LAPACK  3.7.1
LAPACK: Linear Algebra PACKage
dlatps.f
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1 *> \brief \b DLATPS 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 DLATPS + 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/dlatps.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatps.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLATPS( 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 AP( * ), CNORM( * ), X( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> DLATPS solves one of the triangular systems
40 *>
41 *> A *x = s*b or A**T*x = s*b
42 *>
43 *> with scaling to prevent overflow, where A is an upper or lower
44 *> triangular matrix stored in packed form. Here A**T denotes the
45 *> transpose of A, x and b are n-element vectors, and s is a scaling
46 *> factor, usually less than or equal to 1, chosen so that the
47 *> components of x will be less than the overflow threshold. If the
48 *> unscaled problem will not cause overflow, the Level 2 BLAS routine
49 *> DTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
50 *> then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
51 *> \endverbatim
52 *
53 * Arguments:
54 * ==========
55 *
56 *> \param[in] UPLO
57 *> \verbatim
58 *> UPLO is CHARACTER*1
59 *> Specifies whether the matrix A is upper or lower triangular.
60 *> = 'U': Upper triangular
61 *> = 'L': Lower triangular
62 *> \endverbatim
63 *>
64 *> \param[in] TRANS
65 *> \verbatim
66 *> TRANS is CHARACTER*1
67 *> Specifies the operation applied to A.
68 *> = 'N': Solve A * x = s*b (No transpose)
69 *> = 'T': Solve A**T* x = s*b (Transpose)
70 *> = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose)
71 *> \endverbatim
72 *>
73 *> \param[in] DIAG
74 *> \verbatim
75 *> DIAG is CHARACTER*1
76 *> Specifies whether or not the matrix A is unit triangular.
77 *> = 'N': Non-unit triangular
78 *> = 'U': Unit triangular
79 *> \endverbatim
80 *>
81 *> \param[in] NORMIN
82 *> \verbatim
83 *> NORMIN is CHARACTER*1
84 *> Specifies whether CNORM has been set or not.
85 *> = 'Y': CNORM contains the column norms on entry
86 *> = 'N': CNORM is not set on entry. On exit, the norms will
87 *> be computed and stored in CNORM.
88 *> \endverbatim
89 *>
90 *> \param[in] N
91 *> \verbatim
92 *> N is INTEGER
93 *> The order of the matrix A. N >= 0.
94 *> \endverbatim
95 *>
96 *> \param[in] AP
97 *> \verbatim
98 *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
99 *> The upper or lower triangular matrix A, packed columnwise in
100 *> a linear array. The j-th column of A is stored in the array
101 *> AP as follows:
102 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
103 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
104 *> \endverbatim
105 *>
106 *> \param[in,out] X
107 *> \verbatim
108 *> X is DOUBLE PRECISION array, dimension (N)
109 *> On entry, the right hand side b of the triangular system.
110 *> On exit, X is overwritten by the solution vector x.
111 *> \endverbatim
112 *>
113 *> \param[out] SCALE
114 *> \verbatim
115 *> SCALE is DOUBLE PRECISION
116 *> The scaling factor s for the triangular system
117 *> A * x = s*b or A**T* x = s*b.
118 *> If SCALE = 0, the matrix A is singular or badly scaled, and
119 *> the vector x is an exact or approximate solution to A*x = 0.
120 *> \endverbatim
121 *>
122 *> \param[in,out] CNORM
123 *> \verbatim
124 *> CNORM is DOUBLE PRECISION array, dimension (N)
125 *>
126 *> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
127 *> contains the norm of the off-diagonal part of the j-th column
128 *> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
129 *> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
130 *> must be greater than or equal to the 1-norm.
131 *>
132 *> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
133 *> returns the 1-norm of the offdiagonal part of the j-th column
134 *> of A.
135 *> \endverbatim
136 *>
137 *> \param[out] INFO
138 *> \verbatim
139 *> INFO is INTEGER
140 *> = 0: successful exit
141 *> < 0: if INFO = -k, the k-th argument had an illegal value
142 *> \endverbatim
143 *
144 * Authors:
145 * ========
146 *
147 *> \author Univ. of Tennessee
148 *> \author Univ. of California Berkeley
149 *> \author Univ. of Colorado Denver
150 *> \author NAG Ltd.
