LAPACK  3.11.0
LAPACK: Linear Algebra PACKage
zlatps.f
1 *> \brief \b ZLATPS solves a triangular system of equations with the matrix held in packed storage.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download ZLATPS + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlatps.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlatps.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
22 * CNORM, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER DIAG, NORMIN, TRANS, UPLO
26 * INTEGER INFO, N
27 * DOUBLE PRECISION SCALE
28 * ..
29 * .. Array Arguments ..
30 * DOUBLE PRECISION CNORM( * )
31 * COMPLEX*16 AP( * ), X( * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> ZLATPS solves one of the triangular systems
41 *>
42 *> A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
43 *>
44 *> with scaling to prevent overflow, where A is an upper or lower
45 *> triangular matrix stored in packed form. Here A**T denotes the
46 *> transpose of A, A**H denotes the conjugate transpose of A, x and b
47 *> are n-element vectors, and s is a scaling factor, usually less than
48 *> or equal to 1, chosen so that the components of x will be less than
49 *> the overflow threshold. If the unscaled problem will not cause
50 *> overflow, the Level 2 BLAS routine ZTPSV is called. If the matrix A
51 *> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
52 *> non-trivial solution to A*x = 0 is returned.
53 *> \endverbatim
54 *
55 * Arguments:
56 * ==========
57 *
58 *> \param[in] UPLO
59 *> \verbatim
60 *> UPLO is CHARACTER*1
61 *> Specifies whether the matrix A is upper or lower triangular.
62 *> = 'U': Upper triangular
63 *> = 'L': Lower triangular
64 *> \endverbatim
65 *>
66 *> \param[in] TRANS
67 *> \verbatim
68 *> TRANS is CHARACTER*1
69 *> Specifies the operation applied to A.
70 *> = 'N': Solve A * x = s*b (No transpose)
71 *> = 'T': Solve A**T * x = s*b (Transpose)
72 *> = 'C': Solve A**H * x = s*b (Conjugate transpose)
73 *> \endverbatim
74 *>
75 *> \param[in] DIAG
76 *> \verbatim
77 *> DIAG is CHARACTER*1
78 *> Specifies whether or not the matrix A is unit triangular.
79 *> = 'N': Non-unit triangular
80 *> = 'U': Unit triangular
81 *> \endverbatim
82 *>
83 *> \param[in] NORMIN
84 *> \verbatim
85 *> NORMIN is CHARACTER*1
86 *> Specifies whether CNORM has been set or not.
87 *> = 'Y': CNORM contains the column norms on entry
88 *> = 'N': CNORM is not set on entry. On exit, the norms will
89 *> be computed and stored in CNORM.
90 *> \endverbatim
91 *>
92 *> \param[in] N
93 *> \verbatim
94 *> N is INTEGER
95 *> The order of the matrix A. N >= 0.
96 *> \endverbatim
97 *>
98 *> \param[in] AP
99 *> \verbatim
100 *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
101 *> The upper or lower triangular matrix A, packed columnwise in
102 *> a linear array. The j-th column of A is stored in the array
103 *> AP as follows:
104 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
105 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
106 *> \endverbatim
107 *>
108 *> \param[in,out] X
109 *> \verbatim
110 *> X is COMPLEX*16 array, dimension (N)
111 *> On entry, the right hand side b of the triangular system.
112 *> On exit, X is overwritten by the solution vector x.
113 *> \endverbatim
114 *>
115 *> \param[out] SCALE
116 *> \verbatim
117 *> SCALE is DOUBLE PRECISION
118 *> The scaling factor s for the triangular system
119 *> A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
120 *> If SCALE = 0, the matrix A is singular or badly scaled, and
121 *> the vector x is an exact or approximate solution to A*x = 0.
