LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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dsysvxx.f
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1*> \brief \b DSYSVXX
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> Download DSYSVXX + dependencies
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10*> [TGZ]</a>
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12*> [ZIP]</a>
13*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysvxx.f">
14*> [TXT]</a>
15*
16* Definition:
17* ===========
18*
19* SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
20* EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
21* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
22* NPARAMS, PARAMS, WORK, IWORK, INFO )
23*
24* .. Scalar Arguments ..
25* CHARACTER EQUED, FACT, UPLO
26* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
27* $ N_ERR_BNDS
28* DOUBLE PRECISION RCOND, RPVGRW
29* ..
30* .. Array Arguments ..
31* INTEGER IPIV( * ), IWORK( * )
32* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
33* $ X( LDX, * ), WORK( * )
34* DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
35* $ ERR_BNDS_NORM( NRHS, * ),
36* $ ERR_BNDS_COMP( NRHS, * )
37* ..
38*
39*
40*> \par Purpose:
41* =============
42*>
43*> \verbatim
44*>
45*> DSYSVXX uses the diagonal pivoting factorization to compute the
46*> solution to a double precision system of linear equations A * X = B, where A
47*> is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.
48*>
49*> If requested, both normwise and maximum componentwise error bounds
50*> are returned. DSYSVXX will return a solution with a tiny
51*> guaranteed error (O(eps) where eps is the working machine
52*> precision) unless the matrix is very ill-conditioned, in which
53*> case a warning is returned. Relevant condition numbers also are
54*> calculated and returned.
55*>
56*> DSYSVXX accepts user-provided factorizations and equilibration
57*> factors; see the definitions of the FACT and EQUED options.
58*> Solving with refinement and using a factorization from a previous
59*> DSYSVXX call will also produce a solution with either O(eps)
60*> errors or warnings, but we cannot make that claim for general
61*> user-provided factorizations and equilibration factors if they
62*> differ from what DSYSVXX would itself produce.
63*> \endverbatim
64*
65*> \par Description:
66* =================
67*>
68*> \verbatim
69*>
70*> The following steps are performed:
71*>
72*> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
73*> the system:
74*>
75*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
76*>
77*> Whether or not the system will be equilibrated depends on the
78*> scaling of the matrix A, but if equilibration is used, A is
79*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
80*>
81*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
82*> the matrix A (after equilibration if FACT = 'E') as
83*>
84*> A = U * D * U**T, if UPLO = 'U', or
85*> A = L * D * L**T, if UPLO = 'L',
86*>
87*> where U (or L) is a product of permutation and unit upper (lower)
88*> triangular matrices, and D is symmetric and block diagonal with
89*> 1-by-1 and 2-by-2 diagonal blocks.
90*>
91*> 3. If some D(i,i)=0, so that D is exactly singular, then the
92*> routine returns with INFO = i. Otherwise, the factored form of A
93*> is used to estimate the condition number of the matrix A (see
94*> argument RCOND). If the reciprocal of the condition number is
95*> 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(R) 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 and IPIV contain 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, AF, and IPIV 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 DOUBLE PRECISION array, dimension (LDA,N)
161*> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
162*> upper triangular part of A contains the upper triangular
163*> part of the matrix A, and the strictly lower triangular
164*> part of A is not referenced. If UPLO = 'L', the leading
165*> N-by-N lower triangular part of A contains the lower
166*> triangular part of the matrix A, and the strictly upper
167*> triangular part of A is not referenced.
168*>
169*> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
170*> diag(S)*A*diag(S).
171*> \endverbatim
172*>
173*> \param[in] LDA
174*> \verbatim
175*> LDA is INTEGER
176*> The leading dimension of the array A. LDA >= max(1,N).
177*> \endverbatim
178*>
179*> \param[in,out] AF
180*> \verbatim
181*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
182*> If FACT = 'F', then AF is an input argument and on entry
183*> contains the block diagonal matrix D and the multipliers
184*> used to obtain the factor U or L from the factorization A =
185*> U*D*U**T or A = L*D*L**T as computed by DSYTRF.