151 *
152 *> \date December 2016
153 *
154 *> \ingroup doubleOTHERauxiliary
155 *
156 *> \par Further Details:
157 * =====================
158 *>
159 *> \verbatim
160 *>
161 *> A rough bound on x is computed; if that is less than overflow, DTPSV
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 DTPSV 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. The basic
205 *> algorithm for A upper triangular is
206 *>
207 *> for j = 1, ..., n
208 *> x(j) := ( b(j) - A[1:j-1,j]**T * 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]**T * 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 DTPSV if 1/M(n) and 1/G(n) are both greater
225 *> than max(underflow, 1/overflow).
226 *> \endverbatim
227 *>
228 * =====================================================================
229  SUBROUTINE dlatps( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE,
230  $ CNORM, INFO )
231 *
232 * -- LAPACK auxiliary routine (version 3.7.0) --
233 * -- LAPACK is a software package provided by Univ. of Tennessee, --
234 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
235 * December 2016
236 *
237 * .. Scalar Arguments ..
238  CHARACTER DIAG, NORMIN, TRANS, UPLO
239  INTEGER INFO, N
240  DOUBLE PRECISION SCALE
241 * ..
242 * .. Array Arguments ..
243  DOUBLE PRECISION AP( * ), CNORM( * ), X( * )
244 * ..
245 *
246 * =====================================================================
247 *
248 * .. Parameters ..
249  DOUBLE PRECISION ZERO, HALF, ONE
250  parameter( zero = 0.0d+0, half = 0.5d+0, one = 1.0d+0 )
251 * ..
252 * .. Local Scalars ..
253  LOGICAL NOTRAN, NOUNIT, UPPER
254  INTEGER I, IMAX, IP, J, JFIRST, JINC, JLAST, JLEN
255  DOUBLE PRECISION BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
256  $ tmax, tscal, uscal, xbnd, xj, xmax
257 * ..
258 * .. External Functions ..
259  LOGICAL LSAME
260  INTEGER IDAMAX
261  DOUBLE PRECISION DASUM, DDOT, DLAMCH
262  EXTERNAL lsame, idamax, dasum, ddot, dlamch
263 * ..
264 * .. External Subroutines ..
265  EXTERNAL daxpy, dscal, dtpsv, xerbla
266 * ..
267 * .. Intrinsic Functions ..
268  INTRINSIC abs, max, min
269 * ..
270 * .. Executable Statements ..
271 *
272  info = 0
273  upper = lsame( uplo, 'U' )
274  notran = lsame( trans, 'N' )
275  nounit = lsame( diag, 'N' )
276 *
277 * Test the input parameters.
278 *
279  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
280  info = -1
281  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
282  $ lsame( trans, 'C' ) ) THEN
283  info = -2
284  ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
285  info = -3
286  ELSE IF( .NOT.lsame( normin, 'Y' ) .AND. .NOT.
287  $ lsame( normin, 'N' ) ) THEN
288  info = -4
289  ELSE IF( n.LT.0 ) THEN
290  info = -5
291  END IF
292  IF( info.NE.0 ) THEN
293  CALL xerbla( 'DLATPS', -info )
294  RETURN
295  END IF
296 *
297 * Quick return if possible
298 *
299  IF( n.EQ.0 )
300  $ RETURN
301 *
302 * Determine machine dependent parameters to control overflow.
303 *
304  smlnum = dlamch( 'Safe minimum' ) / dlamch( 'Precision' )
305  bignum = one / smlnum
306  scale = one
307 *
308  IF( lsame( normin, 'N' ) ) THEN
309 *
310 * Compute the 1-norm of each column, not including the diagonal.
311 *
312  IF( upper ) THEN
313 *
314 * A is upper triangular.
315 *
316  ip = 1
317  DO 10 j = 1, n
318  cnorm( j ) = dasum( j-1, ap( ip ), 1 )
319  ip = ip + j
320  10 CONTINUE
321  ELSE
322 *
323 * A is lower triangular.
324 *
325  ip = 1
326  DO 20 j = 1, n - 1
327  cnorm( j ) = dasum( n-j, ap( ip+1 ), 1 )
328  ip = ip + n - j + 1
329  20 CONTINUE
330  cnorm( n ) = zero
331  END IF
332  END IF
333 *
334 * Scale the column norms by TSCAL if the maximum element in CNORM is
335 * greater than BIGNUM.
336 *
337  imax = idamax( n, cnorm, 1 )
338  tmax = cnorm( imax )
339  IF( tmax.LE.bignum ) THEN
340  tscal = one
341  ELSE
342  tscal = one / ( smlnum*tmax )
343  CALL dscal( n, tscal, cnorm, 1 )
344  END IF
345 *
346 * Compute a bound on the computed solution vector to see if the
347 * Level 2 BLAS routine DTPSV can be used.