122 *> \endverbatim
123 *>
124 *> \param[in,out] CNORM
125 *> \verbatim
126 *> CNORM is DOUBLE PRECISION array, dimension (N)
127 *>
128 *> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
129 *> contains the norm of the off-diagonal part of the j-th column
130 *> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
131 *> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
132 *> must be greater than or equal to the 1-norm.
133 *>
134 *> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
135 *> returns the 1-norm of the offdiagonal part of the j-th column
136 *> of A.
137 *> \endverbatim
138 *>
139 *> \param[out] INFO
140 *> \verbatim
141 *> INFO is INTEGER
142 *> = 0: successful exit
143 *> < 0: if INFO = -k, the k-th argument had an illegal value
144 *> \endverbatim
145 *
146 * Authors:
147 * ========
148 *
149 *> \author Univ. of Tennessee
150 *> \author Univ. of California Berkeley
151 *> \author Univ. of Colorado Denver
152 *> \author NAG Ltd.
153 *
154 *> \ingroup latps
155 *
156 *> \par Further Details:
157 * =====================
158 *>
159 *> \verbatim
160 *>
161 *> A rough bound on x is computed; if that is less than overflow, ZTPSV
162 *> is called, otherwise, specific code is used which checks for possible
163 *> overflow or divide-by-zero at every operation.
164 *>
165 *> A columnwise scheme is used for solving A*x = b. The basic algorithm
166 *> if A is lower triangular is
167 *>
168 *> x[1:n] := b[1:n]
169 *> for j = 1, ..., n
170 *> x(j) := x(j) / A(j,j)
171 *> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
172 *> end
173 *>
174 *> Define bounds on the components of x after j iterations of the loop:
175 *> M(j) = bound on x[1:j]
176 *> G(j) = bound on x[j+1:n]
177 *> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
178 *>
179 *> Then for iteration j+1 we have
180 *> M(j+1) <= G(j) / | A(j+1,j+1) |
181 *> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
182 *> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
183 *>
184 *> where CNORM(j+1) is greater than or equal to the infinity-norm of
185 *> column j+1 of A, not counting the diagonal. Hence
186 *>
187 *> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
188 *> 1<=i<=j
189 *> and
190 *>
191 *> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
192 *> 1<=i< j
193 *>
194 *> Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the
195 *> reciprocal of the largest M(j), j=1,..,n, is larger than
196 *> max(underflow, 1/overflow).
197 *>
198 *> The bound on x(j) is also used to determine when a step in the
199 *> columnwise method can be performed without fear of overflow. If
200 *> the computed bound is greater than a large constant, x is scaled to
201 *> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
202 *> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
203 *>
204 *> Similarly, a row-wise scheme is used to solve A**T *x = b or
205 *> A**H *x = b. The basic algorithm for A upper triangular is
206 *>
207 *> for j = 1, ..., n
208 *> x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
209 *> end
210 *>
211 *> We simultaneously compute two bounds
212 *> G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
213 *> M(j) = bound on x(i), 1<=i<=j
214 *>
215 *> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
216 *> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
217 *> Then the bound on x(j) is
218 *>
219 *> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
220 *>
221 *> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
222 *> 1<=i<=j
223 *>
224 *> and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater
225 *> than max(underflow, 1/overflow).
226 *> \endverbatim
227 *>
228 * =====================================================================
229  SUBROUTINE zlatps( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
230  $ CNORM, INFO )
231 *
232 * -- LAPACK auxiliary routine --
233 * -- LAPACK is a software package provided by Univ. of Tennessee, --
234 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
235 *
236 * .. Scalar Arguments ..
237  CHARACTER DIAG, NORMIN, TRANS, UPLO
238  INTEGER INFO, N
239  DOUBLE PRECISION SCALE
240 * ..
241 * .. Array Arguments ..
242  DOUBLE PRECISION CNORM( * )
243  COMPLEX*16 AP( * ), X( * )
244 * ..
245 *
246 * =====================================================================
247 *
248 * .. Parameters ..
