND, J )
429 TJJ = CABS1( TJJS )
430 *
431 IF( TJJ.GE.SMLNUM ) THEN
432 *
433 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
434 *
435 IF( XJ.GT.TJJ )
436 $ XBND = XBND*( TJJ / XJ )
437 ELSE
438 *
439 * M(j) could overflow, set XBND to 0.
440 *
441 XBND = ZERO
442 END IF
443 70 CONTINUE
444 GROW = MIN( GROW, XBND )
445 ELSE
446 *
447 * A is unit triangular.
448 *
449 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
450 *
451 GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
452 DO 80 J = JFIRST, JLAST, JINC
453 *
454 * Exit the loop if the growth factor is too small.
455 *
456 IF( GROW.LE.SMLNUM )
457 $ GO TO 90
458 *
459 * G(j) = ( 1 + CNORM(j) )*G(j-1)
460 *
461 XJ = ONE + CNORM( J )
462 GROW = GROW / XJ
463 80 CONTINUE
464 END IF
465 90 CONTINUE
466 END IF
467 *
468 IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
469 *
470 * Use the Level 2 BLAS solve if the reciprocal of the bound on
471 * elements of X is not too small.
472 *
473 CALL ZTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 )
474 ELSE
475 *
476 * Use a Level 1 BLAS solve, scaling intermediate results.
477 *
478 IF( XMAX.GT.BIGNUM*HALF ) THEN
479 *
480 * Scale X so that its components are less than or equal to
481 * BIGNUM in absolute value.
482 *
483 SCALE = ( BIGNUM*HALF ) / XMAX
484 CALL ZDSCAL( N, SCALE, X, 1 )
485 XMAX = BIGNUM
486 ELSE
487 XMAX = XMAX*TWO
488 END IF
489 *
490 IF( NOTRAN ) THEN
491 *
492 * Solve A * x = b
493 *
494 DO 120 J = JFIRST, JLAST, JINC
495 *
496 * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
497 *
498 XJ = CABS1( X( J ) )
499 IF( NOUNIT ) THEN
500 TJJS = AB( MAIND, J )*TSCAL
501 ELSE
502 TJJS = TSCAL
503 IF( TSCAL.EQ.ONE )
504 $ GO TO 110
505 END IF
506 TJJ = CABS1( TJJS )
507 IF( TJJ.GT.SMLNUM ) THEN
508 *
509 * abs(A(j,j)) > SMLNUM:
510 *
511 IF( TJJ.LT.ONE ) THEN
512 IF( XJ.GT.TJJ*BIGNUM ) THEN
513 *
514 * Scale x by 1/b(j).
515 *
516 REC = ONE / XJ
517 CALL ZDSCAL( N, REC, X, 1 )
518 SCALE = SCALE*REC
519 XMAX = XMAX*REC
520 END IF
521 END IF
522 X( J ) = ZLADIV( X( J ), TJJS )
523 XJ = CABS1( X( J ) )
524 ELSE IF( TJJ.GT.ZERO ) THEN
525 *
526 * 0 < abs(A(j,j)) <= SMLNUM:
527 *
528 IF( XJ.GT.TJJ*BIGNUM ) THEN
529 *
530 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
531 * to avoid overflow when dividing by A(j,j).
532 *
533 REC = ( TJJ*BIGNUM ) / XJ
534 IF( CNORM( J ).GT.ONE ) THEN
535 *
536 * Scale by 1/CNORM(j) to avoid overflow when
537 * multiplying x(j) times column j.
538 *
539 REC = REC / CNORM( J )
540 END IF
541 CALL ZDSCAL( N, REC, X, 1 )
542 SCALE = SCALE*REC
543 XMAX = XMAX*REC
544 END IF
545 X( J ) = ZLADIV( X( J ), TJJS )
546 XJ = CABS1( X( J ) )
547 ELSE
548 *
549 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
550 * scale = 0, and compute a solution to A*x = 0.
551 *
552 DO 100 I = 1, N
553 X( I ) = ZERO
554 100 CONTINUE
555 X( J ) = ONE
556 XJ = ONE
557 SCALE = ZERO
558 XMAX = ZERO
559 END IF
560 110 CONTINUE
561 *
562 * Scale x if necessary to avoid overflow when adding a
563 * multiple of column j of A.
564 *
565 IF( XJ.GT.ONE ) THEN
566 REC = ONE / XJ
567 IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
568 *
569 * Scale x by 1/(2*abs(x(j))).
570 *
571 REC = REC*HALF
572 CALL ZDSCAL( N, REC, X, 1 )
573 SCALE = SCALE*REC
574 END IF
575 ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
576 *
577 * Scale x by 1/2.
578 *
579 CALL ZDSCAL( N, HALF, X, 1 )
580 SCALE = SCALE*HALF
581 END IF
582 *
583 IF( UPPER ) THEN
584 IF( J.GT.1 ) THEN
585 *
586 * Compute the update
587 * x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) -
588 * x(j)* A(max(1,j-kd):j-1,j)
589 *
590 JLEN = MIN( KD, J-1 )
591 CALL ZAXPY( JLEN, -X( J )*TSCAL,
592 $ AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 )
593 I = IZAMAX( J-1, X, 1 )
594 XMAX = CABS1( X( I ) )
595 END IF
596 ELSE IF( J.LT.N ) THEN
597 *
598 * Compute the update
599 * x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) -
600 * x(j) * A(j+1:min(j+kd,n),j)
601 *
602 JLEN = MIN( KD, N-J )
603 IF( JLEN.GT.0 )
604 $ CALL ZAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1,
605 $ X( J+1 ), 1 )
606 I = J + IZAMAX( N-J, X( J+1 ), 1 )
607 XMAX = CABS1( X( I ) )
608 END IF
609 120 CONTINUE
610 *
611 ELSE IF( LSAME( TRANS, 'T' ) ) THEN
612 *
613 * Solve A**T * x = b
614 *
615 DO 170 J = JFIRST, JLAST, JINC
616 *
617 * Compute x(j) = b(j) - sum A(k,j)*x(k).
