LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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sposvxx.f
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1*> \brief <b> SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices</b>
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
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17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
22* S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
23* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
24* NPARAMS, PARAMS, WORK, IWORK, INFO )
25*
26* .. Scalar Arguments ..
27* CHARACTER EQUED, FACT, UPLO
28* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
29* \$ N_ERR_BNDS
30* REAL RCOND, RPVGRW
31* ..
32* .. Array Arguments ..
33* INTEGER IWORK( * )
34* REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35* \$ X( LDX, * ), WORK( * )
36* REAL S( * ), PARAMS( * ), BERR( * ),
37* \$ ERR_BNDS_NORM( NRHS, * ),
38* \$ ERR_BNDS_COMP( NRHS, * )
39* ..
40*
41*
42*> \par Purpose:
43* =============
44*>
45*> \verbatim
46*>
47*> SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
48*> to compute the solution to a real system of linear equations
49*> A * X = B, where A is an N-by-N symmetric positive definite matrix
50*> and X and B are N-by-NRHS matrices.
51*>
52*> If requested, both normwise and maximum componentwise error bounds
53*> are returned. SPOSVXX will return a solution with a tiny
54*> guaranteed error (O(eps) where eps is the working machine
55*> precision) unless the matrix is very ill-conditioned, in which
56*> case a warning is returned. Relevant condition numbers also are
57*> calculated and returned.
58*>
59*> SPOSVXX accepts user-provided factorizations and equilibration
60*> factors; see the definitions of the FACT and EQUED options.
61*> Solving with refinement and using a factorization from a previous
62*> SPOSVXX call will also produce a solution with either O(eps)
63*> errors or warnings, but we cannot make that claim for general
64*> user-provided factorizations and equilibration factors if they
65*> differ from what SPOSVXX would itself produce.
66*> \endverbatim
67*
68*> \par Description:
69* =================
70*>
71*> \verbatim
72*>
73*> The following steps are performed:
74*>
75*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
76*> the system:
77*>
78*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
79*>
80*> Whether or not the system will be equilibrated depends on the
81*> scaling of the matrix A, but if equilibration is used, A is
82*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
83*>
84*> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
85*> factor the matrix A (after equilibration if FACT = 'E') as
86*> A = U**T* U, if UPLO = 'U', or
87*> A = L * L**T, if UPLO = 'L',
88*> where U is an upper triangular matrix and L is a lower triangular
89*> matrix.
90*>
91*> 3. If the leading i-by-i principal minor is not positive definite,
92*> then the routine returns with INFO = i. Otherwise, the factored
93*> form of A is used to estimate the condition number of the matrix
94*> A (see argument RCOND). If the reciprocal of the condition number
95*> is less than machine precision, the routine still goes on to solve
96*> for X and compute error bounds as described below.
97*>
98*> 4. The system of equations is solved for X using the factored form
99*> of A.
100*>
101*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
102*> the routine will use iterative refinement to try to get a small
103*> error and error bounds. Refinement calculates the residual to at
104*> least twice the working precision.
105*>
106*> 6. If equilibration was used, the matrix X is premultiplied by
107*> diag(S) so that it solves the original system before
108*> equilibration.
109*> \endverbatim
110*
111* Arguments:
112* ==========
113*
114*> \verbatim
115*> Some optional parameters are bundled in the PARAMS array. These
116*> settings determine how refinement is performed, but often the
117*> defaults are acceptable. If the defaults are acceptable, users
118*> can pass NPARAMS = 0 which prevents the source code from accessing
119*> the PARAMS argument.
120*> \endverbatim
121*>
122*> \param[in] FACT
123*> \verbatim
124*> FACT is CHARACTER*1
125*> Specifies whether or not the factored form of the matrix A is
126*> supplied on entry, and if not, whether the matrix A should be
127*> equilibrated before it is factored.
128*> = 'F': On entry, AF contains the factored form of A.
129*> If EQUED is not 'N', the matrix A has been
130*> equilibrated with scaling factors given by S.
131*> A and AF are not modified.
132*> = 'N': The matrix A will be copied to AF and factored.