249  DOUBLE PRECISION ZERO, HALF, ONE, TWO
250  parameter( zero = 0.0d+0, half = 0.5d+0, one = 1.0d+0,
251  $ two = 2.0d+0 )
252 * ..
253 * .. Local Scalars ..
254  LOGICAL NOTRAN, NOUNIT, UPPER
255  INTEGER I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
256  DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
257  $ xbnd, xj, xmax
258  COMPLEX*16 CSUMJ, TJJS, USCAL, ZDUM
259 * ..
260 * .. External Functions ..
261  LOGICAL LSAME
262  INTEGER IDAMAX, IZAMAX
263  DOUBLE PRECISION DLAMCH, DZASUM
264  COMPLEX*16 ZDOTC, ZDOTU, ZLADIV
265  EXTERNAL lsame, idamax, izamax, dlamch, dzasum, zdotc,
266  $ zdotu, zladiv
267 * ..
268 * .. External Subroutines ..
269  EXTERNAL dscal, xerbla, zaxpy, zdscal, ztpsv, dlabad
270 * ..
271 * .. Intrinsic Functions ..
272  INTRINSIC abs, dble, dcmplx, dconjg, dimag, max, min
273 * ..
274 * .. Statement Functions ..
275  DOUBLE PRECISION CABS1, CABS2
276 * ..
277 * .. Statement Function definitions ..
278  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
279  cabs2( zdum ) = abs( dble( zdum ) / 2.d0 ) +
280  $ abs( dimag( zdum ) / 2.d0 )
281 * ..
282 * .. Executable Statements ..
283 *
284  info = 0
285  upper = lsame( uplo, 'U' )
286  notran = lsame( trans, 'N' )
287  nounit = lsame( diag, 'N' )
288 *
289 * Test the input parameters.
290 *
291  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
292  info = -1
293  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
294  $ lsame( trans, 'C' ) ) THEN
295  info = -2
296  ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
297  info = -3
298  ELSE IF( .NOT.lsame( normin, 'Y' ) .AND. .NOT.
299  $ lsame( normin, 'N' ) ) THEN
300  info = -4
301  ELSE IF( n.LT.0 ) THEN
302  info = -5
303  END IF
304  IF( info.NE.0 ) THEN
305  CALL xerbla( 'ZLATPS', -info )
306  RETURN
307  END IF
308 *
309 * Quick return if possible
310 *
311  IF( n.EQ.0 )
312  $ RETURN
313 *
314 * Determine machine dependent parameters to control overflow.
315 *
316  smlnum = dlamch( 'Safe minimum' )
317  bignum = one / smlnum
318  CALL dlabad( smlnum, bignum )
319  smlnum = smlnum / dlamch( 'Precision' )
320  bignum = one / smlnum
321  scale = one
322 *
323  IF( lsame( normin, 'N' ) ) THEN
324 *
325 * Compute the 1-norm of each column, not including the diagonal.
326 *
327  IF( upper ) THEN
328 *
329 * A is upper triangular.
330 *
331  ip = 1
332  DO 10 j = 1, n
333  cnorm( j ) = dzasum( j-1, ap( ip ), 1 )
334  ip = ip + j
335  10 CONTINUE
336  ELSE
337 *
338 * A is lower triangular.
339 *
340  ip = 1
341  DO 20 j = 1, n - 1
342  cnorm( j ) = dzasum( n-j, ap( ip+1 ), 1 )
343  ip = ip + n - j + 1
344  20 CONTINUE
345  cnorm( n ) = zero
346  END IF
347  END IF
348 *
349 * Scale the column norms by TSCAL if the maximum element in CNORM is
350 * greater than BIGNUM/2.
351 *
352  imax = idamax( n, cnorm, 1 )
353  tmax = cnorm( imax )
354  IF( tmax.LE.bignum*half ) THEN
355  tscal = one
356  ELSE
357  tscal = half / ( smlnum*tmax )
358  CALL dscal( n, tscal, cnorm, 1 )
359  END IF
360 *
361 * Compute a bound on the computed solution vector to see if the
362 * Level 2 BLAS routine ZTPSV can be used.