618 * k<>j
619 *
620 XJ = CABS1( X( J ) )
621 USCAL = TSCAL
622 REC = ONE / MAX( XMAX, ONE )
623 IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
624 *
625 * If x(j) could overflow, scale x by 1/(2*XMAX).
626 *
627 REC = REC*HALF
628 IF( NOUNIT ) THEN
629 TJJS = AB( MAIND, J )*TSCAL
630 ELSE
631 TJJS = TSCAL
632 END IF
633 TJJ = CABS1( TJJS )
634 IF( TJJ.GT.ONE ) THEN
635 *
636 * Divide by A(j,j) when scaling x if A(j,j) > 1.
637 *
638 REC = MIN( ONE, REC*TJJ )
639 USCAL = ZLADIV( USCAL, TJJS )
640 END IF
641 IF( REC.LT.ONE ) THEN
642 CALL ZDSCAL( N, REC, X, 1 )
643 SCALE = SCALE*REC
644 XMAX = XMAX*REC
645 END IF
646 END IF
647 *
648 CSUMJ = ZERO
649 IF( USCAL.EQ.DCMPLX( ONE ) ) THEN
650 *
651 * If the scaling needed for A in the dot product is 1,
652 * call ZDOTU to perform the dot product.
653 *
654 IF( UPPER ) THEN
655 JLEN = MIN( KD, J-1 )
656 CSUMJ = ZDOTU( JLEN, AB( KD+1-JLEN, J ), 1,
657 $ X( J-JLEN ), 1 )
658 ELSE
659 JLEN = MIN( KD, N-J )
660 IF( JLEN.GT.1 )
661 $ CSUMJ = ZDOTU( JLEN, AB( 2, J ), 1, X( J+1 ),
662 $ 1 )
663 END IF
664 ELSE
665 *
666 * Otherwise, use in-line code for the dot product.
667 *
668 IF( UPPER ) THEN
669 JLEN = MIN( KD, J-1 )
670 DO 130 I = 1, JLEN
671 CSUMJ = CSUMJ + ( AB( KD+I-JLEN, J )*USCAL )*
672 $ X( J-JLEN-1+I )
673 130 CONTINUE
674 ELSE
675 JLEN = MIN( KD, N-J )
676 DO 140 I = 1, JLEN
677 CSUMJ = CSUMJ + ( AB( I+1, J )*USCAL )*X( J+I )
678 140 CONTINUE
679 END IF
680 END IF
681 *
682 IF( USCAL.EQ.DCMPLX( TSCAL ) ) THEN
683 *
684 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
685 * was not used to scale the dotproduct.
686 *
687 X( J ) = X( J ) - CSUMJ
688 XJ = CABS1( X( J ) )
689 IF( NOUNIT ) THEN
690 *
691 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
692 *
693 TJJS = AB( MAIND, J )*TSCAL
694 ELSE
695 TJJS = TSCAL
696 IF( TSCAL.EQ.ONE )
697 $ GO TO 160
698 END IF
699 TJJ = CABS1( TJJS )
700 IF( TJJ.GT.SMLNUM ) THEN
701 *
702 * abs(A(j,j)) > SMLNUM:
703 *
704 IF( TJJ.LT.ONE ) THEN
705 IF( XJ.GT.TJJ*BIGNUM ) THEN
706 *
707 * Scale X by 1/abs(x(j)).
708 *
709 REC = ONE / XJ
710 CALL ZDSCAL( N, REC, X, 1 )
711 SCALE = SCALE*REC
712 XMAX = XMAX*REC
713 END IF
714 END IF
715 X( J ) = ZLADIV( X( J ), TJJS )
716 ELSE IF( TJJ.GT.ZERO ) THEN
717 *
718 * 0 < abs(A(j,j)) <= SMLNUM:
719 *
720 IF( XJ.GT.TJJ*BIGNUM ) THEN
721 *
722 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
723 *
724 REC = ( TJJ*BIGNUM ) / XJ
725 CALL ZDSCAL( N, REC, X, 1 )
726 SCALE = SCALE*REC
727 XMAX = XMAX*REC
728 END IF
729 X( J ) = ZLADIV( X( J ), TJJS )
730 ELSE
731 *
732 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
733 * scale = 0 and compute a solution to A**T *x = 0.
734 *
735 DO 150 I = 1, N
736 X( I ) = ZERO
737 150 CONTINUE
738 X( J ) = ONE
739 SCALE = ZERO
740 XMAX = ZERO
741 END IF
742 160 CONTINUE
743 ELSE
744 *
745 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
746 * product has already been divided by 1/A(j,j).
747 *
748 X( J ) = ZLADIV( X( J ), TJJS ) - CSUMJ
749 END IF
750 XMAX = MAX( XMAX, CABS1( X( J ) ) )
751 170 CONTINUE
752 *
753 ELSE
754 *
755 * Solve A**H * x = b
756 *
757 DO 220 J = JFIRST, JLAST, JINC
758 *
759 * Compute x(j) = b(j) - sum A(k,j)*x(k).
760 * k<>j
761 *
762 XJ = CABS1( X( J ) )
763 USCAL = TSCAL
764 REC = ONE / MAX( XMAX, ONE )
765 IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
766 *
767 * If x(j) could overflow, scale x by 1/(2*XMAX).
768 *
769 REC = REC*HALF
770 IF( NOUNIT ) THEN
771 TJJS = DCONJG( AB( MAIND, J ) )*TSCAL
772 ELSE
773 TJJS = TSCAL
774 END IF
775 TJJ = CABS1( TJJS )
776 IF( TJJ.GT.ONE ) THEN
777 *
778 * Divide by A(j,j) when scaling x if A(j,j) > 1.