133*> = 'E': The matrix A will be equilibrated if necessary, then
134*> copied to AF and factored.
135*> \endverbatim
136*>
137*> \param[in] UPLO
138*> \verbatim
139*> UPLO is CHARACTER*1
140*> = 'U': Upper triangle of A is stored;
141*> = 'L': Lower triangle of A is stored.
142*> \endverbatim
143*>
144*> \param[in] N
145*> \verbatim
146*> N is INTEGER
147*> The number of linear equations, i.e., the order of the
148*> matrix A. N >= 0.
149*> \endverbatim
150*>
151*> \param[in] NRHS
152*> \verbatim
153*> NRHS is INTEGER
154*> The number of right hand sides, i.e., the number of columns
155*> of the matrices B and X. NRHS >= 0.
156*> \endverbatim
157*>
158*> \param[in,out] A
159*> \verbatim
160*> A is REAL array, dimension (LDA,N)
161*> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
162*> 'Y', then A must contain the equilibrated matrix
163*> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
164*> triangular part of A contains the upper triangular part of the
165*> matrix A, and the strictly lower triangular part of A is not
166*> referenced. If UPLO = 'L', the leading N-by-N lower triangular
167*> part of A contains the lower triangular part of the matrix A, and
168*> the strictly upper triangular part of A is not referenced. A is
169*> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
170*> 'N' on exit.
171*>
172*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
173*> diag(S)*A*diag(S).
174*> \endverbatim
175*>
176*> \param[in] LDA
177*> \verbatim
178*> LDA is INTEGER
179*> The leading dimension of the array A. LDA >= max(1,N).
180*> \endverbatim
181*>
182*> \param[in,out] AF
183*> \verbatim
184*> AF is REAL array, dimension (LDAF,N)
185*> If FACT = 'F', then AF is an input argument and on entry
186*> contains the triangular factor U or L from the Cholesky
187*> factorization A = U**T*U or A = L*L**T, in the same storage
188*> format as A. If EQUED .ne. 'N', then AF is the factored
189*> form of the equilibrated matrix diag(S)*A*diag(S).
190*>
191*> If FACT = 'N', then AF is an output argument and on exit
192*> returns the triangular factor U or L from the Cholesky
193*> factorization A = U**T*U or A = L*L**T of the original
194*> matrix A.
195*>
196*> If FACT = 'E', then AF is an output argument and on exit
197*> returns the triangular factor U or L from the Cholesky
198*> factorization A = U**T*U or A = L*L**T of the equilibrated
199*> matrix A (see the description of A for the form of the
200*> equilibrated matrix).
201*> \endverbatim
202*>
203*> \param[in] LDAF
204*> \verbatim
205*> LDAF is INTEGER
206*> The leading dimension of the array AF. LDAF >= max(1,N).
207*> \endverbatim
208*>
209*> \param[in,out] EQUED
210*> \verbatim
211*> EQUED is CHARACTER*1
212*> Specifies the form of equilibration that was done.
213*> = 'N': No equilibration (always true if FACT = 'N').
214*> = 'Y': Both row and column equilibration, i.e., A has been
215*> replaced by diag(S) * A * diag(S).
216*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
217*> output argument.
218*> \endverbatim
219*>
220*> \param[in,out] S
221*> \verbatim
222*> S is REAL array, dimension (N)
223*> The row scale factors for A. If EQUED = 'Y', A is multiplied on
224*> the left and right by diag(S). S is an input argument if FACT =
225*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
226*> = 'Y', each element of S must be positive. If S is output, each
227*> element of S is a power of the radix. If S is input, each element
228*> of S should be a power of the radix to ensure a reliable solution
229*> and error estimates. Scaling by powers of the radix does not cause
230*> rounding errors unless the result underflows or overflows.
231*> Rounding errors during scaling lead to refining with a matrix that
232*> is not equivalent to the input matrix, producing error estimates
233*> that may not be reliable.
234*> \endverbatim
235*>
236*> \param[in,out] B
237*> \verbatim
238*> B is REAL array, dimension (LDB,NRHS)
239*> On entry, the N-by-NRHS right hand side matrix B.