363 *
364  xmax = zero
365  DO 30 j = 1, n
366  xmax = max( xmax, cabs2( x( j ) ) )
367  30 CONTINUE
368  xbnd = xmax
369  IF( notran ) THEN
370 *
371 * Compute the growth in A * x = b.
372 *
373  IF( upper ) THEN
374  jfirst = n
375  jlast = 1
376  jinc = -1
377  ELSE
378  jfirst = 1
379  jlast = n
380  jinc = 1
381  END IF
382 *
383  IF( tscal.NE.one ) THEN
384  grow = zero
385  GO TO 60
386  END IF
387 *
388  IF( nounit ) THEN
389 *
390 * A is non-unit triangular.
391 *
392 * Compute GROW = 1/G(j) and XBND = 1/M(j).
393 * Initially, G(0) = max{x(i), i=1,...,n}.
394 *
395  grow = half / max( xbnd, smlnum )
396  xbnd = grow
397  ip = jfirst*( jfirst+1 ) / 2
398  jlen = n
399  DO 40 j = jfirst, jlast, jinc
400 *
401 * Exit the loop if the growth factor is too small.
402 *
403  IF( grow.LE.smlnum )
404  $ GO TO 60
405 *
406  tjjs = ap( ip )
407  tjj = cabs1( tjjs )
408 *
409  IF( tjj.GE.smlnum ) THEN
410 *
411 * M(j) = G(j-1) / abs(A(j,j))
412 *
413  xbnd = min( xbnd, min( one, tjj )*grow )
414  ELSE
415 *
416 * M(j) could overflow, set XBND to 0.
417 *
418  xbnd = zero
419  END IF
420 *
421  IF( tjj+cnorm( j ).GE.smlnum ) THEN
422 *
423 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
424 *
425  grow = grow*( tjj / ( tjj+cnorm( j ) ) )
426  ELSE
427 *
428 * G(j) could overflow, set GROW to 0.
429 *
430  grow = zero
431  END IF
432  ip = ip + jinc*jlen
433  jlen = jlen - 1
434  40 CONTINUE
435  grow = xbnd
436  ELSE
437 *
438 * A is unit triangular.
439 *
440 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
441 *
442  grow = min( one, half / max( xbnd, smlnum ) )
443  DO 50 j = jfirst, jlast, jinc
444 *
445 * Exit the loop if the growth factor is too small.
446 *
447  IF( grow.LE.smlnum )
448  $ GO TO 60
449 *
450 * G(j) = G(j-1)*( 1 + CNORM(j) )
451 *
452  grow = grow*( one / ( one+cnorm( j ) ) )
453  50 CONTINUE
454  END IF
455  60 CONTINUE
456 *
457  ELSE
458 *
459 * Compute the growth in A**T * x = b or A**H * x = b.
460 *
461  IF( upper ) THEN
462  jfirst = 1
463  jlast = n
464  jinc = 1
465  ELSE
466  jfirst = n
467  jlast = 1
468  jinc = -1
469  END IF
470 *
471  IF( tscal.NE.one ) THEN
472  grow = zero
473  GO TO 90
474  END IF
475 *
476  IF( nounit ) THEN
477 *
478 * A is non-unit triangular.
479 *
480 * Compute GROW = 1/G(j) and XBND = 1/M(j).
481 * Initially, M(0) = max{x(i), i=1,...,n}.
482 *
483  grow = half / max( xbnd, smlnum )
484  xbnd = grow
485  ip = jfirst*( jfirst+1 ) / 2
486  jlen = 1
487  DO 70 j = jfirst, jlast, jinc
488 *
489 * Exit the loop if the growth factor is too small.
