1: SUBROUTINE ZLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
2: $ 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, LDA, N
12: DOUBLE PRECISION SCALE
13: * ..
14: * .. Array Arguments ..
15: DOUBLE PRECISION CNORM( * )
16: COMPLEX*16 A( LDA, * ), X( * )
17: * ..
18: *
19: * Purpose
20: * =======
21: *
22: * ZLATRS 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. Here A is an upper or lower
27: * triangular matrix, A**T denotes the transpose of A, A**H denotes the
28: * conjugate transpose of A, x and b are n-element vectors, and s is a
29: * scaling factor, usually less than or equal to 1, chosen so that the
30: * components of x will be less than the overflow threshold. If the
31: * unscaled problem will not cause overflow, the Level 2 BLAS routine
32: * ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
33: * then s is set to 0 and a 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: * A (input) COMPLEX*16 array, dimension (LDA,N)
64: * The triangular matrix A. If UPLO = 'U', the leading n by n
65: * upper triangular part of the array A contains the upper
66: * triangular matrix, and the strictly lower triangular part of
67: * A is not referenced. If UPLO = 'L', the leading n by n lower
68: * triangular part of the array A contains the lower triangular
69: * matrix, and the strictly upper triangular part of A is not
70: * referenced. If DIAG = 'U', the diagonal elements of A are
71: * also not referenced and are assumed to be 1.
72: *
73: * LDA (input) INTEGER
74: * The leading dimension of the array A. LDA >= max (1,N).
75: *
76: * X (input/output) COMPLEX*16 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, A**T * x = s*b, or A**H * 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, ZTRSV
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 ZTRSV 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 or
149: * A**H *x = b. The basic 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 ZTRSV 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, TWO
175: PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0,
176: $ TWO = 2.0D+0 )
177: * ..
178: * .. Local Scalars ..
179: LOGICAL NOTRAN, NOUNIT, UPPER
180: INTEGER I, IMAX, J, JFIRST, JINC, JLAST
181: DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
182: $ XBND, XJ, XMAX
183: COMPLEX*16 CSUMJ, TJJS, USCAL, ZDUM
184: * ..
185: * .. External Functions ..
186: LOGICAL LSAME
187: INTEGER IDAMAX, IZAMAX
188: DOUBLE PRECISION DLAMCH, DZASUM
189: COMPLEX*16 ZDOTC, ZDOTU, ZLADIV
190: EXTERNAL LSAME, IDAMAX, IZAMAX, DLAMCH, DZASUM, ZDOTC,
191: $ ZDOTU, ZLADIV
192: * ..
193: * .. External Subroutines ..
194: EXTERNAL DSCAL, XERBLA, ZAXPY, ZDSCAL, ZTRSV
195: * ..
196: * .. Intrinsic Functions ..
197: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
198: * ..
199: * .. Statement Functions ..
200: DOUBLE PRECISION CABS1, CABS2
201: * ..
202: * .. Statement Function definitions ..
203: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
204: CABS2( ZDUM ) = ABS( DBLE( ZDUM ) / 2.D0 ) +
205: $ ABS( DIMAG( ZDUM ) / 2.D0 )
206: * ..
207: * .. Executable Statements ..
208: *
209: INFO = 0
210: UPPER = LSAME( UPLO, 'U' )
211: NOTRAN = LSAME( TRANS, 'N' )
212: NOUNIT = LSAME( DIAG, 'N' )
213: *
214: * Test the input parameters.
215: *
216: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
217: INFO = -1
218: ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
219: $ LSAME( TRANS, 'C' ) ) THEN
220: INFO = -2
221: ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
222: INFO = -3
223: ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
224: $ LSAME( NORMIN, 'N' ) ) THEN
225: INFO = -4
226: ELSE IF( N.LT.0 ) THEN
227: INFO = -5
228: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
229: INFO = -7
230: END IF
231: IF( INFO.NE.0 ) THEN
232: CALL XERBLA( 'ZLATRS', -INFO )
233: RETURN
234: END IF
235: *
236: * Quick return if possible
237: *
238: IF( N.EQ.0 )
239: $ RETURN
240: *
241: * Determine machine dependent parameters to control overflow.