240*> On exit,
241*> if EQUED = 'N', B is not modified;
242*> if EQUED = 'Y', B is overwritten by diag(S)*B;
243*> \endverbatim
244*>
245*> \param[in] LDB
246*> \verbatim
247*> LDB is INTEGER
248*> The leading dimension of the array B. LDB >= max(1,N).
249*> \endverbatim
250*>
251*> \param[out] X
252*> \verbatim
253*> X is REAL array, dimension (LDX,NRHS)
254*> If INFO = 0, the N-by-NRHS solution matrix X to the original
255*> system of equations. Note that A and B are modified on exit if
256*> EQUED .ne. 'N', and the solution to the equilibrated system is
257*> inv(diag(S))*X.
258*> \endverbatim
259*>
260*> \param[in] LDX
261*> \verbatim
262*> LDX is INTEGER
263*> The leading dimension of the array X. LDX >= max(1,N).
264*> \endverbatim
265*>
266*> \param[out] RCOND
267*> \verbatim
268*> RCOND is REAL
269*> Reciprocal scaled condition number. This is an estimate of the
270*> reciprocal Skeel condition number of the matrix A after
271*> equilibration (if done). If this is less than the machine
272*> precision (in particular, if it is zero), the matrix is singular
273*> to working precision. Note that the error may still be small even
274*> if this number is very small and the matrix appears ill-
275*> conditioned.
276*> \endverbatim
277*>
278*> \param[out] RPVGRW
279*> \verbatim
280*> RPVGRW is REAL
281*> Reciprocal pivot growth. On exit, this contains the reciprocal
282*> pivot growth factor norm(A)/norm(U). The "max absolute element"
283*> norm is used. If this is much less than 1, then the stability of
284*> the LU factorization of the (equilibrated) matrix A could be poor.
285*> This also means that the solution X, estimated condition numbers,
286*> and error bounds could be unreliable. If factorization fails with
287*> 0<INFO<=N, then this contains the reciprocal pivot growth factor
288*> for the leading INFO columns of A.
289*> \endverbatim
290*>
291*> \param[out] BERR
292*> \verbatim
293*> BERR is REAL array, dimension (NRHS)
294*> Componentwise relative backward error. This is the
295*> componentwise relative backward error of each solution vector X(j)
296*> (i.e., the smallest relative change in any element of A or B that
297*> makes X(j) an exact solution).
298*> \endverbatim
299*>
300*> \param[in] N_ERR_BNDS
301*> \verbatim
302*> N_ERR_BNDS is INTEGER
303*> Number of error bounds to return for each right hand side
304*> and each type (normwise or componentwise). See ERR_BNDS_NORM and
305*> ERR_BNDS_COMP below.
306*> \endverbatim
307*>
308*> \param[out] ERR_BNDS_NORM
309*> \verbatim
310*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
311*> For each right-hand side, this array contains information about
312*> various error bounds and condition numbers corresponding to the
313*> normwise relative error, which is defined as follows:
314*>
315*> Normwise relative error in the ith solution vector:
316*> max_j (abs(XTRUE(j,i) - X(j,i)))
317*> ------------------------------
318*> max_j abs(X(j,i))
319*>
320*> The array is indexed by the type of error information as described
321*> below. There currently are up to three pieces of information
322*> returned.
323*>
324*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
325*> right-hand side.
326*>
327*> The second index in ERR_BNDS_NORM(:,err) contains the following
328*> three fields:
329*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
330*> reciprocal condition number is less than the threshold
331*> sqrt(n) * slamch('Epsilon').
332*>
333*> err = 2 "Guaranteed" error bound: The estimated forward error,
334*> almost certainly within a factor of 10 of the true error
335*> so long as the next entry is greater than the threshold
336*> sqrt(n) * slamch('Epsilon'). This error bound should only
337*> be trusted if the previous boolean is true.
338*>
339*> err = 3 Reciprocal condition number: Estimated normwise
340*> reciprocal condition number. Compared with the threshold
341*> sqrt(n) * slamch('Epsilon') to determine if the error
342*> estimate is "guaranteed". These reciprocal condition
343*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
344*> appropriately scaled matrix Z.