186*>
187*> If FACT = 'N', then AF is an output argument and on exit
188*> returns the block diagonal matrix D and the multipliers
189*> used to obtain the factor U or L from the factorization A =
190*> U*D*U**T or A = L*D*L**T.
191*> \endverbatim
192*>
193*> \param[in] LDAF
194*> \verbatim
195*> LDAF is INTEGER
196*> The leading dimension of the array AF. LDAF >= max(1,N).
197*> \endverbatim
198*>
199*> \param[in,out] IPIV
200*> \verbatim
201*> IPIV is INTEGER array, dimension (N)
202*> If FACT = 'F', then IPIV is an input argument and on entry
203*> contains details of the interchanges and the block
204*> structure of D, as determined by DSYTRF. If IPIV(k) > 0,
205*> then rows and columns k and IPIV(k) were interchanged and
206*> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and
207*> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
208*> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
209*> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
210*> then rows and columns k+1 and -IPIV(k) were interchanged
211*> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
212*>
213*> If FACT = 'N', then IPIV is an output argument and on exit
214*> contains details of the interchanges and the block
215*> structure of D, as determined by DSYTRF.
216*> \endverbatim
217*>
218*> \param[in,out] EQUED
219*> \verbatim
220*> EQUED is CHARACTER*1
221*> Specifies the form of equilibration that was done.
222*> = 'N': No equilibration (always true if FACT = 'N').
223*> = 'Y': Both row and column equilibration, i.e., A has been
224*> replaced by diag(S) * A * diag(S).
225*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
226*> output argument.
227*> \endverbatim
228*>
229*> \param[in,out] S
230*> \verbatim
231*> S is DOUBLE PRECISION array, dimension (N)
232*> The scale factors for A. If EQUED = 'Y', A is multiplied on
233*> the left and right by diag(S). S is an input argument if FACT =
234*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
235*> = 'Y', each element of S must be positive. If S is output, each
236*> element of S is a power of the radix. If S is input, each element
237*> of S should be a power of the radix to ensure a reliable solution
238*> and error estimates. Scaling by powers of the radix does not cause
239*> rounding errors unless the result underflows or overflows.
240*> Rounding errors during scaling lead to refining with a matrix that
241*> is not equivalent to the input matrix, producing error estimates
242*> that may not be reliable.
243*> \endverbatim
244*>
245*> \param[in,out] B
246*> \verbatim
247*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
248*> On entry, the N-by-NRHS right hand side matrix B.
249*> On exit,
250*> if EQUED = 'N', B is not modified;
251*> if EQUED = 'Y', B is overwritten by diag(S)*B;
252*> \endverbatim
253*>
254*> \param[in] LDB
255*> \verbatim
256*> LDB is INTEGER
257*> The leading dimension of the array B. LDB >= max(1,N).
258*> \endverbatim
259*>
260*> \param[out] X
261*> \verbatim
262*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
263*> If INFO = 0, the N-by-NRHS solution matrix X to the original
264*> system of equations. Note that A and B are modified on exit if
265*> EQUED .ne. 'N', and the solution to the equilibrated system is
266*> inv(diag(S))*X.
267*> \endverbatim
268*>
269*> \param[in] LDX
270*> \verbatim
271*> LDX is INTEGER
272*> The leading dimension of the array X. LDX >= max(1,N).
273*> \endverbatim
274*>
275*> \param[out] RCOND
276*> \verbatim
277*> RCOND is DOUBLE PRECISION
278*> Reciprocal scaled condition number. This is an estimate of the
279*> reciprocal Skeel condition number of the matrix A after
280*> equilibration (if done). If this is less than the machine
281*> precision (in particular, if it is zero), the matrix is singular
282*> to working precision. Note that the error may still be small even
283*> if this number is very small and the matrix appears ill-
284*> conditioned.