242: *
243: SMLNUM = DLAMCH( 'Safe minimum' )
244: BIGNUM = ONE / SMLNUM
245: CALL DLABAD( SMLNUM, BIGNUM )
246: SMLNUM = SMLNUM / DLAMCH( 'Precision' )
247: BIGNUM = ONE / SMLNUM
248: SCALE = ONE
249: *
250: IF( LSAME( NORMIN, 'N' ) ) THEN
251: *
252: * Compute the 1-norm of each column, not including the diagonal.
253: *
254: IF( UPPER ) THEN
255: *
256: * A is upper triangular.
257: *
258: DO 10 J = 1, N
259: CNORM( J ) = DZASUM( J-1, A( 1, J ), 1 )
260: 10 CONTINUE
261: ELSE
262: *
263: * A is lower triangular.
264: *
265: DO 20 J = 1, N - 1
266: CNORM( J ) = DZASUM( N-J, A( J+1, J ), 1 )
267: 20 CONTINUE
268: CNORM( N ) = ZERO
269: END IF
270: END IF
271: *
272: * Scale the column norms by TSCAL if the maximum element in CNORM is
273: * greater than BIGNUM/2.
274: *
275: IMAX = IDAMAX( N, CNORM, 1 )
276: TMAX = CNORM( IMAX )
277: IF( TMAX.LE.BIGNUM*HALF ) THEN
278: TSCAL = ONE
279: ELSE
280: TSCAL = HALF / ( SMLNUM*TMAX )
281: CALL DSCAL( N, TSCAL, CNORM, 1 )
282: END IF
283: *
284: * Compute a bound on the computed solution vector to see if the
285: * Level 2 BLAS routine ZTRSV can be used.
286: *
287: XMAX = ZERO
288: DO 30 J = 1, N
289: XMAX = MAX( XMAX, CABS2( X( J ) ) )
290: 30 CONTINUE
291: XBND = XMAX
292: *
293: IF( NOTRAN ) THEN
294: *
295: * Compute the growth in A * x = b.
296: *
297: IF( UPPER ) THEN
298: JFIRST = N
299: JLAST = 1
300: JINC = -1
301: ELSE
302: JFIRST = 1
303: JLAST = N
304: JINC = 1
305: END IF
306: *
307: IF( TSCAL.NE.ONE ) THEN
308: GROW = ZERO
309: GO TO 60
310: END IF
311: *
312: IF( NOUNIT ) THEN
313: *
314: * A is non-unit triangular.
315: *
316: * Compute GROW = 1/G(j) and XBND = 1/M(j).
317: * Initially, G(0) = max{x(i), i=1,...,n}.
318: *
319: GROW = HALF / MAX( XBND, SMLNUM )
320: XBND = GROW
321: DO 40 J = JFIRST, JLAST, JINC
322: *
323: * Exit the loop if the growth factor is too small.
324: *
325: IF( GROW.LE.SMLNUM )
326: $ GO TO 60
327: *
328: TJJS = A( J, J )
329: TJJ = CABS1( TJJS )
330: *
331: IF( TJJ.GE.SMLNUM ) THEN
332: *
333: * M(j) = G(j-1) / abs(A(j,j))
334: *
335: XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
336: ELSE
337: *
338: * M(j) could overflow, set XBND to 0.
339: *
340: XBND = ZERO
341: END IF
342: *
343: IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
344: *
345: * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
346: *
347: GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
348: ELSE
349: *
350: * G(j) could overflow, set GROW to 0.
351: *
352: GROW = ZERO
353: END IF
354: 40 CONTINUE
355: GROW = XBND
356: ELSE
357: *
358: * A is unit triangular.
359: *
360: * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
361: *
362: GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
363: DO 50 J = JFIRST, JLAST, JINC
364: *
365: * Exit the loop if the growth factor is too small.
366: *
367: IF( GROW.LE.SMLNUM )
368: $ GO TO 60
369: *
370: * G(j) = G(j-1)*( 1 + CNORM(j) )
371: *
372: GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
373: 50 CONTINUE
374: END IF
375: 60 CONTINUE
376: *
377: ELSE
378: *
379: * Compute the growth in A**T * x = b or A**H * x = b.