779 *
780 REC = MIN( ONE, REC*TJJ )
781 USCAL = ZLADIV( USCAL, TJJS )
782 END IF
783 IF( REC.LT.ONE ) THEN
784 CALL ZDSCAL( N, REC, X, 1 )
785 SCALE = SCALE*REC
786 XMAX = XMAX*REC
787 END IF
788 END IF
789 *
790 CSUMJ = ZERO
791 IF( USCAL.EQ.DCMPLX( ONE ) ) THEN
792 *
793 * If the scaling needed for A in the dot product is 1,
794 * call ZDOTC to perform the dot product.
795 *
796 IF( UPPER ) THEN
797 JLEN = MIN( KD, J-1 )
798 CSUMJ = ZDOTC( JLEN, AB( KD+1-JLEN, J ), 1,
799 $ X( J-JLEN ), 1 )
800 ELSE
801 JLEN = MIN( KD, N-J )
802 IF( JLEN.GT.1 )
803 $ CSUMJ = ZDOTC( JLEN, AB( 2, J ), 1, X( J+1 ),
804 $ 1 )
805 END IF
806 ELSE
807 *
808 * Otherwise, use in-line code for the dot product.
809 *
810 IF( UPPER ) THEN
811 JLEN = MIN( KD, J-1 )
812 DO 180 I = 1, JLEN
813 CSUMJ = CSUMJ + ( DCONJG( AB( KD+I-JLEN, J ) )*
814 $ USCAL )*X( J-JLEN-1+I )
815 180 CONTINUE
816 ELSE
817 JLEN = MIN( KD, N-J )
818 DO 190 I = 1, JLEN
819 CSUMJ = CSUMJ + ( DCONJG( AB( I+1, J ) )*USCAL )
820 $ *X( J+I )
821 190 CONTINUE
822 END IF
823 END IF
824 *
825 IF( USCAL.EQ.DCMPLX( TSCAL ) ) THEN
826 *
827 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
828 * was not used to scale the dotproduct.
829 *
830 X( J ) = X( J ) - CSUMJ
831 XJ = CABS1( X( J ) )
832 IF( NOUNIT ) THEN
833 *
834 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
835 *
836 TJJS = DCONJG( AB( MAIND, J ) )*TSCAL
837 ELSE
838 TJJS = TSCAL
839 IF( TSCAL.EQ.ONE )
840 $ GO TO 210
841 END IF
842 TJJ = CABS1( TJJS )
843 IF( TJJ.GT.SMLNUM ) THEN
844 *
845 * abs(A(j,j)) > SMLNUM:
846 *
847 IF( TJJ.LT.ONE ) THEN
848 IF( XJ.GT.TJJ*BIGNUM ) THEN
849 *
850 * Scale X by 1/abs(x(j)).
851 *
852 REC = ONE / XJ
853 CALL ZDSCAL( N, REC, X, 1 )
854 SCALE = SCALE*REC
855 XMAX = XMAX*REC
856 END IF
857 END IF
858 X( J ) = ZLADIV( X( J ), TJJS )
859 ELSE IF( TJJ.GT.ZERO ) THEN
860 *
861 * 0 < abs(A(j,j)) <= SMLNUM:
862 *
863 IF( XJ.GT.TJJ*BIGNUM ) THEN
864 *
865 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
866 *
867 REC = ( TJJ*BIGNUM ) / XJ
868 CALL ZDSCAL( N, REC, X, 1 )
869 SCALE = SCALE*REC
870 XMAX = XMAX*REC
871 END IF
872 X( J ) = ZLADIV( X( J ), TJJS )
873 ELSE
874 *
875 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
876 * scale = 0 and compute a solution to A**H *x = 0.
877 *
878 DO 200 I = 1, N
879 X( I ) = ZERO
880 200 CONTINUE
881 X( J ) = ONE
882 SCALE = ZERO
883 XMAX = ZERO
884 END IF
885 210 CONTINUE
886 ELSE
887 *
888 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
889 * product has already been divided by 1/A(j,j).
890 *
891 X( J ) = ZLADIV( X( J ), TJJS ) - CSUMJ
892 END IF
893 XMAX = MAX( XMAX, CABS1( X( J ) ) )
894 220 CONTINUE
895 END IF
896 SCALE = SCALE / TSCAL
897 END IF
898 *
899 * Scale the column norms by 1/TSCAL for return.
900 *
901 IF( TSCAL.NE.ONE ) THEN
902 CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
903 END IF
904 *
905 RETURN
906 *
907 * End of ZLATBS
908 *
909 END
2 $ SCALE, CNORM, INFO )
3 *
4 * -- LAPACK auxiliary routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * .. Scalar Arguments ..
10 CHARACTER DIAG, NORMIN, TRANS, UPLO
11 INTEGER INFO, KD, LDAB, N
12 DOUBLE PRECISION SCALE
13 * ..
14 * .. Array Arguments ..
15 DOUBLE PRECISION CNORM( * )
16 COMPLEX*16 AB( LDAB, * ), X( * )
17 * ..
18 *
19 * Purpose
20 * =======
21 *
22 * ZLATBS solves one of the triangular systems
23 *
24 * A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
25 *
26 * with scaling to prevent overflow, where A is an upper or lower
27 * triangular band matrix. Here A**T denotes the transpose of A, x and b
28 * are n-element vectors, and s is a scaling factor, usually less than
29 * or equal to 1, chosen so that the components of x will be less than
30 * the overflow threshold. If the unscaled problem will not cause
31 * overflow, the Level 2 BLAS routine ZTBSV is called. If the matrix A
32 * is singular (A(j,j) = 0 for some j), then s is set to 0 and a
33 * non-trivial solution to A*x = 0 is returned.
34 *
35 * Arguments
36 * =========
37 *
38 * UPLO (input) CHARACTER*1
39 * Specifies whether the matrix A is upper or lower triangular.