348 *
349  j = idamax( n, x, 1 )
350  xmax = abs( x( j ) )
351  xbnd = xmax
352  IF( notran ) THEN
353 *
354 * Compute the growth in A * x = b.
355 *
356  IF( upper ) THEN
357  jfirst = n
358  jlast = 1
359  jinc = -1
360  ELSE
361  jfirst = 1
362  jlast = n
363  jinc = 1
364  END IF
365 *
366  IF( tscal.NE.one ) THEN
367  grow = zero
368  GO TO 50
369  END IF
370 *
371  IF( nounit ) THEN
372 *
373 * A is non-unit triangular.
374 *
375 * Compute GROW = 1/G(j) and XBND = 1/M(j).
376 * Initially, G(0) = max{x(i), i=1,...,n}.
377 *
378  grow = one / max( xbnd, smlnum )
379  xbnd = grow
380  ip = jfirst*( jfirst+1 ) / 2
381  jlen = n
382  DO 30 j = jfirst, jlast, jinc
383 *
384 * Exit the loop if the growth factor is too small.
385 *
386  IF( grow.LE.smlnum )
387  $ GO TO 50
388 *
389 * M(j) = G(j-1) / abs(A(j,j))
390 *
391  tjj = abs( ap( ip ) )
392  xbnd = min( xbnd, min( one, tjj )*grow )
393  IF( tjj+cnorm( j ).GE.smlnum ) THEN
394 *
395 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
396 *
397  grow = grow*( tjj / ( tjj+cnorm( j ) ) )
398  ELSE
399 *
400 * G(j) could overflow, set GROW to 0.
401 *
402  grow = zero
403  END IF
404  ip = ip + jinc*jlen
405  jlen = jlen - 1
406  30 CONTINUE
407  grow = xbnd
408  ELSE
409 *
410 * A is unit triangular.
411 *
412 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
413 *
414  grow = min( one, one / max( xbnd, smlnum ) )
415  DO 40 j = jfirst, jlast, jinc
416 *
417 * Exit the loop if the growth factor is too small.
418 *
419  IF( grow.LE.smlnum )
420  $ GO TO 50
421 *
422 * G(j) = G(j-1)*( 1 + CNORM(j) )
423 *
424  grow = grow*( one / ( one+cnorm( j ) ) )
425  40 CONTINUE
426  END IF
427  50 CONTINUE
428 *
429  ELSE
430 *
431 * Compute the growth in A**T * x = b.
432 *
433  IF( upper ) THEN
434  jfirst = 1
435  jlast = n
436  jinc = 1
437  ELSE
438  jfirst = n
439  jlast = 1
440  jinc = -1
441  END IF
442 *
443  IF( tscal.NE.one ) THEN
444  grow = zero
445  GO TO 80
446  END IF
447 *
448  IF( nounit ) THEN
449 *
450 * A is non-unit triangular.
451 *
452 * Compute GROW = 1/G(j) and XBND = 1/M(j).
453 * Initially, M(0) = max{x(i), i=1,...,n}.
454 *
455  grow = one / max( xbnd, smlnum )
456  xbnd = grow
457  ip = jfirst*( jfirst+1 ) / 2
458  jlen = 1
459  DO 60 j = jfirst, jlast, jinc
460 *
461 * Exit the loop if the growth factor is too small.
462 *
463  IF( grow.LE.smlnum )
464  $ GO TO 80
465 *
466 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
467 *
468  xj = one + cnorm( j )
469  grow = min( grow, xbnd / xj )
470 *
471 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
472 *
473  tjj = abs( ap( ip ) )
474  IF( xj.GT.tjj )
475  $ xbnd = xbnd*( tjj / xj )
476  jlen = jlen + 1
477  ip = ip + jinc*jlen
478  60 CONTINUE
479  grow = min( grow, xbnd )
480  ELSE
481 *
482 * A is unit triangular.
483 *
484 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
485 *
486  grow = min( one, one / max( xbnd, smlnum ) )
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 80
493 *
494 * G(j) = ( 1 + CNORM(j) )*G(j-1)
495 *
496  xj = one + cnorm( j )
497  grow = grow / xj
498  70 CONTINUE
499  END IF
500  80 CONTINUE
501  END IF
502 *
503  IF( ( grow*tscal ).GT.smlnum ) THEN
504 *
505 * Use the Level 2 BLAS solve if the reciprocal of the bound on
506 * elements of X is not too small.
507 *
508  CALL dtpsv( uplo, trans, diag, n, ap, x, 1 )
509  ELSE
510 *
511 * Use a Level 1 BLAS solve, scaling intermediate results.