380: *
381: IF( UPPER ) THEN
382: JFIRST = 1
383: JLAST = N
384: JINC = 1
385: ELSE
386: JFIRST = N
387: JLAST = 1
388: JINC = -1
389: END IF
390: *
391: IF( TSCAL.NE.ONE ) THEN
392: GROW = ZERO
393: GO TO 90
394: END IF
395: *
396: IF( NOUNIT ) THEN
397: *
398: * A is non-unit triangular.
399: *
400: * Compute GROW = 1/G(j) and XBND = 1/M(j).
401: * Initially, M(0) = max{x(i), i=1,...,n}.
402: *
403: GROW = HALF / MAX( XBND, SMLNUM )
404: XBND = GROW
405: DO 70 J = JFIRST, JLAST, JINC
406: *
407: * Exit the loop if the growth factor is too small.
408: *
409: IF( GROW.LE.SMLNUM )
410: $ GO TO 90
411: *
412: * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
413: *
414: XJ = ONE + CNORM( J )
415: GROW = MIN( GROW, XBND / XJ )
416: *
417: TJJS = A( J, J )
418: TJJ = CABS1( TJJS )
419: *
420: IF( TJJ.GE.SMLNUM ) THEN
421: *
422: * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
423: *
424: IF( XJ.GT.TJJ )
425: $ XBND = XBND*( TJJ / XJ )
426: ELSE
427: *
428: * M(j) could overflow, set XBND to 0.
429: *
430: XBND = ZERO
431: END IF
432: 70 CONTINUE
433: GROW = MIN( GROW, XBND )
434: ELSE
435: *
436: * A is unit triangular.
437: *
438: * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
439: *
440: GROW = MIN( ONE, HALF / MAX( XBND, SMLNUM ) )
441: DO 80 J = JFIRST, JLAST, JINC
442: *
443: * Exit the loop if the growth factor is too small.
444: *
445: IF( GROW.LE.SMLNUM )
446: $ GO TO 90
447: *
448: * G(j) = ( 1 + CNORM(j) )*G(j-1)
449: *
450: XJ = ONE + CNORM( J )
451: GROW = GROW / XJ
452: 80 CONTINUE
453: END IF
454: 90 CONTINUE
455: END IF
456: *
457: IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
458: *
459: * Use the Level 2 BLAS solve if the reciprocal of the bound on
460: * elements of X is not too small.
461: *
462: CALL ZTRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
463: ELSE
464: *
465: * Use a Level 1 BLAS solve, scaling intermediate results.
466: *
467: IF( XMAX.GT.BIGNUM*HALF ) THEN
468: *
469: * Scale X so that its components are less than or equal to
470: * BIGNUM in absolute value.
471: *
472: SCALE = ( BIGNUM*HALF ) / XMAX
473: CALL ZDSCAL( N, SCALE, X, 1 )
474: XMAX = BIGNUM
475: ELSE
476: XMAX = XMAX*TWO
477: END IF
478: *
479: IF( NOTRAN ) THEN
480: *
481: * Solve A * x = b
482: *
483: DO 120 J = JFIRST, JLAST, JINC
484: *
485: * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
486: *
487: XJ = CABS1( X( J ) )
488: IF( NOUNIT ) THEN
489: TJJS = A( J, J )*TSCAL
490: ELSE
491: TJJS = TSCAL
492: IF( TSCAL.EQ.ONE )
493: $ GO TO 110
494: END IF
495: TJJ = CABS1( TJJS )
496: IF( TJJ.GT.SMLNUM ) THEN
497: *
498: * abs(A(j,j)) > SMLNUM:
499: *
500: IF( TJJ.LT.ONE ) THEN
501: IF( XJ.GT.TJJ*BIGNUM ) THEN
502: *
503: * Scale x by 1/b(j).
504: *
505: REC = ONE / XJ
506: CALL ZDSCAL( N, REC, X, 1 )
507: SCALE = SCALE*REC
508: XMAX = XMAX*REC
509: END IF
510: END IF
511: X( J ) = ZLADIV( X( J ), TJJS )
512: XJ = CABS1( X( J ) )
513: ELSE IF( TJJ.GT.ZERO ) THEN
514: *
515: * 0 < abs(A(j,j)) <= SMLNUM:
516: *
517: IF( XJ.GT.TJJ*BIGNUM ) THEN
518: *
519: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
520: * to avoid overflow when dividing by A(j,j).