490 *
491  IF( grow.LE.smlnum )
492  $ GO TO 90
493 *
494 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
495 *
496  xj = one + cnorm( j )
497  grow = min( grow, xbnd / xj )
498 *
499  tjjs = ap( ip )
500  tjj = cabs1( tjjs )
501 *
502  IF( tjj.GE.smlnum ) THEN
503 *
504 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
505 *
506  IF( xj.GT.tjj )
507  $ xbnd = xbnd*( tjj / xj )
508  ELSE
509 *
510 * M(j) could overflow, set XBND to 0.
511 *
512  xbnd = zero
513  END IF
514  jlen = jlen + 1
515  ip = ip + jinc*jlen
516  70 CONTINUE
517  grow = min( grow, xbnd )
518  ELSE
519 *
520 * A is unit triangular.
521 *
522 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
523 *
524  grow = min( one, half / max( xbnd, smlnum ) )
525  DO 80 j = jfirst, jlast, jinc
526 *
527 * Exit the loop if the growth factor is too small.
528 *
529  IF( grow.LE.smlnum )
530  $ GO TO 90
531 *
532 * G(j) = ( 1 + CNORM(j) )*G(j-1)
533 *
534  xj = one + cnorm( j )
535  grow = grow / xj
536  80 CONTINUE
537  END IF
538  90 CONTINUE
539  END IF
540 *
541  IF( ( grow*tscal ).GT.smlnum ) THEN
542 *
543 * Use the Level 2 BLAS solve if the reciprocal of the bound on
544 * elements of X is not too small.
545 *
546  CALL ztpsv( uplo, trans, diag, n, ap, x, 1 )
547  ELSE
548 *
549 * Use a Level 1 BLAS solve, scaling intermediate results.
550 *
551  IF( xmax.GT.bignum*half ) THEN
552 *
553 * Scale X so that its components are less than or equal to
554 * BIGNUM in absolute value.
555 *
556  scale = ( bignum*half ) / xmax
557  CALL zdscal( n, scale, x, 1 )
558  xmax = bignum
559  ELSE
560  xmax = xmax*two
561  END IF
562 *
563  IF( notran ) THEN
564 *
565 * Solve A * x = b
566 *
567  ip = jfirst*( jfirst+1 ) / 2
568  DO 120 j = jfirst, jlast, jinc
569 *
570 * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
571 *
572  xj = cabs1( x( j ) )
573  IF( nounit ) THEN
574  tjjs = ap( ip )*tscal
575  ELSE
576  tjjs = tscal
577  IF( tscal.EQ.one )
578  $ GO TO 110
579  END IF
580  tjj = cabs1( tjjs )
581  IF( tjj.GT.smlnum ) THEN
582 *
583 * abs(A(j,j)) > SMLNUM:
584 *
585  IF( tjj.LT.one ) THEN
586  IF( xj.GT.tjj*bignum ) THEN
587 *
588 * Scale x by 1/b(j).
589 *
590  rec = one / xj
591  CALL zdscal( n, rec, x, 1 )
592  scale = scale*rec
593  xmax = xmax*rec
594  END IF
595  END IF
596  x( j ) = zladiv( x( j ), tjjs )
597  xj = cabs1( x( j ) )
598  ELSE IF( tjj.GT.zero ) THEN
599 *
600 * 0 < abs(A(j,j)) <= SMLNUM:
601 *
602  IF( xj.GT.tjj*bignum ) THEN
603 *
604 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
605 * to avoid overflow when dividing by A(j,j).
606 *
607  rec = ( tjj*bignum ) / xj
608  IF( cnorm( j ).GT.one ) THEN
609 *
610 * Scale by 1/CNORM(j) to avoid overflow when
611 * multiplying x(j) times column j.
612 *
613  rec = rec / cnorm( j )
614  END IF
615  CALL zdscal( n, rec, x, 1 )
616  scale = scale*rec
617  xmax = xmax*rec
618  END IF
619  x( j ) = zladiv( x( j ), tjjs )
620  xj = cabs1( x( j ) )
621  ELSE
622 *
623 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
624 * scale = 0, and compute a solution to A*x = 0.