345*> Let Z = S*A, where S scales each row by a power of the
346*> radix so all absolute row sums of Z are approximately 1.
347*>
348*> See Lapack Working Note 165 for further details and extra
349*> cautions.
350*> \endverbatim
351*>
352*> \param[out] ERR_BNDS_COMP
353*> \verbatim
354*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
355*> For each right-hand side, this array contains information about
356*> various error bounds and condition numbers corresponding to the
357*> componentwise relative error, which is defined as follows:
358*>
359*> Componentwise relative error in the ith solution vector:
360*> abs(XTRUE(j,i) - X(j,i))
361*> max_j ----------------------
362*> abs(X(j,i))
363*>
364*> The array is indexed by the right-hand side i (on which the
365*> componentwise relative error depends), and the type of error
366*> information as described below. There currently are up to three
367*> pieces of information returned for each right-hand side. If
368*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
369*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
370*> the first (:,N_ERR_BNDS) entries are returned.
371*>
372*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
373*> right-hand side.
374*>
375*> The second index in ERR_BNDS_COMP(:,err) contains the following
376*> three fields:
377*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
378*> reciprocal condition number is less than the threshold
379*> sqrt(n) * slamch('Epsilon').
380*>
381*> err = 2 "Guaranteed" error bound: The estimated forward error,
382*> almost certainly within a factor of 10 of the true error
383*> so long as the next entry is greater than the threshold
384*> sqrt(n) * slamch('Epsilon'). This error bound should only
385*> be trusted if the previous boolean is true.
386*>
387*> err = 3 Reciprocal condition number: Estimated componentwise
388*> reciprocal condition number. Compared with the threshold
389*> sqrt(n) * slamch('Epsilon') to determine if the error
390*> estimate is "guaranteed". These reciprocal condition
391*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
392*> appropriately scaled matrix Z.
393*> Let Z = S*(A*diag(x)), where x is the solution for the
394*> current right-hand side and S scales each row of
395*> A*diag(x) by a power of the radix so all absolute row
396*> sums of Z are approximately 1.
397*>
398*> See Lapack Working Note 165 for further details and extra
399*> cautions.
400*> \endverbatim
401*>
402*> \param[in] NPARAMS
403*> \verbatim
404*> NPARAMS is INTEGER
405*> Specifies the number of parameters set in PARAMS. If <= 0, the
406*> PARAMS array is never referenced and default values are used.
407*> \endverbatim
408*>
409*> \param[in,out] PARAMS
410*> \verbatim
411*> PARAMS is REAL array, dimension NPARAMS
412*> Specifies algorithm parameters. If an entry is < 0.0, then
413*> that entry will be filled with default value used for that
414*> parameter. Only positions up to NPARAMS are accessed; defaults
415*> are used for higher-numbered parameters.
416*>
417*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
418*> refinement or not.
419*> Default: 1.0
420*> = 0.0: No refinement is performed, and no error bounds are
421*> computed.
422*> = 1.0: Use the double-precision refinement algorithm,
423*> possibly with doubled-single computations if the
424*> compilation environment does not support DOUBLE
425*> PRECISION.
426*> (other values are reserved for future use)
427*>
428*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
429*> computations allowed for refinement.
430*> Default: 10
431*> Aggressive: Set to 100 to permit convergence using approximate
432*> factorizations or factorizations other than LU. If
433*> the factorization uses a technique other than
434*> Gaussian elimination, the guarantees in
435*> err_bnds_norm and err_bnds_comp may no longer be
436*> trustworthy.
437*>
438*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
439*> will attempt to find a solution with small componentwise
440*> relative error in the double-precision algorithm. Positive
441*> is true, 0.0 is false.
442*> Default: 1.0 (attempt componentwise convergence)
443*> \endverbatim
444*>
445*> \param[out] WORK
446*> \verbatim
447*> WORK is REAL array, dimension (4*N)
448*> \endverbatim
449*>
450*> \param[out] IWORK
451*> \verbatim
452*> IWORK is INTEGER array, dimension (N)
453*> \endverbatim
454*>
455*> \param[out] INFO
456*> \verbatim
457*> INFO is INTEGER
458*> = 0: Successful exit. The solution to every right-hand side is
459*> guaranteed.