521: *
522: REC = ( TJJ*BIGNUM ) / XJ
523: IF( CNORM( J ).GT.ONE ) THEN
524: *
525: * Scale by 1/CNORM(j) to avoid overflow when
526: * multiplying x(j) times column j.
527: *
528: REC = REC / CNORM( J )
529: END IF
530: CALL ZDSCAL( N, REC, X, 1 )
531: SCALE = SCALE*REC
532: XMAX = XMAX*REC
533: END IF
534: X( J ) = ZLADIV( X( J ), TJJS )
535: XJ = CABS1( X( J ) )
536: ELSE
537: *
538: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
539: * scale = 0, and compute a solution to A*x = 0.
540: *
541: DO 100 I = 1, N
542: X( I ) = ZERO
543: 100 CONTINUE
544: X( J ) = ONE
545: XJ = ONE
546: SCALE = ZERO
547: XMAX = ZERO
548: END IF
549: 110 CONTINUE
550: *
551: * Scale x if necessary to avoid overflow when adding a
552: * multiple of column j of A.
553: *
554: IF( XJ.GT.ONE ) THEN
555: REC = ONE / XJ
556: IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
557: *
558: * Scale x by 1/(2*abs(x(j))).
559: *
560: REC = REC*HALF
561: CALL ZDSCAL( N, REC, X, 1 )
562: SCALE = SCALE*REC
563: END IF
564: ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
565: *
566: * Scale x by 1/2.
567: *
568: CALL ZDSCAL( N, HALF, X, 1 )
569: SCALE = SCALE*HALF
570: END IF
571: *
572: IF( UPPER ) THEN
573: IF( J.GT.1 ) THEN
574: *
575: * Compute the update
576: * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
577: *
578: CALL ZAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
579: $ 1 )
580: I = IZAMAX( J-1, X, 1 )
581: XMAX = CABS1( X( I ) )
582: END IF
583: ELSE
584: IF( J.LT.N ) THEN
585: *
586: * Compute the update
587: * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
588: *
589: CALL ZAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
590: $ X( J+1 ), 1 )
591: I = J + IZAMAX( N-J, X( J+1 ), 1 )
592: XMAX = CABS1( X( I ) )
593: END IF
594: END IF
595: 120 CONTINUE
596: *
597: ELSE IF( LSAME( TRANS, 'T' ) ) THEN
598: *
599: * Solve A**T * x = b
600: *
601: DO 170 J = JFIRST, JLAST, JINC
602: *
603: * Compute x(j) = b(j) - sum A(k,j)*x(k).
604: * k<>j
605: *
606: XJ = CABS1( X( J ) )
607: USCAL = TSCAL
608: REC = ONE / MAX( XMAX, ONE )
609: IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
610: *
611: * If x(j) could overflow, scale x by 1/(2*XMAX).
612: *
613: REC = REC*HALF
614: IF( NOUNIT ) THEN
615: TJJS = A( J, J )*TSCAL
616: ELSE
617: TJJS = TSCAL
618: END IF
619: TJJ = CABS1( TJJS )
620: IF( TJJ.GT.ONE ) THEN
621: *
622: * Divide by A(j,j) when scaling x if A(j,j) > 1.
623: *
624: REC = MIN( ONE, REC*TJJ )
625: USCAL = ZLADIV( USCAL, TJJS )
626: END IF
627: IF( REC.LT.ONE ) THEN
628: CALL ZDSCAL( N, REC, X, 1 )
629: SCALE = SCALE*REC
630: XMAX = XMAX*REC
631: END IF
632: END IF
633: *
634: CSUMJ = ZERO
635: IF( USCAL.EQ.DCMPLX( ONE ) ) THEN
636: *
637: * If the scaling needed for A in the dot product is 1,
638: * call ZDOTU to perform the dot product.