625 *
626  DO 100 i = 1, n
627  x( i ) = zero
628  100 CONTINUE
629  x( j ) = one
630  xj = one
631  scale = zero
632  xmax = zero
633  END IF
634  110 CONTINUE
635 *
636 * Scale x if necessary to avoid overflow when adding a
637 * multiple of column j of A.
638 *
639  IF( xj.GT.one ) THEN
640  rec = one / xj
641  IF( cnorm( j ).GT.( bignum-xmax )*rec ) THEN
642 *
643 * Scale x by 1/(2*abs(x(j))).
644 *
645  rec = rec*half
646  CALL zdscal( n, rec, x, 1 )
647  scale = scale*rec
648  END IF
649  ELSE IF( xj*cnorm( j ).GT.( bignum-xmax ) ) THEN
650 *
651 * Scale x by 1/2.
652 *
653  CALL zdscal( n, half, x, 1 )
654  scale = scale*half
655  END IF
656 *
657  IF( upper ) THEN
658  IF( j.GT.1 ) THEN
659 *
660 * Compute the update
661 * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
662 *
663  CALL zaxpy( j-1, -x( j )*tscal, ap( ip-j+1 ), 1, x,
664  $ 1 )
665  i = izamax( j-1, x, 1 )
666  xmax = cabs1( x( i ) )
667  END IF
668  ip = ip - j
669  ELSE
670  IF( j.LT.n ) THEN
671 *
672 * Compute the update
673 * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
674 *
675  CALL zaxpy( n-j, -x( j )*tscal, ap( ip+1 ), 1,
676  $ x( j+1 ), 1 )
677  i = j + izamax( n-j, x( j+1 ), 1 )
678  xmax = cabs1( x( i ) )
679  END IF
680  ip = ip + n - j + 1
681  END IF
682  120 CONTINUE
683 *
684  ELSE IF( lsame( trans, 'T' ) ) THEN
685 *
686 * Solve A**T * x = b
687 *
688  ip = jfirst*( jfirst+1 ) / 2
689  jlen = 1
690  DO 170 j = jfirst, jlast, jinc
691 *
692 * Compute x(j) = b(j) - sum A(k,j)*x(k).
693 * k<>j
694 *
695  xj = cabs1( x( j ) )
696  uscal = tscal
697  rec = one / max( xmax, one )
698  IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
699 *
700 * If x(j) could overflow, scale x by 1/(2*XMAX).
701 *
702  rec = rec*half
703  IF( nounit ) THEN
704  tjjs = ap( ip )*tscal
705  ELSE
706  tjjs = tscal
707  END IF
708  tjj = cabs1( tjjs )
709  IF( tjj.GT.one ) THEN
710 *
711 * Divide by A(j,j) when scaling x if A(j,j) > 1.
712 *
713  rec = min( one, rec*tjj )
714  uscal = zladiv( uscal, tjjs )
715  END IF
716  IF( rec.LT.one ) THEN
717  CALL zdscal( n, rec, x, 1 )
718  scale = scale*rec
719  xmax = xmax*rec
720  END IF
721  END IF
722 *
723  csumj = zero
724  IF( uscal.EQ.dcmplx( one ) ) THEN
725 *
726 * If the scaling needed for A in the dot product is 1,
727 * call ZDOTU to perform the dot product.
728 *
729  IF( upper ) THEN
730  csumj = zdotu( j-1, ap( ip-j+1 ), 1, x, 1 )
731  ELSE IF( j.LT.n ) THEN
732  csumj = zdotu( n-j, ap( ip+1 ), 1, x( j+1 ), 1 )
733  END IF
734  ELSE
735 *
736 * Otherwise, use in-line code for the dot product.