512 *
513  IF( xmax.GT.bignum ) THEN
514 *
515 * Scale X so that its components are less than or equal to
516 * BIGNUM in absolute value.
517 *
518  scale = bignum / xmax
519  CALL dscal( n, scale, x, 1 )
520  xmax = bignum
521  END IF
522 *
523  IF( notran ) THEN
524 *
525 * Solve A * x = b
526 *
527  ip = jfirst*( jfirst+1 ) / 2
528  DO 110 j = jfirst, jlast, jinc
529 *
530 * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
531 *
532  xj = abs( x( j ) )
533  IF( nounit ) THEN
534  tjjs = ap( ip )*tscal
535  ELSE
536  tjjs = tscal
537  IF( tscal.EQ.one )
538  $ GO TO 100
539  END IF
540  tjj = abs( tjjs )
541  IF( tjj.GT.smlnum ) THEN
542 *
543 * abs(A(j,j)) > SMLNUM:
544 *
545  IF( tjj.LT.one ) THEN
546  IF( xj.GT.tjj*bignum ) THEN
547 *
548 * Scale x by 1/b(j).
549 *
550  rec = one / xj
551  CALL dscal( n, rec, x, 1 )
552  scale = scale*rec
553  xmax = xmax*rec
554  END IF
555  END IF
556  x( j ) = x( j ) / tjjs
557  xj = abs( x( j ) )
558  ELSE IF( tjj.GT.zero ) THEN
559 *
560 * 0 < abs(A(j,j)) <= SMLNUM:
561 *
562  IF( xj.GT.tjj*bignum ) THEN
563 *
564 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
565 * to avoid overflow when dividing by A(j,j).
566 *
567  rec = ( tjj*bignum ) / xj
568  IF( cnorm( j ).GT.one ) THEN
569 *
570 * Scale by 1/CNORM(j) to avoid overflow when
571 * multiplying x(j) times column j.
572 *
573  rec = rec / cnorm( j )
574  END IF
575  CALL dscal( n, rec, x, 1 )
576  scale = scale*rec
577  xmax = xmax*rec
578  END IF
579  x( j ) = x( j ) / tjjs
580  xj = abs( x( j ) )
581  ELSE
582 *
583 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
584 * scale = 0, and compute a solution to A*x = 0.
585 *
586  DO 90 i = 1, n
587  x( i ) = zero
588  90 CONTINUE
589  x( j ) = one
590  xj = one
591  scale = zero
592  xmax = zero
593  END IF
594  100 CONTINUE
595 *
596 * Scale x if necessary to avoid overflow when adding a
597 * multiple of column j of A.
598 *
599  IF( xj.GT.one ) THEN
600  rec = one / xj
601  IF( cnorm( j ).GT.( bignum-xmax )*rec ) THEN
602 *
603 * Scale x by 1/(2*abs(x(j))).
604 *
605  rec = rec*half
606  CALL dscal( n, rec, x, 1 )
607  scale = scale*rec
608  END IF
609  ELSE IF( xj*cnorm( j ).GT.( bignum-xmax ) ) THEN
610 *
611 * Scale x by 1/2.
612 *
613  CALL dscal( n, half, x, 1 )
614  scale = scale*half
615  END IF
616 *
617  IF( upper ) THEN
618  IF( j.GT.1 ) THEN
619 *
620 * Compute the update
621 * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
622 *
623  CALL daxpy( j-1, -x( j )*tscal, ap( ip-j+1 ), 1, x,
624  $ 1 )
625  i = idamax( j-1, x, 1 )
626  xmax = abs( x( i ) )
627  END IF
628  ip = ip - j
629  ELSE
630  IF( j.LT.n ) THEN
631 *
632 * Compute the update
633 * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
634 *
635  CALL daxpy( n-j, -x( j )*tscal, ap( ip+1 ), 1,
636  $ x( j+1 ), 1 )
637  i = j + idamax( n-j, x( j+1 ), 1 )
638  xmax = abs( x( i ) )
639  END IF
640  ip = ip + n - j + 1
641  END IF
642  110 CONTINUE
643 *
644  ELSE
645 *
646 * Solve A**T * x = b
647 *
648  ip = jfirst*( jfirst+1 ) / 2
649  jlen = 1
650  DO 160 j = jfirst, jlast, jinc
651 *
652 * Compute x(j) = b(j) - sum A(k,j)*x(k).
653 * k<>j
654 *
655  xj = abs( x( j ) )
656  uscal = tscal
657  rec = one / max( xmax, one )
658  IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
659 *
660 * If x(j) could overflow, scale x by 1/(2*XMAX).
