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dlatrs.f
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1 *> \brief \b DLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download DLATRS + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrs.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrs.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrs.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
22 * CNORM, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER DIAG, NORMIN, TRANS, UPLO
26 * INTEGER INFO, LDA, N
27 * DOUBLE PRECISION SCALE
28 * ..
29 * .. Array Arguments ..
30 * DOUBLE PRECISION A( LDA, * ), CNORM( * ), X( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> DLATRS 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. Here A is an upper or lower
44 *> triangular matrix, A**T denotes the transpose of A, x and b are
45 *> n-element vectors, and s is a scaling factor, usually less than
46 *> or equal to 1, chosen so that the components of x will be less than
47 *> the overflow threshold. If the unscaled problem will not cause
48 *> overflow, the Level 2 BLAS routine DTRSV is called. If the matrix A
49 *> is singular (A(j,j) = 0 for some j), then s is set to 0 and a
50 *> 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] A
97 *> \verbatim
98 *> A is DOUBLE PRECISION array, dimension (LDA,N)
99 *> The triangular matrix A. If UPLO = 'U', the leading n by n
100 *> upper triangular part of the array A contains the upper
101 *> triangular matrix, and the strictly lower triangular part of
102 *> A is not referenced. If UPLO = 'L', the leading n by n lower
103 *> triangular part of the array A contains the lower triangular
104 *> matrix, and the strictly upper triangular part of A is not
105 *> referenced. If DIAG = 'U', the diagonal elements of A are
106 *> also not referenced and are assumed to be 1.
107 *> \endverbatim
108 *>
109 *> \param[in] LDA
110 *> \verbatim
111 *> LDA is INTEGER
112 *> The leading dimension of the array A. LDA >= max (1,N).
113 *> \endverbatim
114 *>
115 *> \param[in,out] X
116 *> \verbatim
117 *> X is DOUBLE PRECISION array, dimension (N)
118 *> On entry, the right hand side b of the triangular system.
119 *> On exit, X is overwritten by the solution vector x.
120 *> \endverbatim
121 *>
122 *> \param[out] SCALE
123 *> \verbatim
124 *> SCALE is DOUBLE PRECISION
125 *> The scaling factor s for the triangular system
126 *> A * x = s*b or A**T* x = s*b.
127 *> If SCALE = 0, the matrix A is singular or badly scaled, and
128 *> the vector x is an exact or approximate solution to A*x = 0.
129 *> \endverbatim
130 *>
131 *> \param[in,out] CNORM
132 *> \verbatim
133 *> CNORM is DOUBLE PRECISION array, dimension (N)
134 *>
135 *> If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
136 *> contains the norm of the off-diagonal part of the j-th column
137 *> of A. If TRANS = 'N', CNORM(j) must be greater than or equal
138 *> to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
139 *> must be greater than or equal to the 1-norm.
140 *>
141 *> If NORMIN = 'N', CNORM is an output argument and CNORM(j)
142 *> returns the 1-norm of the offdiagonal part of the j-th column
143 *> of A.
144 *> \endverbatim
145 *>
146 *> \param[out] INFO
147 *> \verbatim
148 *> INFO is INTEGER
149 *> = 0: successful exit
150 *> < 0: if INFO = -k, the k-th argument had an illegal value
151 *> \endverbatim
152 *
153 * Authors:
154 * ========
155 *
156 *> \author Univ. of Tennessee
157 *> \author Univ. of California Berkeley
158 *> \author Univ. of Colorado Denver
159 *> \author NAG Ltd.
160 *
161 *> \date September 2012
162 *
163 *> \ingroup doubleOTHERauxiliary
164 *
165 *> \par Further Details:
166 * =====================
167 *>
168 *> \verbatim
169 *>
170 *> A rough bound on x is computed; if that is less than overflow, DTRSV
171 *> is called, otherwise, specific code is used which checks for possible
172 *> overflow or divide-by-zero at every operation.
173 *>
174 *> A columnwise scheme is used for solving A*x = b. The basic algorithm
175 *> if A is lower triangular is
176 *>
177 *> x[1:n] := b[1:n]
178 *> for j = 1, ..., n
179 *> x(j) := x(j) / A(j,j)
180 *> x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
181 *> end
182 *>
183 *> Define bounds on the components of x after j iterations of the loop:
184 *> M(j) = bound on x[1:j]
185 *> G(j) = bound on x[j+1:n]
186 *> Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
187 *>
188 *> Then for iteration j+1 we have
189 *> M(j+1) <= G(j) / | A(j+1,j+1) |
190 *> G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
191 *> <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
192 *>
193 *> where CNORM(j+1) is greater than or equal to the infinity-norm of
194 *> column j+1 of A, not counting the diagonal. Hence
195 *>
196 *> G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
197 *> 1<=i<=j
198 *> and
199 *>
200 *> |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
201 *> 1<=i< j
202 *>
203 *> Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
204 *> reciprocal of the largest M(j), j=1,..,n, is larger than
205 *> max(underflow, 1/overflow).