460*> < 0: If INFO = -i, the i-th argument had an illegal value
461*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
462*> has been completed, but the factor U is exactly singular, so
463*> the solution and error bounds could not be computed. RCOND = 0
464*> is returned.
465*> = N+J: The solution corresponding to the Jth right-hand side is
466*> not guaranteed. The solutions corresponding to other right-
467*> hand sides K with K > J may not be guaranteed as well, but
468*> only the first such right-hand side is reported. If a small
469*> componentwise error is not requested (PARAMS(3) = 0.0) then
470*> the Jth right-hand side is the first with a normwise error
471*> bound that is not guaranteed (the smallest J such
472*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
473*> the Jth right-hand side is the first with either a normwise or
474*> componentwise error bound that is not guaranteed (the smallest
475*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
476*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
477*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
478*> about all of the right-hand sides check ERR_BNDS_NORM or
479*> ERR_BNDS_COMP.
480*> \endverbatim
481*
482* Authors:
483* ========
484*
485*> \author Univ. of Tennessee
486*> \author Univ. of California Berkeley
487*> \author Univ. of Colorado Denver
488*> \author NAG Ltd.
489*
490*> \ingroup realPOsolve
491*
492* =====================================================================
493 SUBROUTINE sposvxx( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
494 \$ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
495 \$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
496 \$ NPARAMS, PARAMS, WORK, IWORK, INFO )
497*
498* -- LAPACK driver routine --
499* -- LAPACK is a software package provided by Univ. of Tennessee, --
500* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
501*
502* .. Scalar Arguments ..
503 CHARACTER EQUED, FACT, UPLO
504 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
505 \$ N_ERR_BNDS
506 REAL RCOND, RPVGRW
507* ..
508* .. Array Arguments ..
509 INTEGER IWORK( * )
510 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
511 \$ X( LDX, * ), WORK( * )
512 REAL S( * ), PARAMS( * ), BERR( * ),
513 \$ err_bnds_norm( nrhs, * ),
514 \$ err_bnds_comp( nrhs, * )
515* ..
516*
517* ==================================================================
518*
519* .. Parameters ..
520 REAL ZERO, ONE
521 PARAMETER ( ZERO = 0.0e+0, one = 1.0e+0 )
522 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
523 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
524 INTEGER CMP_ERR_I, PIV_GROWTH_I
525 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
526 \$ berr_i = 3 )
527 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
528 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
529 \$ piv_growth_i = 9 )
530* ..
531* .. Local Scalars ..
532 LOGICAL EQUIL, NOFACT, RCEQU
533 INTEGER INFEQU, J
534 REAL AMAX, BIGNUM, SMIN, SMAX,
535 \$ scond, smlnum
536* ..
537* .. External Functions ..
538 EXTERNAL lsame, slamch, sla_porpvgrw
539 LOGICAL LSAME
540 REAL SLAMCH, SLA_PORPVGRW
541* ..
542* .. External Subroutines ..
543 EXTERNAL spoequb, spotrf, spotrs, slacpy, slaqsy,
545* ..
546* .. Intrinsic Functions ..
547 INTRINSIC max, min
548* ..
549* .. Executable Statements ..
550*
551 info = 0
552 nofact = lsame( fact, 'N' )
553 equil = lsame( fact, 'E' )
554 smlnum = slamch( 'Safe minimum' )
555 bignum = one / smlnum
556 IF( nofact .OR. equil ) THEN
557 equed = 'N'
558 rcequ = .false.
559 ELSE
560 rcequ = lsame( equed, 'Y' )
561 ENDIF
562*
563* Default is failure. If an input parameter is wrong or
564* factorization fails, make everything look horrible. Only the
565* pivot growth is set here, the rest is initialized in SPORFSX.
566*
567 rpvgrw = zero
568*
569* Test the input parameters. PARAMS is not tested until SPORFSX.
