LAPACK  3.7.1
LAPACK: Linear Algebra PACKage
zlatps.f
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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>
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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 *> \date December 2016
155 *
156 *> \ingroup complex16OTHERauxiliary
157 *
158 *> \par Further Details:
159 * =====================
160 *>
161 *> \verbatim
162 *>
163 *> A rough bound on x is computed; if that is less than overflow, ZTPSV
164 *> is called, otherwise, specific code is used which checks for possible
165 *> overflow or divide-by-zero at every operation.
166 *>
167 *> A columnwise scheme is used for solving A*x = b. The basic algorithm
168 *> if A is lower triangular is
169 *>
170 *> x[1:n] := b[1:n]
171 *> for j = 1, ..., n
172 *> x(j) := x(j) / A(j,j)
173 *> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
174 *> end
175 *>
176 *> Define bounds on the components of x after j iterations of the loop:
177 *> M(j) = bound on x[1:j]
178 *> G(j) = bound on x[j+1:n]
179 *> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
180 *>
181 *> Then for iteration j+1 we have
182 *> M(j+1) <= G(j) / | A(j+1,j+1) |
183 *> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
184 *> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
185 *>
186 *> where CNORM(j+1) is greater than or equal to the infinity-norm of
187 *> column j+1 of A, not counting the diagonal. Hence
188 *>
189 *> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
190 *> 1<=i<=j
191 *> and
192 *>
193 *> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
194 *> 1<=i< j
195 *>
196 *> Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the
197 *> reciprocal of the largest M(j), j=1,..,n, is larger than
198 *> max(underflow, 1/overflow).
199 *>
200 *> The bound on x(j) is also used to determine when a step in the
201 *> columnwise method can be performed without fear of overflow. If
202 *> the computed bound is greater than a large constant, x is scaled to
203 *> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
204 *> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
205 *>
206 *> Similarly, a row-wise scheme is used to solve A**T *x = b or
207 *> A**H *x = b. The basic algorithm for A upper triangular is
208 *>
209 *> for j = 1, ..., n
210 *> x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
211 *> end
212 *>
213 *> We simultaneously compute two bounds
214 *> G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
215 *> M(j) = bound on x(i), 1<=i<=j
216 *>
217 *> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
218 *> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
219 *> Then the bound on x(j) is
220 *>
221 *> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
222 *>
223 *> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
224 *> 1<=i<=j
225 *>
226 *> and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater
227 *> than max(underflow, 1/overflow).
228 *> \endverbatim
229 *>
230 * =====================================================================
231  SUBROUTINE zlatps( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
232  $ CNORM, INFO )
233 *
234 * -- LAPACK auxiliary routine (version 3.7.0) --
235 * -- LAPACK is a software package provided by Univ. of Tennessee, --
236 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
237 * December 2016
238 *
239 * .. Scalar Arguments ..
240  CHARACTER DIAG, NORMIN, TRANS, UPLO
241  INTEGER INFO, N
242  DOUBLE PRECISION SCALE
243 * ..
244 * .. Array Arguments ..
245  DOUBLE PRECISION CNORM( * )
246  COMPLEX*16 AP( * ), X( * )
247 * ..
248 *
249 * =====================================================================
250 *
251 * .. Parameters ..
252  DOUBLE PRECISION ZERO, HALF, ONE, TWO
253  parameter( zero = 0.0d+0, half = 0.5d+0, one = 1.0d+0,
254  $ two = 2.0d+0 )
255 * ..
256 * .. Local Scalars ..
257  LOGICAL NOTRAN, NOUNIT, UPPER
258  INTEGER I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
259  DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, TJJ, TMAX, TSCAL,
260  $ xbnd, xj, xmax
261  COMPLEX*16 CSUMJ, TJJS, USCAL, ZDUM
262 * ..
263 * .. External Functions ..
264  LOGICAL LSAME
265  INTEGER IDAMAX, IZAMAX
266  DOUBLE PRECISION DLAMCH, DZASUM
267  COMPLEX*16 ZDOTC, ZDOTU, ZLADIV
268  EXTERNAL lsame, idamax, izamax, dlamch, dzasum, zdotc,
269  $ zdotu, zladiv
270 * ..
271 * .. External Subroutines ..
272  EXTERNAL dscal, xerbla, zaxpy, zdscal, ztpsv
273 * ..
274 * .. Intrinsic Functions ..
275  INTRINSIC abs, dble, dcmplx, dconjg, dimag, max, min
276 * ..