40 * = 'U': Upper triangular
41 * = 'L': Lower triangular
42 *
43 * TRANS (input) CHARACTER*1
44 * Specifies the operation applied to A.
45 * = 'N': Solve A * x = s*b (No transpose)
46 * = 'T': Solve A**T * x = s*b (Transpose)
47 * = 'C': Solve A**H * x = s*b (Conjugate transpose)
48 *
49 * DIAG (input) CHARACTER*1
50 * Specifies whether or not the matrix A is unit triangular.
51 * = 'N': Non-unit triangular
52 * = 'U': Unit triangular
53 *
54 * NORMIN (input) CHARACTER*1
55 * Specifies whether CNORM has been set or not.
56 * = 'Y': CNORM contains the column norms on entry
57 * = 'N': CNORM is not set on entry. On exit, the norms will
58 * be computed and stored in CNORM.
59 *
60 * N (input) INTEGER
61 * The order of the matrix A. N >= 0.
62 *
63 * KD (input) INTEGER
64 * The number of subdiagonals or superdiagonals in the
65 * triangular matrix A. KD >= 0.
66 *
67 * AB (input) COMPLEX*16 array, dimension (LDAB,N)
68 * The upper or lower triangular band matrix A, stored in the
69 * first KD+1 rows of the array. The j-th column of A is stored
70 * in the j-th column of the array AB as follows:
71 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
72 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
73 *
74 * LDAB (input) INTEGER
75 * The leading dimension of the array AB. LDAB >= KD+1.
76 *
77 * X (input/output) COMPLEX*16 array, dimension (N)
78 * On entry, the right hand side b of the triangular system.
79 * On exit, X is overwritten by the solution vector x.
80 *
81 * SCALE (output) DOUBLE PRECISION
82 * The scaling factor s for the triangular system
83 * A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
84 * If SCALE = 0, the matrix A is singular or badly scaled, and
85 * the vector x is an exact or approximate solution to A*x = 0.
86 *
87 * CNORM (input or output) DOUBLE PRECISION array, dimension (N)
88 *
89 * If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
90 * contains the norm of the off-diagonal part of the j-th column
91 * of A. If TRANS = 'N', CNORM(j) must be greater than or equal
92 * to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
93 * must be greater than or equal to the 1-norm.
94 *
95 * If NORMIN = 'N', CNORM is an output argument and CNORM(j)
96 * returns the 1-norm of the offdiagonal part of the j-th column
97 * of A.
98 *
99 * INFO (output) INTEGER
100 * = 0: successful exit
101 * < 0: if INFO = -k, the k-th argument had an illegal value
102 *
103 * Further Details
104 * ======= =======
105 *
106 * A rough bound on x is computed; if that is less than overflow, ZTBSV
107 * is called, otherwise, specific code is used which checks for possible
108 * overflow or divide-by-zero at every operation.
109 *
110 * A columnwise scheme is used for solving A*x = b. The basic algorithm
111 * if A is lower triangular is
112 *
113 * x[1:n] := b[1:n]
114 * for j = 1, ..., n
115 * x(j) := x(j) / A(j,j)
116 * x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
117 * end
118 *
119 * Define bounds on the components of x after j iterations of the loop:
120 * M(j) = bound on x[1:j]
121 * G(j) = bound on x[j+1:n]
122 * Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
123 *
124 * Then for iteration j+1 we have
125 * M(j+1) <= G(j) / | A(j+1,j+1) |
126 * G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
127 * <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
128 *
129 * where CNORM(j+1) is greater than or equal to the infinity-norm of
130 * column j+1 of A, not counting the diagonal. Hence
131 *
132 * G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
133 * 1<=i<=j
134 * and
135 *
136 * |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
137 * 1<=i< j
138 *
139 * Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTBSV if the
140 * reciprocal of the largest M(j), j=1,..,n, is larger than
141 * max(underflow, 1/overflow).
142 *
143 * The bound on x(j) is also used to determine when a step in the
144 * columnwise method can be performed without fear of overflow. If
145 * the computed bound is greater than a large constant, x is scaled to
146 * prevent overflow, but if the bound overflows, x is set to 0, x(j) to
147 * 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
148 *
149 * Similarly, a row-wise scheme is used to solve A**T *x = b or
150 * A**H *x = b. The basic algorithm for A upper triangular is
151 *
152 * for j = 1, ..., n
153 * x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
154 * end
155 *
156 * We simultaneously compute two bounds
157 * G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
158 * M(j) = bound on x(i), 1<=i<=j
159 *
160 * The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
161 * add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
162 * Then the bound on x(j) is
163 *
164 * M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
165 *
166 * <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
167 * 1<=i<=j
168 *
169 * and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both greater
170 * than max(underflow, 1/overflow).
171 *
172 * =====================================================================
173 *
174 * .. Parameters ..
175 DOUBLE PRECISION ZERO, HALF, ONE, TWO
176 PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
177 $ TWO = 2.0D+0 )
178 * ..
179 * .. Local Scalars ..
180 LOGICAL NOTRAN, NOUNIT, UPPER
181 INTEGER I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
182 DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
183 $ XBND, XJ, XMAX
184 COMPLEX*16 CSUMJ, TJJS, USCAL, ZDUM
185 * ..
186 * .. External Functions ..
187 LOGICAL LSAME
188 INTEGER IDAMAX, IZAMAX
189 DOUBLE PRECISION DLAMCH, DZASUM
190 COMPLEX*16 ZDOTC, ZDOTU, ZLADIV
191 EXTERNAL LSAME, IDAMAX, IZAMAX, DLAMCH, DZASUM, ZDOTC,
192 $ ZDOTU, ZLADIV
193 * ..
194 * .. External Subroutines ..
195 EXTERNAL DSCAL, XERBLA, ZAXPY, ZDSCAL, ZTBSV
196 * ..
197 * .. Intrinsic Functions ..