285*> \endverbatim
286*>
287*> \param[out] RPVGRW
288*> \verbatim
289*> RPVGRW is DOUBLE PRECISION
290*> Reciprocal pivot growth. On exit, this contains the reciprocal
291*> pivot growth factor norm(A)/norm(U). The "max absolute element"
292*> norm is used. If this is much less than 1, then the stability of
293*> the LU factorization of the (equilibrated) matrix A could be poor.
294*> This also means that the solution X, estimated condition numbers,
295*> and error bounds could be unreliable. If factorization fails with
296*> 0<INFO<=N, then this contains the reciprocal pivot growth factor
297*> for the leading INFO columns of A.
298*> \endverbatim
299*>
300*> \param[out] BERR
301*> \verbatim
302*> BERR is DOUBLE PRECISION array, dimension (NRHS)
303*> Componentwise relative backward error. This is the
304*> componentwise relative backward error of each solution vector X(j)
305*> (i.e., the smallest relative change in any element of A or B that
306*> makes X(j) an exact solution).
307*> \endverbatim
308*>
309*> \param[in] N_ERR_BNDS
310*> \verbatim
311*> N_ERR_BNDS is INTEGER
312*> Number of error bounds to return for each right hand side
313*> and each type (normwise or componentwise). See ERR_BNDS_NORM and
314*> ERR_BNDS_COMP below.
315*> \endverbatim
316*>
317*> \param[out] ERR_BNDS_NORM
318*> \verbatim
319*> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
320*> For each right-hand side, this array contains information about
321*> various error bounds and condition numbers corresponding to the
322*> normwise relative error, which is defined as follows:
323*>
324*> Normwise relative error in the ith solution vector:
325*> max_j (abs(XTRUE(j,i) - X(j,i)))
326*> ------------------------------
327*> max_j abs(X(j,i))
328*>
329*> The array is indexed by the type of error information as described
330*> below. There currently are up to three pieces of information
331*> returned.
332*>
333*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
334*> right-hand side.
335*>
336*> The second index in ERR_BNDS_NORM(:,err) contains the following
337*> three fields:
338*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
339*> reciprocal condition number is less than the threshold
340*> sqrt(n) * dlamch('Epsilon').
341*>
342*> err = 2 "Guaranteed" error bound: The estimated forward error,
343*> almost certainly within a factor of 10 of the true error
344*> so long as the next entry is greater than the threshold
345*> sqrt(n) * dlamch('Epsilon'). This error bound should only
346*> be trusted if the previous boolean is true.
347*>
348*> err = 3 Reciprocal condition number: Estimated normwise
349*> reciprocal condition number. Compared with the threshold
350*> sqrt(n) * dlamch('Epsilon') to determine if the error
351*> estimate is "guaranteed". These reciprocal condition
352*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
353*> appropriately scaled matrix Z.
354*> Let Z = S*A, where S scales each row by a power of the
355*> radix so all absolute row sums of Z are approximately 1.
356*>
357*> See Lapack Working Note 165 for further details and extra
358*> cautions.
359*> \endverbatim
360*>
361*> \param[out] ERR_BNDS_COMP
362*> \verbatim
363*> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
364*> For each right-hand side, this array contains information about
365*> various error bounds and condition numbers corresponding to the
366*> componentwise relative error, which is defined as follows:
367*>
368*> Componentwise relative error in the ith solution vector:
369*> abs(XTRUE(j,i) - X(j,i))
370*> max_j ----------------------
371*> abs(X(j,i))
372*>
373*> The array is indexed by the right-hand side i (on which the
374*> componentwise relative error depends), and the type of error
375*> information as described below. There currently are up to three
376*> pieces of information returned for each right-hand side. If
377*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
378*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
379*> the first (:,N_ERR_BNDS) entries are returned.
380*>
381*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
382*> right-hand side.
