```001:       SUBROUTINE SLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
002:      \$                   SCALE, CNORM, INFO )
003: *
004: *  -- LAPACK auxiliary routine (version 3.2) --
005: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
006: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
007: *     November 2006
008: *
009: *     .. Scalar Arguments ..
010:       CHARACTER          DIAG, NORMIN, TRANS, UPLO
011:       INTEGER            INFO, KD, LDAB, N
012:       REAL               SCALE
013: *     ..
014: *     .. Array Arguments ..
015:       REAL               AB( LDAB, * ), CNORM( * ), X( * )
016: *     ..
017: *
018: *  Purpose
019: *  =======
020: *
021: *  SLATBS solves one of the triangular systems
022: *
023: *     A *x = s*b  or  A'*x = s*b
024: *
025: *  with scaling to prevent overflow, where A is an upper or lower
026: *  triangular band matrix.  Here A' denotes the transpose of A, x and b
027: *  are n-element vectors, and s is a scaling factor, usually less than
028: *  or equal to 1, chosen so that the components of x will be less than
029: *  the overflow threshold.  If the unscaled problem will not cause
030: *  overflow, the Level 2 BLAS routine STBSV is called.  If the matrix A
031: *  is singular (A(j,j) = 0 for some j), then s is set to 0 and a
032: *  non-trivial solution to A*x = 0 is returned.
033: *
034: *  Arguments
035: *  =========
036: *
037: *  UPLO    (input) CHARACTER*1
038: *          Specifies whether the matrix A is upper or lower triangular.
039: *          = 'U':  Upper triangular
040: *          = 'L':  Lower triangular
041: *
042: *  TRANS   (input) CHARACTER*1
043: *          Specifies the operation applied to A.
044: *          = 'N':  Solve A * x = s*b  (No transpose)
045: *          = 'T':  Solve A'* x = s*b  (Transpose)
046: *          = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)
047: *
048: *  DIAG    (input) CHARACTER*1
049: *          Specifies whether or not the matrix A is unit triangular.
050: *          = 'N':  Non-unit triangular
051: *          = 'U':  Unit triangular
052: *
053: *  NORMIN  (input) CHARACTER*1
054: *          Specifies whether CNORM has been set or not.
055: *          = 'Y':  CNORM contains the column norms on entry
056: *          = 'N':  CNORM is not set on entry.  On exit, the norms will
057: *                  be computed and stored in CNORM.
058: *
059: *  N       (input) INTEGER
060: *          The order of the matrix A.  N >= 0.
061: *
062: *  KD      (input) INTEGER
063: *          The number of subdiagonals or superdiagonals in the
064: *          triangular matrix A.  KD >= 0.
065: *
066: *  AB      (input) REAL array, dimension (LDAB,N)
067: *          The upper or lower triangular band matrix A, stored in the
068: *          first KD+1 rows of the array. The j-th column of A is stored
069: *          in the j-th column of the array AB as follows:
070: *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
071: *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
072: *
073: *  LDAB    (input) INTEGER
074: *          The leading dimension of the array AB.  LDAB >= KD+1.
075: *
076: *  X       (input/output) REAL array, dimension (N)
077: *          On entry, the right hand side b of the triangular system.
078: *          On exit, X is overwritten by the solution vector x.
079: *
080: *  SCALE   (output) REAL
081: *          The scaling factor s for the triangular system
082: *             A * x = s*b  or  A'* x = s*b.
083: *          If SCALE = 0, the matrix A is singular or badly scaled, and
084: *          the vector x is an exact or approximate solution to A*x = 0.
085: *
086: *  CNORM   (input or output) REAL array, dimension (N)
087: *
088: *          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
089: *          contains the norm of the off-diagonal part of the j-th column
090: *          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
091: *          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
092: *          must be greater than or equal to the 1-norm.
093: *
094: *          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
095: *          returns the 1-norm of the offdiagonal part of the j-th column
096: *          of A.
097: *
098: *  INFO    (output) INTEGER
099: *          = 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, STBSV
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 STBSV 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'*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]' * 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]' * 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 STBSV if 1/M(n) and 1/G(n) are both greater
169: *  than max(underflow, 1/overflow).
170: *
171: *  =====================================================================
172: *
173: *     .. Parameters ..
174:       REAL               ZERO, HALF, ONE
175:       PARAMETER          ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
176: *     ..
177: *     .. Local Scalars ..
178:       LOGICAL            NOTRAN, NOUNIT, UPPER
179:       INTEGER            I, IMAX, J, JFIRST, JINC, JLAST, JLEN, MAIND
180:       REAL               BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
181:      \$                   TMAX, TSCAL, USCAL, XBND, XJ, XMAX
182: *     ..
183: *     .. External Functions ..
184:       LOGICAL            LSAME
185:       INTEGER            ISAMAX
186:       REAL               SASUM, SDOT, SLAMCH
187:       EXTERNAL           LSAME, ISAMAX, SASUM, SDOT, SLAMCH
188: *     ..
189: *     .. External Subroutines ..
190:       EXTERNAL           SAXPY, SSCAL, STBSV, 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( 'SLATBS', -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 = SLAMCH( 'Safe minimum' ) / SLAMCH( '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 ) = SASUM( 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 ) = SASUM( 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 = ISAMAX( 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 SSCAL( 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 STBSV can be used.
278: *
279:       J = ISAMAX( 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' * 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 STBSV( 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 SSCAL( N, SCALE, X, 1 )
446:             XMAX = BIGNUM
447:          END IF
448: *
449:          IF( NOTRAN ) THEN
450: *
451: *           Solve A * x = b
452: *
453:             DO 100 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 95
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 SSCAL( 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 SSCAL( 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:    95          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 SSCAL( 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 SSCAL( 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 SAXPY( JLEN, -X( J )*TSCAL,
551:      \$                           AB( KD+1-JLEN, J ), 1, X( J-JLEN ), 1 )
552:                      I = ISAMAX( 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 SAXPY( JLEN, -X( J )*TSCAL, AB( 2, J ), 1,
564:      \$                           X( J+1 ), 1 )
565:                   I = J + ISAMAX( N-J, X( J+1 ), 1 )
566:                   XMAX = ABS( X( I ) )
567:                END IF
568:   100       CONTINUE
569: *
570:          ELSE
571: *
572: *           Solve A' * x = b
573: *
574:             DO 140 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 SSCAL( 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 SDOT to perform the dot product.
612: *
613:                   IF( UPPER ) THEN
614:                      JLEN = MIN( KD, J-1 )
615:                      SUMJ = SDOT( 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 = SDOT( 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 110 I = 1, JLEN
629:                         SUMJ = SUMJ + ( AB( KD+I-JLEN, J )*USCAL )*
630:      \$                         X( J-JLEN-1+I )
631:   110                CONTINUE
632:                   ELSE
633:                      JLEN = MIN( KD, N-J )
634:                      DO 120 I = 1, JLEN
635:                         SUMJ = SUMJ + ( AB( I+1, J )*USCAL )*X( J+I )
636:   120                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 135
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 SSCAL( 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 SSCAL( 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'*x = 0.
692: *
693:                         DO 130 I = 1, N
694:                            X( I ) = ZERO
695:   130                   CONTINUE
696:                         X( J ) = ONE
697:                         SCALE = ZERO
698:                         XMAX = ZERO
699:                      END IF
700:   135             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:   140       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 SSCAL( N, ONE / TSCAL, CNORM, 1 )
718:       END IF
719: *
720:       RETURN
721: *
722: *     End of SLATBS
723: *
724:       END
725: ```