277 * .. Statement Functions ..
278  DOUBLE PRECISION CABS1, CABS2
279 * ..
280 * .. Statement Function definitions ..
281  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
282  cabs2( zdum ) = abs( dble( zdum ) / 2.d0 ) +
283  $ abs( dimag( zdum ) / 2.d0 )
284 * ..
285 * .. Executable Statements ..
286 *
287  info = 0
288  upper = lsame( uplo, 'U' )
289  notran = lsame( trans, 'N' )
290  nounit = lsame( diag, 'N' )
291 *
292 * Test the input parameters.
293 *
294  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
295  info = -1
296  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
297  $ lsame( trans, 'C' ) ) THEN
298  info = -2
299  ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
300  info = -3
301  ELSE IF( .NOT.lsame( normin, 'Y' ) .AND. .NOT.
302  $ lsame( normin, 'N' ) ) THEN
303  info = -4
304  ELSE IF( n.LT.0 ) THEN
305  info = -5
306  END IF
307  IF( info.NE.0 ) THEN
308  CALL xerbla( 'ZLATPS', -info )
309  RETURN
310  END IF
311 *
312 * Quick return if possible
313 *
314  IF( n.EQ.0 )
315  $ RETURN
316 *
317 * Determine machine dependent parameters to control overflow.
318 *
319  smlnum = dlamch( 'Safe minimum' )
320  bignum = one / smlnum
321  CALL dlabad( smlnum, bignum )
322  smlnum = smlnum / dlamch( 'Precision' )
323  bignum = one / smlnum
324  scale = one
325 *
326  IF( lsame( normin, 'N' ) ) THEN
327 *
328 * Compute the 1-norm of each column, not including the diagonal.
329 *
330  IF( upper ) THEN
331 *
332 * A is upper triangular.
333 *
334  ip = 1
335  DO 10 j = 1, n
336  cnorm( j ) = dzasum( j-1, ap( ip ), 1 )
337  ip = ip + j
338  10 CONTINUE
339  ELSE
340 *
341 * A is lower triangular.
342 *
343  ip = 1
344  DO 20 j = 1, n - 1
345  cnorm( j ) = dzasum( n-j, ap( ip+1 ), 1 )
346  ip = ip + n - j + 1
347  20 CONTINUE
348  cnorm( n ) = zero
349  END IF
350  END IF
351 *
352 * Scale the column norms by TSCAL if the maximum element in CNORM is
353 * greater than BIGNUM/2.
354 *
355  imax = idamax( n, cnorm, 1 )
356  tmax = cnorm( imax )
357  IF( tmax.LE.bignum*half ) THEN
358  tscal = one
359  ELSE
360  tscal = half / ( smlnum*tmax )
361  CALL dscal( n, tscal, cnorm, 1 )
362  END IF
363 *
364 * Compute a bound on the computed solution vector to see if the
365 * Level 2 BLAS routine ZTPSV can be used.
366 *
367  xmax = zero
368  DO 30 j = 1, n
369  xmax = max( xmax, cabs2( x( j ) ) )
370  30 CONTINUE
371  xbnd = xmax
372  IF( notran ) THEN
373 *
374 * Compute the growth in A * x = b.
375 *
376  IF( upper ) THEN
377  jfirst = n
378  jlast = 1
379  jinc = -1
380  ELSE
381  jfirst = 1
382  jlast = n
383  jinc = 1
384  END IF
385 *
386  IF( tscal.NE.one ) THEN
387  grow = zero
388  GO TO 60
389  END IF
390 *
391  IF( nounit ) THEN
392 *
393 * A is non-unit triangular.
394 *
395 * Compute GROW = 1/G(j) and XBND = 1/M(j).
396 * Initially, G(0) = max{x(i), i=1,...,n}.
397 *
398  grow = half / max( xbnd, smlnum )
399  xbnd = grow
400  ip = jfirst*( jfirst+1 ) / 2
401  jlen = n
402  DO 40 j = jfirst, jlast, jinc
403 *
404 * Exit the loop if the growth factor is too small.
405 *
406  IF( grow.LE.smlnum )
407  $ GO TO 60
408 *
409  tjjs = ap( ip )
410  tjj = cabs1( tjjs )
411 *
412  IF( tjj.GE.smlnum ) THEN
413 *
414 * M(j) = G(j-1) / abs(A(j,j))
415 *
416  xbnd = min( xbnd, min( one, tjj )*grow )
417  ELSE
418 *
419 * M(j) could overflow, set XBND to 0.