639: *
640: IF( UPPER ) THEN
641: CSUMJ = ZDOTU( J-1, A( 1, J ), 1, X, 1 )
642: ELSE IF( J.LT.N ) THEN
643: CSUMJ = ZDOTU( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
644: END IF
645: ELSE
646: *
647: * Otherwise, use in-line code for the dot product.
648: *
649: IF( UPPER ) THEN
650: DO 130 I = 1, J - 1
651: CSUMJ = CSUMJ + ( A( I, J )*USCAL )*X( I )
652: 130 CONTINUE
653: ELSE IF( J.LT.N ) THEN
654: DO 140 I = J + 1, N
655: CSUMJ = CSUMJ + ( A( I, J )*USCAL )*X( I )
656: 140 CONTINUE
657: END IF
658: END IF
659: *
660: IF( USCAL.EQ.DCMPLX( TSCAL ) ) THEN
661: *
662: * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
663: * was not used to scale the dotproduct.
664: *
665: X( J ) = X( J ) - CSUMJ
666: XJ = CABS1( X( J ) )
667: IF( NOUNIT ) THEN
668: TJJS = A( J, J )*TSCAL
669: ELSE
670: TJJS = TSCAL
671: IF( TSCAL.EQ.ONE )
672: $ GO TO 160
673: END IF
674: *
675: * Compute x(j) = x(j) / A(j,j), scaling if necessary.
676: *
677: TJJ = CABS1( TJJS )
678: IF( TJJ.GT.SMLNUM ) THEN
679: *
680: * abs(A(j,j)) > SMLNUM:
681: *
682: IF( TJJ.LT.ONE ) THEN
683: IF( XJ.GT.TJJ*BIGNUM ) THEN
684: *
685: * Scale X by 1/abs(x(j)).
686: *
687: REC = ONE / XJ
688: CALL ZDSCAL( N, REC, X, 1 )
689: SCALE = SCALE*REC
690: XMAX = XMAX*REC
691: END IF
692: END IF
693: X( J ) = ZLADIV( X( J ), TJJS )
694: ELSE IF( TJJ.GT.ZERO ) THEN
695: *
696: * 0 < abs(A(j,j)) <= SMLNUM:
697: *
698: IF( XJ.GT.TJJ*BIGNUM ) THEN
699: *
700: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
701: *
702: REC = ( TJJ*BIGNUM ) / XJ
703: CALL ZDSCAL( N, REC, X, 1 )
704: SCALE = SCALE*REC
705: XMAX = XMAX*REC
706: END IF
707: X( J ) = ZLADIV( X( J ), TJJS )
708: ELSE
709: *
710: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
711: * scale = 0 and compute a solution to A**T *x = 0.
712: *
713: DO 150 I = 1, N
714: X( I ) = ZERO
715: 150 CONTINUE
716: X( J ) = ONE
717: SCALE = ZERO
718: XMAX = ZERO
719: END IF
720: 160 CONTINUE
721: ELSE
722: *
723: * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
724: * product has already been divided by 1/A(j,j).
725: *
726: X( J ) = ZLADIV( X( J ), TJJS ) - CSUMJ
727: END IF
728: XMAX = MAX( XMAX, CABS1( X( J ) ) )
729: 170 CONTINUE
730: *
731: ELSE
732: *
733: * Solve A**H * x = b
734: *
735: DO 220 J = JFIRST, JLAST, JINC
736: *
737: * Compute x(j) = b(j) - sum A(k,j)*x(k).
738: * k<>j
739: *
740: XJ = CABS1( X( J ) )
741: USCAL = TSCAL
742: REC = ONE / MAX( XMAX, ONE )
743: IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
744: *
745: * If x(j) could overflow, scale x by 1/(2*XMAX).
746: *
747: REC = REC*HALF
748: IF( NOUNIT ) THEN
749: TJJS = DCONJG( A( J, J ) )*TSCAL
750: ELSE
751: TJJS = TSCAL
752: END IF
753: TJJ = CABS1( TJJS )
754: IF( TJJ.GT.ONE ) THEN
755: *
756: * Divide by A(j,j) when scaling x if A(j,j) > 1.