383*>
384*> The second index in ERR_BNDS_COMP(:,err) contains the following
385*> three fields:
386*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
387*> reciprocal condition number is less than the threshold
388*> sqrt(n) * dlamch('Epsilon').
389*>
390*> err = 2 "Guaranteed" error bound: The estimated forward error,
391*> almost certainly within a factor of 10 of the true error
392*> so long as the next entry is greater than the threshold
393*> sqrt(n) * dlamch('Epsilon'). This error bound should only
394*> be trusted if the previous boolean is true.
395*>
396*> err = 3 Reciprocal condition number: Estimated componentwise
397*> reciprocal condition number. Compared with the threshold
398*> sqrt(n) * dlamch('Epsilon') to determine if the error
399*> estimate is "guaranteed". These reciprocal condition
400*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
401*> appropriately scaled matrix Z.
402*> Let Z = S*(A*diag(x)), where x is the solution for the
403*> current right-hand side and S scales each row of
404*> A*diag(x) by a power of the radix so all absolute row
405*> sums of Z are approximately 1.
406*>
407*> See Lapack Working Note 165 for further details and extra
408*> cautions.
409*> \endverbatim
410*>
411*> \param[in] NPARAMS
412*> \verbatim
413*> NPARAMS is INTEGER
414*> Specifies the number of parameters set in PARAMS. If <= 0, the
415*> PARAMS array is never referenced and default values are used.
416*> \endverbatim
417*>
418*> \param[in,out] PARAMS
419*> \verbatim
420*> PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
421*> Specifies algorithm parameters. If an entry is < 0.0, then
422*> that entry will be filled with default value used for that
423*> parameter. Only positions up to NPARAMS are accessed; defaults
424*> are used for higher-numbered parameters.
425*>
426*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
427*> refinement or not.
428*> Default: 1.0D+0
429*> = 0.0: No refinement is performed, and no error bounds are
430*> computed.
431*> = 1.0: Use the extra-precise refinement algorithm.
432*> (other values are reserved for future use)
433*>
434*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
435*> computations allowed for refinement.
436*> Default: 10
437*> Aggressive: Set to 100 to permit convergence using approximate
438*> factorizations or factorizations other than LU. If
439*> the factorization uses a technique other than
440*> Gaussian elimination, the guarantees in
441*> err_bnds_norm and err_bnds_comp may no longer be
442*> trustworthy.
443*>
444*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
445*> will attempt to find a solution with small componentwise
446*> relative error in the double-precision algorithm. Positive
447*> is true, 0.0 is false.
448*> Default: 1.0 (attempt componentwise convergence)
449*> \endverbatim
450*>
451*> \param[out] WORK
452*> \verbatim
453*> WORK is DOUBLE PRECISION array, dimension (4*N)
454*> \endverbatim
455*>
456*> \param[out] IWORK
457*> \verbatim
458*> IWORK is INTEGER array, dimension (N)
459*> \endverbatim
460*>
461*> \param[out] INFO
462*> \verbatim
463*> INFO is INTEGER
464*> = 0: Successful exit. The solution to every right-hand side is
465*> guaranteed.
466*> < 0: If INFO = -i, the i-th argument had an illegal value
467*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
468*> has been completed, but the factor U is exactly singular, so
469*> the solution and error bounds could not be computed. RCOND = 0
470*> is returned.
471*> = N+J: The solution corresponding to the Jth right-hand side is
472*> not guaranteed. The solutions corresponding to other right-
473*> hand sides K with K > J may not be guaranteed as well, but
474*> only the first such right-hand side is reported. If a small
475*> componentwise error is not requested (PARAMS(3) = 0.0) then
476*> the Jth right-hand side is the first with a normwise error
477*> bound that is not guaranteed (the smallest J such
478*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
479*> the Jth right-hand side is the first with either a normwise or
480*> componentwise error bound that is not guaranteed (the smallest
481*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
482*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
483*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
484*> about all of the right-hand sides check ERR_BNDS_NORM or
485*> ERR_BNDS_COMP.