206 *>
207 *> The bound on x(j) is also used to determine when a step in the
208 *> columnwise method can be performed without fear of overflow. If
209 *> the computed bound is greater than a large constant, x is scaled to
210 *> prevent overflow, but if the bound overflows, x is set to 0, x(j) to
211 *> 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
212 *>
213 *> Similarly, a row-wise scheme is used to solve A**T*x = b. The basic
214 *> algorithm for A upper triangular is
215 *>
216 *> for j = 1, ..., n
217 *> x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
218 *> end
219 *>
220 *> We simultaneously compute two bounds
221 *> G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j
222 *> M(j) = bound on x(i), 1<=i<=j
223 *>
224 *> The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
225 *> add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
226 *> Then the bound on x(j) is
227 *>
228 *> M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
229 *>
230 *> <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
231 *> 1<=i<=j
232 *>
233 *> and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
234 *> than max(underflow, 1/overflow).
235 *> \endverbatim
236 *>
237 * =====================================================================
238  SUBROUTINE dlatrs( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
239  $ cnorm, info )
240 *
241 * -- LAPACK auxiliary routine (version 3.4.2) --
242 * -- LAPACK is a software package provided by Univ. of Tennessee, --
243 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
244 * September 2012
245 *
246 * .. Scalar Arguments ..
247  CHARACTER diag, normin, trans, uplo
248  INTEGER info, lda, n
249  DOUBLE PRECISION scale
250 * ..
251 * .. Array Arguments ..
252  DOUBLE PRECISION a( lda, * ), cnorm( * ), x( * )
253 * ..
254 *
255 * =====================================================================
256 *
257 * .. Parameters ..
258  DOUBLE PRECISION zero, half, one
259  parameter( zero = 0.0d+0, half = 0.5d+0, one = 1.0d+0 )
260 * ..
261 * .. Local Scalars ..
262  LOGICAL notran, nounit, upper
263  INTEGER i, imax, j, jfirst, jinc, jlast
264  DOUBLE PRECISION bignum, grow, rec, smlnum, sumj, tjj, tjjs,
265  $ tmax, tscal, uscal, xbnd, xj, xmax
266 * ..
267 * .. External Functions ..
268  LOGICAL lsame
269  INTEGER idamax
270  DOUBLE PRECISION dasum, ddot, dlamch
271  EXTERNAL lsame, idamax, dasum, ddot, dlamch
272 * ..
273 * .. External Subroutines ..
274  EXTERNAL daxpy, dscal, dtrsv, xerbla
275 * ..
276 * .. Intrinsic Functions ..
277  INTRINSIC abs, max, min
278 * ..
279 * .. Executable Statements ..
280 *
281  info = 0
282  upper = lsame( uplo, 'U' )
283  notran = lsame( trans, 'N' )
284  nounit = lsame( diag, 'N' )
285 *
286 * Test the input parameters.
287 *
288  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
289  info = -1
290  ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
291  $ lsame( trans, 'C' ) ) THEN
292  info = -2
293  ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
294  info = -3
295  ELSE IF( .NOT.lsame( normin, 'Y' ) .AND. .NOT.
296  $ lsame( normin, 'N' ) ) THEN
297  info = -4
298  ELSE IF( n.LT.0 ) THEN
299  info = -5
300  ELSE IF( lda.LT.max( 1, n ) ) THEN
301  info = -7
302  END IF
303  IF( info.NE.0 ) THEN
304  CALL xerbla( 'DLATRS', -info )
305  return
306  END IF
307 *
308 * Quick return if possible
309 *
310  IF( n.EQ.0 )
311  $ return
312 *
313 * Determine machine dependent parameters to control overflow.
314 *
315  smlnum = dlamch( 'Safe minimum' ) / dlamch( 'Precision' )
316  bignum = one / smlnum
317  scale = one
318 *
319  IF( lsame( normin, 'N' ) ) THEN
320 *
321 * Compute the 1-norm of each column, not including the diagonal.
322 *
323  IF( upper ) THEN
324 *
325 * A is upper triangular.
326 *
327  DO 10 j = 1, n
328  cnorm( j ) = dasum( j-1, a( 1, j ), 1 )
329  10 continue
330  ELSE
331 *
332 * A is lower triangular.