661 *
662  rec = rec*half
663  IF( nounit ) THEN
664  tjjs = ap( ip )*tscal
665  ELSE
666  tjjs = tscal
667  END IF
668  tjj = abs( tjjs )
669  IF( tjj.GT.one ) THEN
670 *
671 * Divide by A(j,j) when scaling x if A(j,j) > 1.
672 *
673  rec = min( one, rec*tjj )
674  uscal = uscal / tjjs
675  END IF
676  IF( rec.LT.one ) THEN
677  CALL dscal( n, rec, x, 1 )
678  scale = scale*rec
679  xmax = xmax*rec
680  END IF
681  END IF
682 *
683  sumj = zero
684  IF( uscal.EQ.one ) THEN
685 *
686 * If the scaling needed for A in the dot product is 1,
687 * call DDOT to perform the dot product.
688 *
689  IF( upper ) THEN
690  sumj = ddot( j-1, ap( ip-j+1 ), 1, x, 1 )
691  ELSE IF( j.LT.n ) THEN
692  sumj = ddot( n-j, ap( ip+1 ), 1, x( j+1 ), 1 )
693  END IF
694  ELSE
695 *
696 * Otherwise, use in-line code for the dot product.
697 *
698  IF( upper ) THEN
699  DO 120 i = 1, j - 1
700  sumj = sumj + ( ap( ip-j+i )*uscal )*x( i )
701  120 CONTINUE
702  ELSE IF( j.LT.n ) THEN
703  DO 130 i = 1, n - j
704  sumj = sumj + ( ap( ip+i )*uscal )*x( j+i )
705  130 CONTINUE
706  END IF
707  END IF
708 *
709  IF( uscal.EQ.tscal ) THEN
710 *
711 * Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
712 * was not used to scale the dotproduct.
713 *
714  x( j ) = x( j ) - sumj
715  xj = abs( x( j ) )
716  IF( nounit ) THEN
717 *
718 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
719 *
720  tjjs = ap( ip )*tscal
721  ELSE
722  tjjs = tscal
723  IF( tscal.EQ.one )
724  $ GO TO 150
725  END IF
726  tjj = abs( tjjs )
727  IF( tjj.GT.smlnum ) THEN
728 *
729 * abs(A(j,j)) > SMLNUM:
730 *
731  IF( tjj.LT.one ) THEN
732  IF( xj.GT.tjj*bignum ) THEN
733 *
734 * Scale X by 1/abs(x(j)).
735 *
736  rec = one / xj
737  CALL dscal( n, rec, x, 1 )
738  scale = scale*rec
739  xmax = xmax*rec
740  END IF
741  END IF
742  x( j ) = x( j ) / tjjs
743  ELSE IF( tjj.GT.zero ) THEN
744 *
745 * 0 < abs(A(j,j)) <= SMLNUM:
746 *
747  IF( xj.GT.tjj*bignum ) THEN
748 *
749 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
750 *
751  rec = ( tjj*bignum ) / xj
752  CALL dscal( n, rec, x, 1 )
753  scale = scale*rec
754  xmax = xmax*rec
755  END IF
756  x( j ) = x( j ) / tjjs
757  ELSE
758 *
759 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
760 * scale = 0, and compute a solution to A**T*x = 0.
761 *
762  DO 140 i = 1, n
763  x( i ) = zero
764  140 CONTINUE
765  x( j ) = one
766  scale = zero
767  xmax = zero
768  END IF
769  150 CONTINUE
770  ELSE
771 *
772 * Compute x(j) := x(j) / A(j,j) - sumj if the dot
773 * product has already been divided by 1/A(j,j).
774 *
775  x( j ) = x( j ) / tjjs - sumj
776  END IF
777  xmax = max( xmax, abs( x( j ) ) )
778  jlen = jlen + 1
779  ip = ip + jinc*jlen
780  160 CONTINUE
781  END IF
782  scale = scale / tscal
783  END IF
784 *
785 * Scale the column norms by 1/TSCAL for return.
786 *
787  IF( tscal.NE.one ) THEN
788  CALL dscal( n, one / tscal, cnorm, 1 )
789  END IF
790 *
791  RETURN
792 *
793 * End of DLATPS
794 *
795  END
dlatps
subroutine dlatps(UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
DLATPS solves a triangular system of equations with the matrix held in packed storage.
Definition: dlatps.f:231
daxpy
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:91
dtpsv
subroutine dtpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPSV
Definition: dtpsv.f:146
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
dscal
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:81