420 *
421  xbnd = zero
422  END IF
423 *
424  IF( tjj+cnorm( j ).GE.smlnum ) THEN
425 *
426 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
427 *
428  grow = grow*( tjj / ( tjj+cnorm( j ) ) )
429  ELSE
430 *
431 * G(j) could overflow, set GROW to 0.
432 *
433  grow = zero
434  END IF
435  ip = ip + jinc*jlen
436  jlen = jlen - 1
437  40 CONTINUE
438  grow = xbnd
439  ELSE
440 *
441 * A is unit triangular.
442 *
443 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
444 *
445  grow = min( one, half / max( xbnd, smlnum ) )
446  DO 50 j = jfirst, jlast, jinc
447 *
448 * Exit the loop if the growth factor is too small.
449 *
450  IF( grow.LE.smlnum )
451  $ GO TO 60
452 *
453 * G(j) = G(j-1)*( 1 + CNORM(j) )
454 *
455  grow = grow*( one / ( one+cnorm( j ) ) )
456  50 CONTINUE
457  END IF
458  60 CONTINUE
459 *
460  ELSE
461 *
462 * Compute the growth in A**T * x = b or A**H * x = b.
463 *
464  IF( upper ) THEN
465  jfirst = 1
466  jlast = n
467  jinc = 1
468  ELSE
469  jfirst = n
470  jlast = 1
471  jinc = -1
472  END IF
473 *
474  IF( tscal.NE.one ) THEN
475  grow = zero
476  GO TO 90
477  END IF
478 *
479  IF( nounit ) THEN
480 *
481 * A is non-unit triangular.
482 *
483 * Compute GROW = 1/G(j) and XBND = 1/M(j).
484 * Initially, M(0) = max{x(i), i=1,...,n}.
485 *
486  grow = half / max( xbnd, smlnum )
487  xbnd = grow
488  ip = jfirst*( jfirst+1 ) / 2
489  jlen = 1
490  DO 70 j = jfirst, jlast, jinc
491 *
492 * Exit the loop if the growth factor is too small.
493 *
494  IF( grow.LE.smlnum )
495  $ GO TO 90
496 *
497 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
498 *
499  xj = one + cnorm( j )
500  grow = min( grow, xbnd / xj )
501 *
502  tjjs = ap( ip )
503  tjj = cabs1( tjjs )
504 *
505  IF( tjj.GE.smlnum ) THEN
506 *
507 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
508 *
509  IF( xj.GT.tjj )
510  $ xbnd = xbnd*( tjj / xj )
511  ELSE
512 *
513 * M(j) could overflow, set XBND to 0.
514 *
515  xbnd = zero
516  END IF
517  jlen = jlen + 1
518  ip = ip + jinc*jlen
519  70 CONTINUE
520  grow = min( grow, xbnd )
521  ELSE
522 *
523 * A is unit triangular.
524 *
525 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
526 *
527  grow = min( one, half / max( xbnd, smlnum ) )
528  DO 80 j = jfirst, jlast, jinc
529 *
530 * Exit the loop if the growth factor is too small.
531 *
532  IF( grow.LE.smlnum )
533  $ GO TO 90
534 *
535 * G(j) = ( 1 + CNORM(j) )*G(j-1)
536 *
537  xj = one + cnorm( j )
538  grow = grow / xj
539  80 CONTINUE
540  END IF
541  90 CONTINUE
542  END IF
543 *
544  IF( ( grow*tscal ).GT.smlnum ) THEN
545 *
546 * Use the Level 2 BLAS solve if the reciprocal of the bound on
547 * elements of X is not too small.
548 *
549  CALL ztpsv( uplo, trans, diag, n, ap, x, 1 )
550  ELSE
551 *
552 * Use a Level 1 BLAS solve, scaling intermediate results.
553 *
554  IF( xmax.GT.bignum*half ) THEN
555 *
556 * Scale X so that its components are less than or equal to
557 * BIGNUM in absolute value.