198 INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
199 * ..
200 * .. Statement Functions ..
201 DOUBLE PRECISION CABS1, CABS2
202 * ..
203 * .. Statement Function definitions ..
204 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
205 CABS2( ZDUM ) = ABS( DBLE( ZDUM ) / 2.D0 ) +
206 $ ABS( DIMAG( ZDUM ) / 2.D0 )
207 * ..
208 * .. Executable Statements ..
209 *
210 INFO = 0
211 UPPER = LSAME( UPLO, 'U' )
212 NOTRAN = LSAME( TRANS, 'N' )
213 NOUNIT = LSAME( DIAG, 'N' )
214 *
215 * Test the input parameters.
216 *
217 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
218 INFO = -1
219 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
220 $ LSAME( TRANS, 'C' ) ) THEN
221 INFO = -2
222 ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
223 INFO = -3
224 ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
225 $ LSAME( NORMIN, 'N' ) ) THEN
226 INFO = -4
227 ELSE IF( N.LT.0 ) THEN
228 INFO = -5
229 ELSE IF( KD.LT.0 ) THEN
230 INFO = -6
231 ELSE IF( LDAB.LT.KD+1 ) THEN
232 INFO = -8
233 END IF
234 IF( INFO.NE.0 ) THEN
235 CALL XERBLA( 'ZLATBS', -INFO )
236 RETURN
237 END IF
238 *
239 * Quick return if possible
240 *
241 IF( N.EQ.0 )
242 $ RETURN
243 *
244 * Determine machine dependent parameters to control overflow.
245 *
246 SMLNUM = DLAMCH( 'Safe minimum' )
247 BIGNUM = ONE / SMLNUM
248 CALL DLABAD( SMLNUM, BIGNUM )
249 SMLNUM = SMLNUM / DLAMCH( 'Precision' )
250 BIGNUM = ONE / SMLNUM
251 SCALE = ONE
252 *
253 IF( LSAME( NORMIN, 'N' ) ) THEN
254 *
255 * Compute the 1-norm of each column, not including the diagonal.
256 *
257 IF( UPPER ) THEN
258 *
259 * A is upper triangular.
260 *
261 DO 10 J = 1, N
262 JLEN = MIN( KD, J-1 )
263 CNORM( J ) = DZASUM( JLEN, AB( KD+1-JLEN, J ), 1 )
264 10 CONTINUE
265 ELSE
266 *
267 * A is lower triangular.
268 *
269 DO 20 J = 1, N
270 JLEN = MIN( KD, N-J )
271 IF( JLEN.GT.0 ) THEN
272 CNORM( J ) = DZASUM( JLEN, AB( 2, J ), 1 )
273 ELSE
274 CNORM( J ) = ZERO
275 END IF
276 20 CONTINUE
277 END IF
278 END IF
279 *
280 * Scale the column norms by TSCAL if the maximum element in CNORM is
281 * greater than BIGNUM/2.
282 *
283 IMAX = IDAMAX( N, CNORM, 1 )
284 TMAX = CNORM( IMAX )
285 IF( TMAX.LE.BIGNUM*HALF ) THEN
286 TSCAL = ONE
287 ELSE
288 TSCAL = HALF / ( SMLNUM*TMAX )
289 CALL DSCAL( N, TSCAL, CNORM, 1 )
290 END IF
291 *
292 * Compute a bound on the computed solution vector to see if the
293 * Level 2 BLAS routine ZTBSV can be used.
294 *
295 XMAX = ZERO
296 DO 30 J = 1, N
297 XMAX = MAX( XMAX, CABS2( X( J ) ) )
298 30 CONTINUE
299 XBND = XMAX
300 IF( NOTRAN ) THEN
301 *
302 * Compute the growth in A * x = b.
303 *
304 IF( UPPER ) THEN
305 JFIRST = N
306 JLAST = 1
307 JINC = -1
308 MAIND = KD + 1
309 ELSE
310 JFIRST = 1
311 JLAST = N
312 JINC = 1
313 MAIND = 1
314 END IF
315 *
316 IF( TSCAL.NE.ONE ) THEN
317 GROW = ZERO
318 GO TO 60
319 END IF
320 *
321 IF( NOUNIT ) THEN
322 *
323 * A is non-unit triangular.
324 *
325 * Compute GROW = 1/G(j) and XBND = 1/M(j).
326 * Initially, G(0) = max{x(i), i=1,...,n}.
327 *
328 GROW = HALF / MAX( XBND, SMLNUM )
329 XBND = GROW
330 DO 40 J = JFIRST, JLAST, JINC
331 *
332 * Exit the loop if the growth factor is too small.
333 *
334 IF( GROW.LE.SMLNUM )
335 $ GO TO 60
336 *
337 TJJS = AB( MAIND, J )
338 TJJ = CABS1( TJJS )
339 *
340 IF( TJJ.GE.SMLNUM ) THEN
341 *
342 * M(j) = G(j-1) / abs(A(j,j))
343 *
344 XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
345 ELSE
346 *
347 * M(j) could overflow, set XBND to 0.
348 *
349 XBND = ZERO
350 END IF
351 *
352 IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
353 *
354 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
355 *
356 GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
357 ELSE
358 *
359 * G(j) could overflow, set GROW to 0.
360 *
361 GROW = ZERO
362 END IF
363 40 CONTINUE
364 GROW = XBND
365 ELSE
366 *
367 * A is unit triangular.
368 *
369 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
370 *
371 GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
372 DO 50 J = JFIRST, JLAST, JINC
373 *
374 * Exit the loop if the growth factor is too small.
375 *
376 IF( GROW.LE.SMLNUM )
377 $ GO TO 60
378 *
379 * G(j) = G(j-1)*( 1 + CNORM(j) )
380 *
381 GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
382 50 CONTINUE
383 END IF
384 60 CONTINUE
385 *
386 ELSE
387 *
388 * Compute the growth in A**T * x = b or A**H * x = b.
