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