737 *
738  IF( upper ) THEN
739  DO 130 i = 1, j - 1
740  csumj = csumj + ( ap( ip-j+i )*uscal )*x( i )
741  130 CONTINUE
742  ELSE IF( j.LT.n ) THEN
743  DO 140 i = 1, n - j
744  csumj = csumj + ( ap( ip+i )*uscal )*x( j+i )
745  140 CONTINUE
746  END IF
747  END IF
748 *
749  IF( uscal.EQ.dcmplx( tscal ) ) THEN
750 *
751 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
752 * was not used to scale the dotproduct.
753 *
754  x( j ) = x( j ) - csumj
755  xj = cabs1( x( j ) )
756  IF( nounit ) THEN
757 *
758 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
759 *
760  tjjs = ap( ip )*tscal
761  ELSE
762  tjjs = tscal
763  IF( tscal.EQ.one )
764  $ GO TO 160
765  END IF
766  tjj = cabs1( tjjs )
767  IF( tjj.GT.smlnum ) THEN
768 *
769 * abs(A(j,j)) > SMLNUM:
770 *
771  IF( tjj.LT.one ) THEN
772  IF( xj.GT.tjj*bignum ) THEN
773 *
774 * Scale X by 1/abs(x(j)).
775 *
776  rec = one / xj
777  CALL zdscal( n, rec, x, 1 )
778  scale = scale*rec
779  xmax = xmax*rec
780  END IF
781  END IF
782  x( j ) = zladiv( x( j ), tjjs )
783  ELSE IF( tjj.GT.zero ) THEN
784 *
785 * 0 < abs(A(j,j)) <= SMLNUM:
786 *
787  IF( xj.GT.tjj*bignum ) THEN
788 *
789 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
790 *
791  rec = ( tjj*bignum ) / xj
792  CALL zdscal( n, rec, x, 1 )
793  scale = scale*rec
794  xmax = xmax*rec
795  END IF
796  x( j ) = zladiv( x( j ), tjjs )
797  ELSE
798 *
799 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
800 * scale = 0 and compute a solution to A**T *x = 0.
801 *
802  DO 150 i = 1, n
803  x( i ) = zero
804  150 CONTINUE
805  x( j ) = one
806  scale = zero
807  xmax = zero
808  END IF
809  160 CONTINUE
810  ELSE
811 *
812 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
813 * product has already been divided by 1/A(j,j).
814 *
815  x( j ) = zladiv( x( j ), tjjs ) - csumj
816  END IF
817  xmax = max( xmax, cabs1( x( j ) ) )
818  jlen = jlen + 1
819  ip = ip + jinc*jlen
820  170 CONTINUE
821 *
822  ELSE
823 *
824 * Solve A**H * x = b
825 *
826  ip = jfirst*( jfirst+1 ) / 2
827  jlen = 1
828  DO 220 j = jfirst, jlast, jinc
829 *
830 * Compute x(j) = b(j) - sum A(k,j)*x(k).
831 * k<>j
832 *
833  xj = cabs1( x( j ) )
834  uscal = tscal
835  rec = one / max( xmax, one )
836  IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
837 *
838 * If x(j) could overflow, scale x by 1/(2*XMAX).
839 *
840  rec = rec*half
841  IF( nounit ) THEN
842  tjjs = dconjg( ap( ip ) )*tscal
843  ELSE
844  tjjs = tscal
845  END IF
846  tjj = cabs1( tjjs )
847  IF( tjj.GT.one ) THEN
848 *
849 * Divide by A(j,j) when scaling x if A(j,j) > 1.
850 *
851  rec = min( one, rec*tjj )
852  uscal = zladiv( uscal, tjjs )
853  END IF
854  IF( rec.LT.one ) THEN
855  CALL zdscal( n, rec, x, 1 )
856  scale = scale*rec
857  xmax = xmax*rec
858  END IF
859  END IF
860 *
861  csumj = zero
862  IF( uscal.EQ.dcmplx( one ) ) THEN
863 *
864 * If the scaling needed for A in the dot product is 1,
865 * call ZDOTC to perform the dot product.