333 *
334  DO 20 j = 1, n - 1
335  cnorm( j ) = dasum( n-j, a( j+1, j ), 1 )
336  20 continue
337  cnorm( n ) = zero
338  END IF
339  END IF
340 *
341 * Scale the column norms by TSCAL if the maximum element in CNORM is
342 * greater than BIGNUM.
343 *
344  imax = idamax( n, cnorm, 1 )
345  tmax = cnorm( imax )
346  IF( tmax.LE.bignum ) THEN
347  tscal = one
348  ELSE
349  tscal = one / ( smlnum*tmax )
350  CALL dscal( n, tscal, cnorm, 1 )
351  END IF
352 *
353 * Compute a bound on the computed solution vector to see if the
354 * Level 2 BLAS routine DTRSV can be used.
355 *
356  j = idamax( n, x, 1 )
357  xmax = abs( x( j ) )
358  xbnd = xmax
359  IF( notran ) THEN
360 *
361 * Compute the growth in A * x = b.
362 *
363  IF( upper ) THEN
364  jfirst = n
365  jlast = 1
366  jinc = -1
367  ELSE
368  jfirst = 1
369  jlast = n
370  jinc = 1
371  END IF
372 *
373  IF( tscal.NE.one ) THEN
374  grow = zero
375  go to 50
376  END IF
377 *
378  IF( nounit ) THEN
379 *
380 * A is non-unit triangular.
381 *
382 * Compute GROW = 1/G(j) and XBND = 1/M(j).
383 * Initially, G(0) = max{x(i), i=1,...,n}.
384 *
385  grow = one / max( xbnd, smlnum )
386  xbnd = grow
387  DO 30 j = jfirst, jlast, jinc
388 *
389 * Exit the loop if the growth factor is too small.
390 *
391  IF( grow.LE.smlnum )
392  $ go to 50
393 *
394 * M(j) = G(j-1) / abs(A(j,j))
395 *
396  tjj = abs( a( j, j ) )
397  xbnd = min( xbnd, min( one, tjj )*grow )
398  IF( tjj+cnorm( j ).GE.smlnum ) THEN
399 *
400 * G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
401 *
402  grow = grow*( tjj / ( tjj+cnorm( j ) ) )
403  ELSE
404 *
405 * G(j) could overflow, set GROW to 0.
406 *
407  grow = zero
408  END IF
409  30 continue
410  grow = xbnd
411  ELSE
412 *
413 * A is unit triangular.
414 *
415 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
416 *
417  grow = min( one, one / max( xbnd, smlnum ) )
418  DO 40 j = jfirst, jlast, jinc
419 *
420 * Exit the loop if the growth factor is too small.
421 *
422  IF( grow.LE.smlnum )
423  $ go to 50
424 *
425 * G(j) = G(j-1)*( 1 + CNORM(j) )
426 *
427  grow = grow*( one / ( one+cnorm( j ) ) )
428  40 continue
429  END IF
430  50 continue
431 *
432  ELSE
433 *
434 * Compute the growth in A**T * x = b.
435 *
436  IF( upper ) THEN
437  jfirst = 1
438  jlast = n
439  jinc = 1
440  ELSE
441  jfirst = n
442  jlast = 1
443  jinc = -1
444  END IF
445 *
446  IF( tscal.NE.one ) THEN
447  grow = zero
448  go to 80
449  END IF
450 *
451  IF( nounit ) THEN
452 *
453 * A is non-unit triangular.
454 *
455 * Compute GROW = 1/G(j) and XBND = 1/M(j).
456 * Initially, M(0) = max{x(i), i=1,...,n}.
457 *
458  grow = one / max( xbnd, smlnum )
459  xbnd = grow
460  DO 60 j = jfirst, jlast, jinc
461 *
462 * Exit the loop if the growth factor is too small.
463 *
464  IF( grow.LE.smlnum )
465  $ go to 80
466 *
467 * G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
468 *
469  xj = one + cnorm( j )
470  grow = min( grow, xbnd / xj )
471 *
472 * M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
473 *
474  tjj = abs( a( j, j ) )
475  IF( xj.GT.tjj )
476  $ xbnd = xbnd*( tjj / xj )
477  60 continue
478  grow = min( grow, xbnd )
479  ELSE
480 *
481 * A is unit triangular.
482 *
483 * Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
484 *
485  grow = min( one, one / max( xbnd, smlnum ) )
486  DO 70 j = jfirst, jlast, jinc
487 *
488 * Exit the loop if the growth factor is too small.
