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1.1 bertrand 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' 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