389 *
390 IF( UPPER ) THEN
391 JFIRST = 1
392 JLAST = N
393 JINC = 1
394 MAIND = KD + 1
395 ELSE
396 JFIRST = N
397 JLAST = 1
398 JINC = -1
399 MAIND = 1
400 END IF
401 *
402 IF( TSCAL.NE.ONE ) THEN
403 GROW = ZERO
404 GO TO 90
405 END IF
406 *
407 IF( NOUNIT ) THEN
408 *
409 * A is non-unit triangular.
410 *
411 * Compute GROW = 1/G(j) and XBND = 1/M(j).
412 * Initially, M(0) = max{x(i), i=1,...,n}.
413 *
414 GROW = HALF / MAX( XBND, SMLNUM )
415 XBND = GROW
416 DO 70 J = JFIRST, JLAST, JINC
417 *
418 * Exit the loop if the growth factor is too small.
419 *
420 IF( GROW.LE.SMLNUM )
421 $ GO TO 90
422 *
423 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
424 *
425 XJ = ONE + CNORM( J )
426 GROW = MIN( GROW, XBND / XJ )
427 *
428 TJJS = AB( MAIND, J )
429 TJJ = CABS1( TJJS )
430 *
431 IF( TJJ.GE.SMLNUM ) THEN
432 *
433 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
434 *
435 IF( XJ.GT.TJJ )
436 $ XBND = XBND*( TJJ / XJ )
437 ELSE
438 *
439 * M(j) could overflow, set XBND to 0.
440 *
441 XBND = ZERO
442 END IF
443 70 CONTINUE
444 GROW = MIN( GROW, XBND )
445 ELSE
446 *
447 * A is unit triangular.
448 *
449 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
450 *
451 GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
452 DO 80 J = JFIRST, JLAST, JINC
453 *
454 * Exit the loop if the growth factor is too small.
455 *
456 IF( GROW.LE.SMLNUM )
457 $ GO TO 90
458 *
459 * G(j) = ( 1 + CNORM(j) )*G(j-1)
460 *
461 XJ = ONE + CNORM( J )
462 GROW = GROW / XJ
463 80 CONTINUE
464 END IF
465 90 CONTINUE
466 END IF
467 *
468 IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
469 *
470 * Use the Level 2 BLAS solve if the reciprocal of the bound on
471 * elements of X is not too small.
472 *
473 CALL ZTBSV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, X, 1 )
474 ELSE
475 *
476 * Use a Level 1 BLAS solve, scaling intermediate results.
477 *
478 IF( XMAX.GT.BIGNUM*HALF ) THEN
479 *
480 * Scale X so that its components are less than or equal to
481 * BIGNUM in absolute value.
482 *
483 SCALE = ( BIGNUM*HALF ) / XMAX
484 CALL ZDSCAL( N, SCALE, X, 1 )
485 XMAX = BIGNUM
486 ELSE
487 XMAX = XMAX*TWO
488 END IF
489 *
490 IF( NOTRAN ) THEN
491 *
492 * Solve A * x = b
493 *
494 DO 120 J = JFIRST, JLAST, JINC
495 *
496 * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
497 *
498 XJ = CABS1( X( J ) )
499 IF( NOUNIT ) THEN
500 TJJS = AB( MAIND, J )*TSCAL
501 ELSE
502 TJJS = TSCAL
503 IF( TSCAL.EQ.ONE )
504 $ GO TO 110
505 END IF
506 TJJ = CABS1( TJJS )
507 IF( TJJ.GT.SMLNUM ) THEN
508 *
509 * abs(A(j,j)) > SMLNUM:
510 *
511 IF( TJJ.LT.ONE ) THEN
512 IF( XJ.GT.TJJ*BIGNUM ) THEN
513 *
514 * Scale x by 1/b(j).
515 *
516 REC = ONE / XJ
517 CALL ZDSCAL( N, REC, X, 1 )
518 SCALE = SCALE*REC
519 XMAX = XMAX*REC
520 END IF
521 END IF
522 X( J ) = ZLADIV( X( J ), TJJS )
523 XJ = CABS1( X( J ) )
524 ELSE IF( TJJ.GT.ZERO ) THEN
525 *
526 * 0 < abs(A(j,j)) <= SMLNUM:
527 *
528 IF( XJ.GT.TJJ*BIGNUM ) THEN
529 *
530 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
531 * to avoid overflow when dividing by A(j,j).
532 *
533 REC = ( TJJ*BIGNUM ) / XJ
534 IF( CNORM( J ).GT.ONE ) THEN
535 *
536 * Scale by 1/CNORM(j) to avoid overflow when
537 * multiplying x(j) times column j.
538 *
539 REC = REC / CNORM( J )
540 END IF
541 CALL ZDSCAL( N, REC, X, 1 )
542 SCALE = SCALE*REC
543 XMAX = XMAX*REC
544 END IF
545 X( J ) = ZLADIV( X( J ), TJJS )
546 XJ = CABS1( X( J ) )
547 ELSE
548 *
549 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
550 * scale = 0, and compute a solution to A*x = 0.
551 *
552 DO 100 I = 1, N
553 X( I ) = ZERO
554 100 CONTINUE
555 X( J ) = ONE
556 XJ = ONE
557 SCALE = ZERO
558 XMAX = ZERO
559 END IF
560 110 CONTINUE
561 *
562 * Scale x if necessary to avoid overflow when adding a
563 * multiple of column j of A.
564 *
565 IF( XJ.GT.ONE ) THEN
566 REC = ONE / XJ
567 IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
568 *
569 * Scale x by 1/(2*abs(x(j))).
570 *
571 REC = REC*HALF
572 CALL ZDSCAL( N, REC, X, 1 )
573 SCALE = SCALE*REC
574 END IF
575 ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
576 *
577 * Scale x by 1/2.