558 *
559  scale = ( bignum*half ) / xmax
560  CALL zdscal( n, scale, x, 1 )
561  xmax = bignum
562  ELSE
563  xmax = xmax*two
564  END IF
565 *
566  IF( notran ) THEN
567 *
568 * Solve A * x = b
569 *
570  ip = jfirst*( jfirst+1 ) / 2
571  DO 120 j = jfirst, jlast, jinc
572 *
573 * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
574 *
575  xj = cabs1( x( j ) )
576  IF( nounit ) THEN
577  tjjs = ap( ip )*tscal
578  ELSE
579  tjjs = tscal
580  IF( tscal.EQ.one )
581  $ GO TO 110
582  END IF
583  tjj = cabs1( tjjs )
584  IF( tjj.GT.smlnum ) THEN
585 *
586 * abs(A(j,j)) > SMLNUM:
587 *
588  IF( tjj.LT.one ) THEN
589  IF( xj.GT.tjj*bignum ) THEN
590 *
591 * Scale x by 1/b(j).
592 *
593  rec = one / xj
594  CALL zdscal( n, rec, x, 1 )
595  scale = scale*rec
596  xmax = xmax*rec
597  END IF
598  END IF
599  x( j ) = zladiv( x( j ), tjjs )
600  xj = cabs1( x( j ) )
601  ELSE IF( tjj.GT.zero ) THEN
602 *
603 * 0 < abs(A(j,j)) <= SMLNUM:
604 *
605  IF( xj.GT.tjj*bignum ) THEN
606 *
607 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
608 * to avoid overflow when dividing by A(j,j).
609 *
610  rec = ( tjj*bignum ) / xj
611  IF( cnorm( j ).GT.one ) THEN
612 *
613 * Scale by 1/CNORM(j) to avoid overflow when
614 * multiplying x(j) times column j.
615 *
616  rec = rec / cnorm( j )
617  END IF
618  CALL zdscal( n, rec, x, 1 )
619  scale = scale*rec
620  xmax = xmax*rec
621  END IF
622  x( j ) = zladiv( x( j ), tjjs )
623  xj = cabs1( x( j ) )
624  ELSE
625 *
626 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
627 * scale = 0, and compute a solution to A*x = 0.
628 *
629  DO 100 i = 1, n
630  x( i ) = zero
631  100 CONTINUE
632  x( j ) = one
633  xj = one
634  scale = zero
635  xmax = zero
636  END IF
637  110 CONTINUE
638 *
639 * Scale x if necessary to avoid overflow when adding a
640 * multiple of column j of A.
641 *
642  IF( xj.GT.one ) THEN
643  rec = one / xj
644  IF( cnorm( j ).GT.( bignum-xmax )*rec ) THEN
645 *
646 * Scale x by 1/(2*abs(x(j))).
647 *
648  rec = rec*half
649  CALL zdscal( n, rec, x, 1 )
650  scale = scale*rec
651  END IF
652  ELSE IF( xj*cnorm( j ).GT.( bignum-xmax ) ) THEN
653 *
654 * Scale x by 1/2.
655 *
656  CALL zdscal( n, half, x, 1 )
657  scale = scale*half
658  END IF
659 *
660  IF( upper ) THEN
661  IF( j.GT.1 ) THEN
662 *
663 * Compute the update
664 * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
665 *
666  CALL zaxpy( j-1, -x( j )*tscal, ap( ip-j+1 ), 1, x,
667  $ 1 )
668  i = izamax( j-1, x, 1 )
669  xmax = cabs1( x( i ) )
670  END IF
671  ip = ip - j
672  ELSE
673  IF( j.LT.n ) THEN
674 *
675 * Compute the update
676 * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
677 *
678  CALL zaxpy( n-j, -x( j )*tscal, ap( ip+1 ), 1,
679  $ x( j+1 ), 1 )
680  i = j + izamax( n-j, x( j+1 ), 1 )
681  xmax = cabs1( x( i ) )
682  END IF
683  ip = ip + n - j + 1
684  END IF
685  120 CONTINUE
686 *
687  ELSE IF( lsame( trans, 'T' ) ) THEN
688 *
689 * Solve A**T * x = b
690 *
691  ip = jfirst*( jfirst+1 ) / 2
692  jlen = 1
693  DO 170 j = jfirst, jlast, jinc
694 *
695 * Compute x(j) = b(j) - sum A(k,j)*x(k).
696 * k<>j
697 *
698  xj = cabs1( x( j ) )
699  uscal = tscal
700  rec = one / max( xmax, one )
701  IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
702 *
703 * If x(j) could overflow, scale x by 1/(2*XMAX).
704 *
705  rec = rec*half
706  IF( nounit ) THEN
707  tjjs = ap( ip )*tscal
708  ELSE
709  tjjs = tscal
710  END IF
711  tjj = cabs1( tjjs )
712  IF( tjj.GT.one ) THEN
713 *
714 * Divide by A(j,j) when scaling x if A(j,j) > 1.