866 *
867  IF( upper ) THEN
868  csumj = zdotc( j-1, ap( ip-j+1 ), 1, x, 1 )
869  ELSE IF( j.LT.n ) THEN
870  csumj = zdotc( n-j, ap( ip+1 ), 1, x( j+1 ), 1 )
871  END IF
872  ELSE
873 *
874 * Otherwise, use in-line code for the dot product.
875 *
876  IF( upper ) THEN
877  DO 180 i = 1, j - 1
878  csumj = csumj + ( dconjg( ap( ip-j+i ) )*uscal )
879  $ *x( i )
880  180 CONTINUE
881  ELSE IF( j.LT.n ) THEN
882  DO 190 i = 1, n - j
883  csumj = csumj + ( dconjg( ap( ip+i ) )*uscal )*
884  $ x( j+i )
885  190 CONTINUE
886  END IF
887  END IF
888 *
889  IF( uscal.EQ.dcmplx( tscal ) ) THEN
890 *
891 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
892 * was not used to scale the dotproduct.
893 *
894  x( j ) = x( j ) - csumj
895  xj = cabs1( x( j ) )
896  IF( nounit ) THEN
897 *
898 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
899 *
900  tjjs = dconjg( ap( ip ) )*tscal
901  ELSE
902  tjjs = tscal
903  IF( tscal.EQ.one )
904  $ GO TO 210
905  END IF
906  tjj = cabs1( tjjs )
907  IF( tjj.GT.smlnum ) THEN
908 *
909 * abs(A(j,j)) > SMLNUM:
910 *
911  IF( tjj.LT.one ) THEN
912  IF( xj.GT.tjj*bignum ) THEN
913 *
914 * Scale X by 1/abs(x(j)).
915 *
916  rec = one / xj
917  CALL zdscal( n, rec, x, 1 )
918  scale = scale*rec
919  xmax = xmax*rec
920  END IF
921  END IF
922  x( j ) = zladiv( x( j ), tjjs )
923  ELSE IF( tjj.GT.zero ) THEN
924 *
925 * 0 < abs(A(j,j)) <= SMLNUM:
926 *
927  IF( xj.GT.tjj*bignum ) THEN
928 *
929 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
930 *
931  rec = ( tjj*bignum ) / xj
932  CALL zdscal( n, rec, x, 1 )
933  scale = scale*rec
934  xmax = xmax*rec
935  END IF
936  x( j ) = zladiv( x( j ), tjjs )
937  ELSE
938 *
939 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
940 * scale = 0 and compute a solution to A**H *x = 0.
941 *
942  DO 200 i = 1, n
943  x( i ) = zero
944  200 CONTINUE
945  x( j ) = one
946  scale = zero
947  xmax = zero
948  END IF
949  210 CONTINUE
950  ELSE
951 *
952 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
953 * product has already been divided by 1/A(j,j).
954 *
955  x( j ) = zladiv( x( j ), tjjs ) - csumj
956  END IF
957  xmax = max( xmax, cabs1( x( j ) ) )
958  jlen = jlen + 1
959  ip = ip + jinc*jlen
960  220 CONTINUE
961  END IF
962  scale = scale / tscal
963  END IF
964 *
965 * Scale the column norms by 1/TSCAL for return.
966 *
967  IF( tscal.NE.one ) THEN
968  CALL dscal( n, one / tscal, cnorm, 1 )
969  END IF
970 *
971  RETURN
972 *
973 * End of ZLATPS
974 *
975  END
zaxpy
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
zdscal
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:78
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
zlatps
subroutine zlatps(UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
ZLATPS solves a triangular system of equations with the matrix held in packed storage.
Definition: zlatps.f:231
ztpsv
subroutine ztpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
ZTPSV
Definition: ztpsv.f:144
dscal
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
dlabad
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74