578 *
579 CALL ZDSCAL( N, HALF, X, 1 )
580 SCALE = SCALE*HALF
581 END IF
582 *
583 IF( UPPER ) THEN
584 IF( J.GT.1 ) THEN
585 *
586 * Compute the update
587 * x(max(1,j-kd):j-1) := x(max(1,j-kd):j-1) -
588 * x(j)* A(max(1,j-kd):j-1,j)
589 *
590 JLEN = MIN( KD, J-1 )
591 CALL ZAXPY( JLEN, -X( J )*TSCAL,
592 $ AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 )
593 I = IZAMAX( J-1, X, 1 )
594 XMAX = CABS1( X( I ) )
595 END IF
596 ELSE IF( J.LT.N ) THEN
597 *
598 * Compute the update
599 * x(j+1:min(j+kd,n)) := x(j+1:min(j+kd,n)) -
600 * x(j) * A(j+1:min(j+kd,n),j)
601 *
602 JLEN = MIN( KD, N-J )
603 IF( JLEN.GT.0 )
604 $ CALL ZAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1,
605 $ X( J+1 ), 1 )
606 I = J + IZAMAX( N-J, X( J+1 ), 1 )
607 XMAX = CABS1( X( I ) )
608 END IF
609 120 CONTINUE
610 *
611 ELSE IF( LSAME( TRANS, 'T' ) ) THEN
612 *
613 * Solve A**T * x = b
614 *
615 DO 170 J = JFIRST, JLAST, JINC
616 *
617 * Compute x(j) = b(j) - sum A(k,j)*x(k).
618 * k<>j
619 *
620 XJ = CABS1( X( J ) )
621 USCAL = TSCAL
622 REC = ONE / MAX( XMAX, ONE )
623 IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
624 *
625 * If x(j) could overflow, scale x by 1/(2*XMAX).
626 *
627 REC = REC*HALF
628 IF( NOUNIT ) THEN
629 TJJS = AB( MAIND, J )*TSCAL
630 ELSE
631 TJJS = TSCAL
632 END IF
633 TJJ = CABS1( TJJS )
634 IF( TJJ.GT.ONE ) THEN
635 *
636 * Divide by A(j,j) when scaling x if A(j,j) > 1.
637 *
638 REC = MIN( ONE, REC*TJJ )
639 USCAL = ZLADIV( USCAL, TJJS )
640 END IF
641 IF( REC.LT.ONE ) THEN
642 CALL ZDSCAL( N, REC, X, 1 )
643 SCALE = SCALE*REC
644 XMAX = XMAX*REC
645 END IF
646 END IF
647 *
648 CSUMJ = ZERO
649 IF( USCAL.EQ.DCMPLX( ONE ) ) THEN
650 *
651 * If the scaling needed for A in the dot product is 1,
652 * call ZDOTU to perform the dot product.
653 *
654 IF( UPPER ) THEN
655 JLEN = MIN( KD, J-1 )
656 CSUMJ = ZDOTU( JLEN, AB( KD+1-JLEN, J ), 1,
657 $ X( J-JLEN ), 1 )
658 ELSE
659 JLEN = MIN( KD, N-J )
660 IF( JLEN.GT.1 )
661 $ CSUMJ = ZDOTU( JLEN, AB( 2, J ), 1, X( J+1 ),
662 $ 1 )
663 END IF
664 ELSE
665 *
666 * Otherwise, use in-line code for the dot product.
667 *
668 IF( UPPER ) THEN
669 JLEN = MIN( KD, J-1 )
670 DO 130 I = 1, JLEN
671 CSUMJ = CSUMJ + ( AB( KD+I-JLEN, J )*USCAL )*
672 $ X( J-JLEN-1+I )
673 130 CONTINUE
674 ELSE
675 JLEN = MIN( KD, N-J )
676 DO 140 I = 1, JLEN
677 CSUMJ = CSUMJ + ( AB( I+1, J )*USCAL )*X( J+I )
678 140 CONTINUE
679 END IF
680 END IF
681 *
682 IF( USCAL.EQ.DCMPLX( TSCAL ) ) THEN
683 *
684 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
685 * was not used to scale the dotproduct.
686 *
687 X( J ) = X( J ) - CSUMJ
688 XJ = CABS1( X( J ) )
689 IF( NOUNIT ) THEN
690 *
691 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
692 *
693 TJJS = AB( MAIND, J )*TSCAL
694 ELSE
695 TJJS = TSCAL
696 IF( TSCAL.EQ.ONE )
697 $ GO TO 160
698 END IF
699 TJJ = CABS1( TJJS )
700 IF( TJJ.GT.SMLNUM ) THEN
701 *
702 * abs(A(j,j)) > SMLNUM:
703 *
704 IF( TJJ.LT.ONE ) THEN
705 IF( XJ.GT.TJJ*BIGNUM ) THEN
706 *
707 * Scale X by 1/abs(x(j)).
708 *
709 REC = ONE / XJ
710 CALL ZDSCAL( N, REC, X, 1 )
711 SCALE = SCALE*REC
712 XMAX = XMAX*REC
713 END IF
714 END IF
715 X( J ) = ZLADIV( X( J ), TJJS )
716 ELSE IF( TJJ.GT.ZERO ) THEN
717 *
718 * 0 < abs(A(j,j)) <= SMLNUM:
719 *
720 IF( XJ.GT.TJJ*BIGNUM ) THEN
721 *
722 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
723 *
724 REC = ( TJJ*BIGNUM ) / XJ
725 CALL ZDSCAL( N, REC, X, 1 )
726 SCALE = SCALE*REC
727 XMAX = XMAX*REC
728 END IF
729 X( J ) = ZLADIV( X( J ), TJJS )
730 ELSE
731 *
732 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
733 * scale = 0 and compute a solution to A**T *x = 0.