715 *
716  rec = min( one, rec*tjj )
717  uscal = zladiv( uscal, tjjs )
718  END IF
719  IF( rec.LT.one ) THEN
720  CALL zdscal( n, rec, x, 1 )
721  scale = scale*rec
722  xmax = xmax*rec
723  END IF
724  END IF
725 *
726  csumj = zero
727  IF( uscal.EQ.dcmplx( one ) ) THEN
728 *
729 * If the scaling needed for A in the dot product is 1,
730 * call ZDOTU to perform the dot product.
731 *
732  IF( upper ) THEN
733  csumj = zdotu( j-1, ap( ip-j+1 ), 1, x, 1 )
734  ELSE IF( j.LT.n ) THEN
735  csumj = zdotu( n-j, ap( ip+1 ), 1, x( j+1 ), 1 )
736  END IF
737  ELSE
738 *
739 * Otherwise, use in-line code for the dot product.
740 *
741  IF( upper ) THEN
742  DO 130 i = 1, j - 1
743  csumj = csumj + ( ap( ip-j+i )*uscal )*x( i )
744  130 CONTINUE
745  ELSE IF( j.LT.n ) THEN
746  DO 140 i = 1, n - j
747  csumj = csumj + ( ap( ip+i )*uscal )*x( j+i )
748  140 CONTINUE
749  END IF
750  END IF
751 *
752  IF( uscal.EQ.dcmplx( tscal ) ) THEN
753 *
754 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
755 * was not used to scale the dotproduct.
756 *
757  x( j ) = x( j ) - csumj
758  xj = cabs1( x( j ) )
759  IF( nounit ) THEN
760 *
761 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
762 *
763  tjjs = ap( ip )*tscal
764  ELSE
765  tjjs = tscal
766  IF( tscal.EQ.one )
767  $ GO TO 160
768  END IF
769  tjj = cabs1( tjjs )
770  IF( tjj.GT.smlnum ) THEN
771 *
772 * abs(A(j,j)) > SMLNUM:
773 *
774  IF( tjj.LT.one ) THEN
775  IF( xj.GT.tjj*bignum ) THEN
776 *
777 * Scale X by 1/abs(x(j)).
778 *
779  rec = one / xj
780  CALL zdscal( n, rec, x, 1 )
781  scale = scale*rec
782  xmax = xmax*rec
783  END IF
784  END IF
785  x( j ) = zladiv( x( j ), tjjs )
786  ELSE IF( tjj.GT.zero ) THEN
787 *
788 * 0 < abs(A(j,j)) <= SMLNUM:
789 *
790  IF( xj.GT.tjj*bignum ) THEN
791 *
792 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
793 *
794  rec = ( tjj*bignum ) / xj
795  CALL zdscal( n, rec, x, 1 )
796  scale = scale*rec
797  xmax = xmax*rec
798  END IF
799  x( j ) = zladiv( x( j ), tjjs )
800  ELSE
801 *
802 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
803 * scale = 0 and compute a solution to A**T *x = 0.
804 *
805  DO 150 i = 1, n
806  x( i ) = zero
807  150 CONTINUE
808  x( j ) = one
809  scale = zero
810  xmax = zero
811  END IF
812  160 CONTINUE
813  ELSE
814 *
815 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
816 * product has already been divided by 1/A(j,j).
817 *
818  x( j ) = zladiv( x( j ), tjjs ) - csumj
819  END IF
820  xmax = max( xmax, cabs1( x( j ) ) )
821  jlen = jlen + 1
822  ip = ip + jinc*jlen
823  170 CONTINUE
824 *
825  ELSE
826 *
827 * Solve A**H * x = b
828 *
829  ip = jfirst*( jfirst+1 ) / 2
830  jlen = 1
831  DO 220 j = jfirst, jlast, jinc
832 *
833 * Compute x(j) = b(j) - sum A(k,j)*x(k).
834 * k<>j
835 *
836  xj = cabs1( x( j ) )
837  uscal = tscal
838  rec = one / max( xmax, one )
839  IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
840 *
841 * If x(j) could overflow, scale x by 1/(2*XMAX).
842 *
843  rec = rec*half
844  IF( nounit ) THEN
845  tjjs = dconjg( ap( ip ) )*tscal
846  ELSE
847  tjjs = tscal
848  END IF
849  tjj = cabs1( tjjs )
850  IF( tjj.GT.one ) THEN
851 *
852 * Divide by A(j,j) when scaling x if A(j,j) > 1.