489 *
490  IF( grow.LE.smlnum )
491  $ go to 80
492 *
493 * G(j) = ( 1 + CNORM(j) )*G(j-1)
494 *
495  xj = one + cnorm( j )
496  grow = grow / xj
497  70 continue
498  END IF
499  80 continue
500  END IF
501 *
502  IF( ( grow*tscal ).GT.smlnum ) THEN
503 *
504 * Use the Level 2 BLAS solve if the reciprocal of the bound on
505 * elements of X is not too small.
506 *
507  CALL dtrsv( uplo, trans, diag, n, a, lda, x, 1 )
508  ELSE
509 *
510 * Use a Level 1 BLAS solve, scaling intermediate results.
511 *
512  IF( xmax.GT.bignum ) THEN
513 *
514 * Scale X so that its components are less than or equal to
515 * BIGNUM in absolute value.
516 *
517  scale = bignum / xmax
518  CALL dscal( n, scale, x, 1 )
519  xmax = bignum
520  END IF
521 *
522  IF( notran ) THEN
523 *
524 * Solve A * x = b
525 *
526  DO 110 j = jfirst, jlast, jinc
527 *
528 * Compute x(j) = b(j) / A(j,j), scaling x if necessary.
529 *
530  xj = abs( x( j ) )
531  IF( nounit ) THEN
532  tjjs = a( j, j )*tscal
533  ELSE
534  tjjs = tscal
535  IF( tscal.EQ.one )
536  $ go to 100
537  END IF
538  tjj = abs( tjjs )
539  IF( tjj.GT.smlnum ) THEN
540 *
541 * abs(A(j,j)) > SMLNUM:
542 *
543  IF( tjj.LT.one ) THEN
544  IF( xj.GT.tjj*bignum ) THEN
545 *
546 * Scale x by 1/b(j).
547 *
548  rec = one / xj
549  CALL dscal( n, rec, x, 1 )
550  scale = scale*rec
551  xmax = xmax*rec
552  END IF
553  END IF
554  x( j ) = x( j ) / tjjs
555  xj = abs( x( j ) )
556  ELSE IF( tjj.GT.zero ) THEN
557 *
558 * 0 < abs(A(j,j)) <= SMLNUM:
559 *
560  IF( xj.GT.tjj*bignum ) THEN
561 *
562 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
563 * to avoid overflow when dividing by A(j,j).
564 *
565  rec = ( tjj*bignum ) / xj
566  IF( cnorm( j ).GT.one ) THEN
567 *
568 * Scale by 1/CNORM(j) to avoid overflow when
569 * multiplying x(j) times column j.
570 *
571  rec = rec / cnorm( j )
572  END IF
573  CALL dscal( n, rec, x, 1 )
574  scale = scale*rec
575  xmax = xmax*rec
576  END IF
577  x( j ) = x( j ) / tjjs
578  xj = abs( x( j ) )
579  ELSE
580 *
581 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
582 * scale = 0, and compute a solution to A*x = 0.
583 *
584  DO 90 i = 1, n
585  x( i ) = zero
586  90 continue
587  x( j ) = one
588  xj = one
589  scale = zero
590  xmax = zero
591  END IF
592  100 continue
593 *
594 * Scale x if necessary to avoid overflow when adding a
595 * multiple of column j of A.
596 *
597  IF( xj.GT.one ) THEN
598  rec = one / xj
599  IF( cnorm( j ).GT.( bignum-xmax )*rec ) THEN
600 *
601 * Scale x by 1/(2*abs(x(j))).
602 *
603  rec = rec*half
604  CALL dscal( n, rec, x, 1 )
605  scale = scale*rec
606  END IF
607  ELSE IF( xj*cnorm( j ).GT.( bignum-xmax ) ) THEN
608 *
609 * Scale x by 1/2.
610 *
611  CALL dscal( n, half, x, 1 )
612  scale = scale*half
613  END IF
614 *
615  IF( upper ) THEN
616  IF( j.GT.1 ) THEN
617 *
618 * Compute the update
619 * x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
620 *
621  CALL daxpy( j-1, -x( j )*tscal, a( 1, j ), 1, x,
622  $ 1 )
623  i = idamax( j-1, x, 1 )
624  xmax = abs( x( i ) )
625  END IF
626  ELSE
627  IF( j.LT.n ) THEN
628 *
629 * Compute the update
630 * x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
631 *
632  CALL daxpy( n-j, -x( j )*tscal, a( j+1, j ), 1,
633  $ x( j+1 ), 1 )
634  i = j + idamax( n-j, x( j+1 ), 1 )
635  xmax = abs( x( i ) )
636  END IF
637  END IF
638  110 continue
639 *
640  ELSE
641 *
642 * Solve A**T * x = b
643 *
644  DO 160 j = jfirst, jlast, jinc
645 *
646 * Compute x(j) = b(j) - sum A(k,j)*x(k).