486*> \endverbatim
487*
488* Authors:
489* ========
490*
491*> \author Univ. of Tennessee
492*> \author Univ. of California Berkeley
493*> \author Univ. of Colorado Denver
494*> \author NAG Ltd.
495*
496*> \ingroup hesvxx
497*
498* =====================================================================
499 SUBROUTINE dsysvxx( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF,
500 $ IPIV,
501 $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
502 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
503 $ NPARAMS, PARAMS, WORK, IWORK, INFO )
504*
505* -- LAPACK driver routine --
506* -- LAPACK is a software package provided by Univ. of Tennessee, --
507* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
508*
509* .. Scalar Arguments ..
510 CHARACTER EQUED, FACT, UPLO
511 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
512 $ N_ERR_BNDS
513 DOUBLE PRECISION RCOND, RPVGRW
514* ..
515* .. Array Arguments ..
516 INTEGER IPIV( * ), IWORK( * )
517 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
518 $ X( LDX, * ), WORK( * )
519 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
520 $ err_bnds_norm( nrhs, * ),
521 $ err_bnds_comp( nrhs, * )
522* ..
523*
524* ==================================================================
525*
526* .. Parameters ..
527 DOUBLE PRECISION ZERO, ONE
528 PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
529 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
530 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
531 INTEGER CMP_ERR_I, PIV_GROWTH_I
532 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
533 $ berr_i = 3 )
534 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
535 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
536 $ piv_growth_i = 9 )
537* ..
538* .. Local Scalars ..
539 LOGICAL EQUIL, NOFACT, RCEQU
540 INTEGER INFEQU, J
541 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
542* ..
543* .. External Functions ..
544 EXTERNAL lsame, dlamch, dla_syrpvgrw
545 LOGICAL LSAME
546 DOUBLE PRECISION DLAMCH, DLA_SYRPVGRW
547* ..
548* .. External Subroutines ..
549 EXTERNAL dsyequb, dsytrf, dsytrs,
551* ..
552* .. Intrinsic Functions ..
553 INTRINSIC max, min
554* ..
555* .. Executable Statements ..
556*
557 info = 0
558 nofact = lsame( fact, 'N' )
559 equil = lsame( fact, 'E' )
560 smlnum = dlamch( 'Safe minimum' )
561 bignum = one / smlnum
562 IF( nofact .OR. equil ) THEN
563 equed = 'N'
564 rcequ = .false.
565 ELSE
566 rcequ = lsame( equed, 'Y' )
567 ENDIF
568*
569* Default is failure. If an input parameter is wrong or
570* factorization fails, make everything look horrible. Only the
571* pivot growth is set here, the rest is initialized in DSYRFSX.
572*
573 rpvgrw = zero
574*
575* Test the input parameters. PARAMS is not tested until DSYRFSX.
576*
577 IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
578 $ lsame( fact, 'F' ) ) THEN
579 info = -1
580 ELSE IF( .NOT.lsame(uplo, 'U') .AND.
581 $ .NOT.lsame(uplo, 'L') ) THEN
582 info = -2
583 ELSE IF( n.LT.0 ) THEN
584 info = -3
585 ELSE IF( nrhs.LT.0 ) THEN
586 info = -4
587 ELSE IF( lda.LT.max( 1, n ) ) THEN
588 info = -6
589 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
590 info = -8
591 ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
592 $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
593 info = -10
594 ELSE
595 IF ( rcequ ) THEN
596 smin = bignum
597 smax = zero
598 DO 10 j = 1, n
599 smin = min( smin, s( j ) )
600 smax = max( smax, s( j ) )
601 10 CONTINUE
602 IF( smin.LE.zero ) THEN
603 info = -11
604 ELSE IF( n.GT.0 ) THEN
605 scond = max( smin, smlnum ) / min( smax, bignum )
606 ELSE
607 scond = one
608 END IF
609 END IF
610 IF( info.EQ.0 ) THEN
611 IF( ldb.LT.max( 1, n ) ) THEN
612 info = -13
613 ELSE IF( ldx.LT.max( 1, n ) ) THEN
614 info = -15
615 END IF
616 END IF
617 END IF
618*
619 IF( info.NE.0 ) THEN
620 CALL xerbla( 'DSYSVXX', -info )
621 RETURN
622 END IF
623*
624 IF( equil ) THEN
625*
626* Compute row and column scalings to equilibrate the matrix A.