853 *
854  rec = min( one, rec*tjj )
855  uscal = zladiv( uscal, tjjs )
856  END IF
857  IF( rec.LT.one ) THEN
858  CALL zdscal( n, rec, x, 1 )
859  scale = scale*rec
860  xmax = xmax*rec
861  END IF
862  END IF
863 *
864  csumj = zero
865  IF( uscal.EQ.dcmplx( one ) ) THEN
866 *
867 * If the scaling needed for A in the dot product is 1,
868 * call ZDOTC to perform the dot product.
869 *
870  IF( upper ) THEN
871  csumj = zdotc( j-1, ap( ip-j+1 ), 1, x, 1 )
872  ELSE IF( j.LT.n ) THEN
873  csumj = zdotc( n-j, ap( ip+1 ), 1, x( j+1 ), 1 )
874  END IF
875  ELSE
876 *
877 * Otherwise, use in-line code for the dot product.
878 *
879  IF( upper ) THEN
880  DO 180 i = 1, j - 1
881  csumj = csumj + ( dconjg( ap( ip-j+i ) )*uscal )
882  $ *x( i )
883  180 CONTINUE
884  ELSE IF( j.LT.n ) THEN
885  DO 190 i = 1, n - j
886  csumj = csumj + ( dconjg( ap( ip+i ) )*uscal )*
887  $ x( j+i )
888  190 CONTINUE
889  END IF
890  END IF
891 *
892  IF( uscal.EQ.dcmplx( tscal ) ) THEN
893 *
894 * Compute x(j) := ( x(j) - CSUMJ ) / A(j,j) if 1/A(j,j)
895 * was not used to scale the dotproduct.
896 *
897  x( j ) = x( j ) - csumj
898  xj = cabs1( x( j ) )
899  IF( nounit ) THEN
900 *
901 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
902 *
903  tjjs = dconjg( ap( ip ) )*tscal
904  ELSE
905  tjjs = tscal
906  IF( tscal.EQ.one )
907  $ GO TO 210
908  END IF
909  tjj = cabs1( tjjs )
910  IF( tjj.GT.smlnum ) THEN
911 *
912 * abs(A(j,j)) > SMLNUM:
913 *
914  IF( tjj.LT.one ) THEN
915  IF( xj.GT.tjj*bignum ) THEN
916 *
917 * Scale X by 1/abs(x(j)).
918 *
919  rec = one / xj
920  CALL zdscal( n, rec, x, 1 )
921  scale = scale*rec
922  xmax = xmax*rec
923  END IF
924  END IF
925  x( j ) = zladiv( x( j ), tjjs )
926  ELSE IF( tjj.GT.zero ) THEN
927 *
928 * 0 < abs(A(j,j)) <= SMLNUM:
929 *
930  IF( xj.GT.tjj*bignum ) THEN
931 *
932 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
933 *
934  rec = ( tjj*bignum ) / xj
935  CALL zdscal( n, rec, x, 1 )
936  scale = scale*rec
937  xmax = xmax*rec
938  END IF
939  x( j ) = zladiv( x( j ), tjjs )
940  ELSE
941 *
942 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
943 * scale = 0 and compute a solution to A**H *x = 0.
944 *
945  DO 200 i = 1, n
946  x( i ) = zero
947  200 CONTINUE
948  x( j ) = one
949  scale = zero
950  xmax = zero
951  END IF
952  210 CONTINUE
953  ELSE
954 *
955 * Compute x(j) := x(j) / A(j,j) - CSUMJ if the dot
956 * product has already been divided by 1/A(j,j).
957 *
958  x( j ) = zladiv( x( j ), tjjs ) - csumj
959  END IF
960  xmax = max( xmax, cabs1( x( j ) ) )
961  jlen = jlen + 1
962  ip = ip + jinc*jlen
963  220 CONTINUE
964  END IF
965  scale = scale / tscal
966  END IF
967 *
968 * Scale the column norms by 1/TSCAL for return.
969 *
970  IF( tscal.NE.one ) THEN
971  CALL dscal( n, one / tscal, cnorm, 1 )
972  END IF
973 *
974  RETURN
975 *
976 * End of ZLATPS
977 *
978  END
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:233
ztpsv
subroutine ztpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
ZTPSV
Definition: ztpsv.f:146
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
dscal
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81
dlabad
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:76
zdscal
subroutine zdscal(N, DA, ZX, INCX)
ZDSCAL
Definition: zdscal.f:80
zaxpy
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:90