570*
571 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
572 \$ lsame( fact, 'F' ) ) THEN
573 info = -1
574 ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
575 \$ .NOT.lsame( uplo, 'L' ) ) THEN
576 info = -2
577 ELSE IF( n.LT.0 ) THEN
578 info = -3
579 ELSE IF( nrhs.LT.0 ) THEN
580 info = -4
581 ELSE IF( lda.LT.max( 1, n ) ) THEN
582 info = -6
583 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
584 info = -8
585 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
586 \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
587 info = -9
588 ELSE
589 IF ( rcequ ) THEN
590 smin = bignum
591 smax = zero
592 DO 10 j = 1, n
593 smin = min( smin, s( j ) )
594 smax = max( smax, s( j ) )
595 10 CONTINUE
596 IF( smin.LE.zero ) THEN
597 info = -10
598 ELSE IF( n.GT.0 ) THEN
599 scond = max( smin, smlnum ) / min( smax, bignum )
600 ELSE
601 scond = one
602 END IF
603 END IF
604 IF( info.EQ.0 ) THEN
605 IF( ldb.LT.max( 1, n ) ) THEN
606 info = -12
607 ELSE IF( ldx.LT.max( 1, n ) ) THEN
608 info = -14
609 END IF
610 END IF
611 END IF
612*
613 IF( info.NE.0 ) THEN
614 CALL xerbla( 'SPOSVXX', -info )
615 RETURN
616 END IF
617*
618 IF( equil ) THEN
619*
620* Compute row and column scalings to equilibrate the matrix A.
621*
622 CALL spoequb( n, a, lda, s, scond, amax, infequ )
623 IF( infequ.EQ.0 ) THEN
624*
625* Equilibrate the matrix.
626*
627 CALL slaqsy( uplo, n, a, lda, s, scond, amax, equed )
628 rcequ = lsame( equed, 'Y' )
629 END IF
630 END IF
631*
632* Scale the right-hand side.
633*
634 IF( rcequ ) CALL slascl2( n, nrhs, s, b, ldb )
635*
636 IF( nofact .OR. equil ) THEN
637*
638* Compute the Cholesky factorization of A.
639*
640 CALL slacpy( uplo, n, n, a, lda, af, ldaf )
641 CALL spotrf( uplo, n, af, ldaf, info )
642*
643* Return if INFO is non-zero.
644*
645 IF( info.NE.0 ) THEN
646*
647* Pivot in column INFO is exactly 0
648* Compute the reciprocal pivot growth factor of the
649* leading rank-deficient INFO columns of A.
650*
651 rpvgrw = sla_porpvgrw( uplo, info, a, lda, af, ldaf, work )
652 RETURN
653 ENDIF
654 END IF
655*
656* Compute the reciprocal growth factor RPVGRW.
657*
658 rpvgrw = sla_porpvgrw( uplo, n, a, lda, af, ldaf, work )
659*
660* Compute the solution matrix X.
661*
662 CALL slacpy( 'Full', n, nrhs, b, ldb, x, ldx )
663 CALL spotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
664*
665* Use iterative refinement to improve the computed solution and
666* compute error bounds and backward error estimates for it.
667*
668 CALL sporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
669 \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
670 \$ err_bnds_comp, nparams, params, work, iwork, info )
671
672*
673* Scale solutions.
674*
675 IF ( rcequ ) THEN
676 CALL slascl2 ( n, nrhs, s, x, ldx )
677 END IF
678*
679 RETURN
680*
681* End of SPOSVXX
682*
683 END
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine slascl2(M, N, D, X, LDX)
SLASCL2 performs diagonal scaling on a matrix.
Definition: slascl2.f:90
subroutine sporfsx(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
SPORFSX
Definition: sporfsx.f:394
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:107
real function sla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition: sla_porpvgrw.f:104
subroutine spotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
SPOTRS
Definition: spotrs.f:110
subroutine spoequb(N, A, LDA, S, SCOND, AMAX, INFO)
SPOEQUB
Definition: spoequb.f:118
subroutine sposvxx(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO)
SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices
Definition: sposvxx.f:497
subroutine slaqsy(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
SLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition: slaqsy.f:133
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68