734 *
735 DO 150 I = 1, N
736 X( I ) = ZERO
737 150 CONTINUE
738 X( J ) = ONE
739 SCALE = ZERO
740 XMAX = ZERO
741 END IF
742 160 CONTINUE
743 ELSE
744 *
745 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
746 * product has already been divided by 1/A(j,j).
747 *
748 X( J ) = ZLADIV( X( J ), TJJS ) - CSUMJ
749 END IF
750 XMAX = MAX( XMAX, CABS1( X( J ) ) )
751 170 CONTINUE
752 *
753 ELSE
754 *
755 * Solve A**H * x = b
756 *
757 DO 220 J = JFIRST, JLAST, JINC
758 *
759 * Compute x(j) = b(j) - sum A(k,j)*x(k).
760 * k<>j
761 *
762 XJ = CABS1( X( J ) )
763 USCAL = TSCAL
764 REC = ONE / MAX( XMAX, ONE )
765 IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
766 *
767 * If x(j) could overflow, scale x by 1/(2*XMAX).
768 *
769 REC = REC*HALF
770 IF( NOUNIT ) THEN
771 TJJS = DCONJG( AB( MAIND, J ) )*TSCAL
772 ELSE
773 TJJS = TSCAL
774 END IF
775 TJJ = CABS1( TJJS )
776 IF( TJJ.GT.ONE ) THEN
777 *
778 * Divide by A(j,j) when scaling x if A(j,j) > 1.
779 *
780 REC = MIN( ONE, REC*TJJ )
781 USCAL = ZLADIV( USCAL, TJJS )
782 END IF
783 IF( REC.LT.ONE ) THEN
784 CALL ZDSCAL( N, REC, X, 1 )
785 SCALE = SCALE*REC
786 XMAX = XMAX*REC
787 END IF
788 END IF
789 *
790 CSUMJ = ZERO
791 IF( USCAL.EQ.DCMPLX( ONE ) ) THEN
792 *
793 * If the scaling needed for A in the dot product is 1,
794 * call ZDOTC to perform the dot product.
795 *
796 IF( UPPER ) THEN
797 JLEN = MIN( KD, J-1 )
798 CSUMJ = ZDOTC( JLEN, AB( KD+1-JLEN, J ), 1,
799 $ X( J-JLEN ), 1 )
800 ELSE
801 JLEN = MIN( KD, N-J )
802 IF( JLEN.GT.1 )
803 $ CSUMJ = ZDOTC( JLEN, AB( 2, J ), 1, X( J+1 ),
804 $ 1 )
805 END IF
806 ELSE
807 *
808 * Otherwise, use in-line code for the dot product.
809 *
810 IF( UPPER ) THEN
811 JLEN = MIN( KD, J-1 )
812 DO 180 I = 1, JLEN
813 CSUMJ = CSUMJ + ( DCONJG( AB( KD+I-JLEN, J ) )*
814 $ USCAL )*X( J-JLEN-1+I )
815 180 CONTINUE
816 ELSE
817 JLEN = MIN( KD, N-J )
818 DO 190 I = 1, JLEN
819 CSUMJ = CSUMJ + ( DCONJG( AB( I+1, J ) )*USCAL )
820 $ *X( J+I )
821 190 CONTINUE
822 END IF
823 END IF
824 *
825 IF( USCAL.EQ.DCMPLX( TSCAL ) ) THEN
826 *
827 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
828 * was not used to scale the dotproduct.
829 *
830 X( J ) = X( J ) - CSUMJ
831 XJ = CABS1( X( J ) )
832 IF( NOUNIT ) THEN
833 *
834 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
835 *
836 TJJS = DCONJG( AB( MAIND, J ) )*TSCAL
837 ELSE
838 TJJS = TSCAL
839 IF( TSCAL.EQ.ONE )
840 $ GO TO 210
841 END IF
842 TJJ = CABS1( TJJS )
843 IF( TJJ.GT.SMLNUM ) THEN
844 *
845 * abs(A(j,j)) > SMLNUM:
846 *
847 IF( TJJ.LT.ONE ) THEN
848 IF( XJ.GT.TJJ*BIGNUM ) THEN
849 *
850 * Scale X by 1/abs(x(j)).
851 *
852 REC = ONE / XJ
853 CALL ZDSCAL( N, REC, X, 1 )
854 SCALE = SCALE*REC
855 XMAX = XMAX*REC
856 END IF
857 END IF
858 X( J ) = ZLADIV( X( J ), TJJS )
859 ELSE IF( TJJ.GT.ZERO ) THEN
860 *
861 * 0 < abs(A(j,j)) <= SMLNUM:
862 *
863 IF( XJ.GT.TJJ*BIGNUM ) THEN
864 *
865 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
866 *
867 REC = ( TJJ*BIGNUM ) / XJ
868 CALL ZDSCAL( N, REC, X, 1 )
869 SCALE = SCALE*REC
870 XMAX = XMAX*REC
871 END IF
872 X( J ) = ZLADIV( X( J ), TJJS )
873 ELSE
874 *
875 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
876 * scale = 0 and compute a solution to A**H *x = 0.
877 *
878 DO 200 I = 1, N
879 X( I ) = ZERO
880 200 CONTINUE
881 X( J ) = ONE
882 SCALE = ZERO
883 XMAX = ZERO
884 END IF
885 210 CONTINUE
886 ELSE
887 *
888 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
889 * product has already been divided by 1/A(j,j).
890 *
891 X( J ) = ZLADIV( X( J ), TJJS ) - CSUMJ
892 END IF
893 XMAX = MAX( XMAX, CABS1( X( J ) ) )
894 220 CONTINUE
895 END IF
896 SCALE = SCALE / TSCAL
897 END IF
898 *
899 * Scale the column norms by 1/TSCAL for return.
900 *
901 IF( TSCAL.NE.ONE ) THEN
902 CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
903 END IF
904 *
905 RETURN
906 *
907 * End of ZLATBS
908 *
909 END