627*
628 CALL dsyequb( uplo, n, a, lda, s, scond, amax, work,
629 $ infequ )
630 IF( infequ.EQ.0 ) THEN
631*
632* Equilibrate the matrix.
633*
634 CALL dlaqsy( uplo, n, a, lda, s, scond, amax, equed )
635 rcequ = lsame( equed, 'Y' )
636 END IF
637 END IF
638*
639* Scale the right-hand side.
640*
641 IF( rcequ ) CALL dlascl2( n, nrhs, s, b, ldb )
642*
643 IF( nofact .OR. equil ) THEN
644*
645* Compute the LDL^T or UDU^T factorization of A.
646*
647 CALL dlacpy( uplo, n, n, a, lda, af, ldaf )
648 CALL dsytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n),
649 $ info )
650*
651* Return if INFO is non-zero.
652*
653 IF( info.GT.0 ) THEN
654*
655* Pivot in column INFO is exactly 0
656* Compute the reciprocal pivot growth factor of the
657* leading rank-deficient INFO columns of A.
658*
659 IF ( n.GT.0 )
660 $ rpvgrw = dla_syrpvgrw(uplo, n, info, a, lda, af,
661 $ ldaf, ipiv, work )
662 RETURN
663 END IF
664 END IF
665*
666* Compute the reciprocal pivot growth factor RPVGRW.
667*
668 IF ( n.GT.0 )
669 $ rpvgrw = dla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
670 $ ipiv, work )
671*
672* Compute the solution matrix X.
673*
674 CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
675 CALL dsytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
676*
677* Use iterative refinement to improve the computed solution and
678* compute error bounds and backward error estimates for it.
679*
680 CALL dsyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
681 $ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
682 $ err_bnds_comp, nparams, params, work, iwork, info )
683*
684* Scale solutions.
685*
686 IF ( rcequ ) THEN
687 CALL dlascl2 ( n, nrhs, s, x, ldx )
688 END IF
689*
690 RETURN
691*
692* End of DSYSVXX
693*
694 END
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dsyequb(uplo, n, a, lda, s, scond, amax, work, info)
DSYEQUB
Definition dsyequb.f:130
subroutine dsyrfsx(uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv, s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
DSYRFSX
Definition dsyrfsx.f:401
subroutine dsysvxx(fact, uplo, n, nrhs, a, lda, af, ldaf, ipiv, equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr, n_err_bnds, err_bnds_norm, err_bnds_comp, nparams, params, work, iwork, info)
DSYSVXX
Definition dsysvxx.f:504
subroutine dsytrf(uplo, n, a, lda, ipiv, work, lwork, info)
DSYTRF
Definition dsytrf.f:180
subroutine dsytrs(uplo, n, nrhs, a, lda, ipiv, b, ldb, info)
DSYTRS
Definition dsytrs.f:118
double precision function dla_syrpvgrw(uplo, n, info, a, lda, af, ldaf, ipiv, work)
DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite m...
subroutine dlacpy(uplo, m, n, a, lda, b, ldb)
DLACPY copies all or part of one two-dimensional array to another.
Definition dlacpy.f:101
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
subroutine dlaqsy(uplo, n, a, lda, s, scond, amax, equed)
DLAQSY scales a symmetric/Hermitian matrix, using scaling factors computed by spoequ.
Definition dlaqsy.f:131
subroutine dlascl2(m, n, d, x, ldx)
DLASCL2 performs diagonal scaling on a matrix.
Definition dlascl2.f:88
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48