647 * k<>j
648 *
649  xj = abs( x( j ) )
650  uscal = tscal
651  rec = one / max( xmax, one )
652  IF( cnorm( j ).GT.( bignum-xj )*rec ) THEN
653 *
654 * If x(j) could overflow, scale x by 1/(2*XMAX).
655 *
656  rec = rec*half
657  IF( nounit ) THEN
658  tjjs = a( j, j )*tscal
659  ELSE
660  tjjs = tscal
661  END IF
662  tjj = abs( tjjs )
663  IF( tjj.GT.one ) THEN
664 *
665 * Divide by A(j,j) when scaling x if A(j,j) > 1.
666 *
667  rec = min( one, rec*tjj )
668  uscal = uscal / tjjs
669  END IF
670  IF( rec.LT.one ) THEN
671  CALL dscal( n, rec, x, 1 )
672  scale = scale*rec
673  xmax = xmax*rec
674  END IF
675  END IF
676 *
677  sumj = zero
678  IF( uscal.EQ.one ) THEN
679 *
680 * If the scaling needed for A in the dot product is 1,
681 * call DDOT to perform the dot product.
682 *
683  IF( upper ) THEN
684  sumj = ddot( j-1, a( 1, j ), 1, x, 1 )
685  ELSE IF( j.LT.n ) THEN
686  sumj = ddot( n-j, a( j+1, j ), 1, x( j+1 ), 1 )
687  END IF
688  ELSE
689 *
690 * Otherwise, use in-line code for the dot product.
691 *
692  IF( upper ) THEN
693  DO 120 i = 1, j - 1
694  sumj = sumj + ( a( i, j )*uscal )*x( i )
695  120 continue
696  ELSE IF( j.LT.n ) THEN
697  DO 130 i = j + 1, n
698  sumj = sumj + ( a( i, j )*uscal )*x( i )
699  130 continue
700  END IF
701  END IF
702 *
703  IF( uscal.EQ.tscal ) THEN
704 *
705 * Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
706 * was not used to scale the dotproduct.
707 *
708  x( j ) = x( j ) - sumj
709  xj = abs( x( j ) )
710  IF( nounit ) THEN
711  tjjs = a( j, j )*tscal
712  ELSE
713  tjjs = tscal
714  IF( tscal.EQ.one )
715  $ go to 150
716  END IF
717 *
718 * Compute x(j) = x(j) / A(j,j), scaling if necessary.
719 *
720  tjj = abs( tjjs )
721  IF( tjj.GT.smlnum ) THEN
722 *
723 * abs(A(j,j)) > SMLNUM:
724 *
725  IF( tjj.LT.one ) THEN
726  IF( xj.GT.tjj*bignum ) THEN
727 *
728 * Scale X by 1/abs(x(j)).
729 *
730  rec = one / xj
731  CALL dscal( n, rec, x, 1 )
732  scale = scale*rec
733  xmax = xmax*rec
734  END IF
735  END IF
736  x( j ) = x( j ) / tjjs
737  ELSE IF( tjj.GT.zero ) THEN
738 *
739 * 0 < abs(A(j,j)) <= SMLNUM:
740 *
741  IF( xj.GT.tjj*bignum ) THEN
742 *
743 * Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
744 *
745  rec = ( tjj*bignum ) / xj
746  CALL dscal( n, rec, x, 1 )
747  scale = scale*rec
748  xmax = xmax*rec
749  END IF
750  x( j ) = x( j ) / tjjs
751  ELSE
752 *
753 * A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
754 * scale = 0, and compute a solution to A**T*x = 0.
755 *
756  DO 140 i = 1, n
757  x( i ) = zero
758  140 continue
759  x( j ) = one
760  scale = zero
761  xmax = zero
762  END IF
763  150 continue
764  ELSE
765 *
766 * Compute x(j) := x(j) / A(j,j) - sumj if the dot
767 * product has already been divided by 1/A(j,j).
768 *
769  x( j ) = x( j ) / tjjs - sumj
770  END IF
771  xmax = max( xmax, abs( x( j ) ) )
772  160 continue
773  END IF
774  scale = scale / tscal
775  END IF
776 *
777 * Scale the column norms by 1/TSCAL for return.
778 *
779  IF( tscal.NE.one ) THEN
780  CALL dscal( n, one / tscal, cnorm, 1 )
781  END IF
782 *
783  return
784 *
785 * End of DLATRS
786 *
787  END