757: *
758: REC = MIN( ONE, REC*TJJ )
759: USCAL = ZLADIV( USCAL, TJJS )
760: END IF
761: IF( REC.LT.ONE ) THEN
762: CALL ZDSCAL( N, REC, X, 1 )
763: SCALE = SCALE*REC
764: XMAX = XMAX*REC
765: END IF
766: END IF
767: *
768: CSUMJ = ZERO
769: IF( USCAL.EQ.DCMPLX( ONE ) ) THEN
770: *
771: * If the scaling needed for A in the dot product is 1,
772: * call ZDOTC to perform the dot product.
773: *
774: IF( UPPER ) THEN
775: CSUMJ = ZDOTC( J-1, A( 1, J ), 1, X, 1 )
776: ELSE IF( J.LT.N ) THEN
777: CSUMJ = ZDOTC( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
778: END IF
779: ELSE
780: *
781: * Otherwise, use in-line code for the dot product.
782: *
783: IF( UPPER ) THEN
784: DO 180 I = 1, J - 1
785: CSUMJ = CSUMJ + ( DCONJG( A( I, J ) )*USCAL )*
786: $ X( I )
787: 180 CONTINUE
788: ELSE IF( J.LT.N ) THEN
789: DO 190 I = J + 1, N
790: CSUMJ = CSUMJ + ( DCONJG( A( I, J ) )*USCAL )*
791: $ X( I )
792: 190 CONTINUE
793: END IF
794: END IF
795: *
796: IF( USCAL.EQ.DCMPLX( TSCAL ) ) THEN
797: *
798: * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
799: * was not used to scale the dotproduct.
800: *
801: X( J ) = X( J ) - CSUMJ
802: XJ = CABS1( X( J ) )
803: IF( NOUNIT ) THEN
804: TJJS = DCONJG( A( J, J ) )*TSCAL
805: ELSE
806: TJJS = TSCAL
807: IF( TSCAL.EQ.ONE )
808: $ GO TO 210
809: END IF
810: *
811: * Compute x(j) = x(j) / A(j,j), scaling if necessary.
812: *
813: TJJ = CABS1( TJJS )
814: IF( TJJ.GT.SMLNUM ) THEN
815: *
816: * abs(A(j,j)) > SMLNUM:
817: *
818: IF( TJJ.LT.ONE ) THEN
819: IF( XJ.GT.TJJ*BIGNUM ) THEN
820: *
821: * Scale X by 1/abs(x(j)).
822: *
823: REC = ONE / XJ
824: CALL ZDSCAL( N, REC, X, 1 )
825: SCALE = SCALE*REC
826: XMAX = XMAX*REC
827: END IF
828: END IF
829: X( J ) = ZLADIV( X( J ), TJJS )
830: ELSE IF( TJJ.GT.ZERO ) THEN
831: *
832: * 0 < abs(A(j,j)) <= SMLNUM:
833: *
834: IF( XJ.GT.TJJ*BIGNUM ) THEN
835: *
836: * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
837: *
838: REC = ( TJJ*BIGNUM ) / XJ
839: CALL ZDSCAL( N, REC, X, 1 )
840: SCALE = SCALE*REC
841: XMAX = XMAX*REC
842: END IF
843: X( J ) = ZLADIV( X( J ), TJJS )
844: ELSE
845: *
846: * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
847: * scale = 0 and compute a solution to A**H *x = 0.
848: *
849: DO 200 I = 1, N
850: X( I ) = ZERO
851: 200 CONTINUE
852: X( J ) = ONE
853: SCALE = ZERO
854: XMAX = ZERO
855: END IF
856: 210 CONTINUE
857: ELSE
858: *
859: * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
860: * product has already been divided by 1/A(j,j).
861: *
862: X( J ) = ZLADIV( X( J ), TJJS ) - CSUMJ
863: END IF
864: XMAX = MAX( XMAX, CABS1( X( J ) ) )
865: 220 CONTINUE
866: END IF
867: SCALE = SCALE / TSCAL
868: END IF
869: *
870: * Scale the column norms by 1/TSCAL for return.
871: *
872: IF( TSCAL.NE.ONE ) THEN
873: CALL DSCAL( N, ONE / TSCAL, CNORM, 1 )
874: END IF
875: *
876: RETURN
877: *
878: * End of ZLATRS
879: *
880: END
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