1: *> \brief \b DSYSVXX
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
22: * EQUED, 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: * DOUBLE PRECISION RCOND, RPVGRW
31: * ..
32: * .. Array Arguments ..
33: * INTEGER IPIV( * ), IWORK( * )
34: * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35: * $ X( LDX, * ), WORK( * )
36: * DOUBLE PRECISION 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: *> DSYSVXX uses the diagonal pivoting factorization to compute the
48: *> solution to a double precision system of linear equations A * X = B, where A
49: *> is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.
50: *>
51: *> If requested, both normwise and maximum componentwise error bounds
52: *> are returned. DSYSVXX will return a solution with a tiny
53: *> guaranteed error (O(eps) where eps is the working machine
54: *> precision) unless the matrix is very ill-conditioned, in which
55: *> case a warning is returned. Relevant condition numbers also are
56: *> calculated and returned.
57: *>
58: *> DSYSVXX accepts user-provided factorizations and equilibration
59: *> factors; see the definitions of the FACT and EQUED options.
60: *> Solving with refinement and using a factorization from a previous
61: *> DSYSVXX call will also produce a solution with either O(eps)
62: *> errors or warnings, but we cannot make that claim for general
63: *> user-provided factorizations and equilibration factors if they
64: *> differ from what DSYSVXX would itself produce.
65: *> \endverbatim
66: *
67: *> \par Description:
68: * =================
69: *>
70: *> \verbatim
71: *>
72: *> The following steps are performed:
73: *>
74: *> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
75: *> the system:
76: *>
77: *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
78: *>
79: *> Whether or not the system will be equilibrated depends on the
80: *> scaling of the matrix A, but if equilibration is used, A is
81: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
82: *>
83: *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
84: *> the matrix A (after equilibration if FACT = 'E') as
85: *>
86: *> A = U * D * U**T, if UPLO = 'U', or
87: *> A = L * D * L**T, if UPLO = 'L',
88: *>
89: *> where U (or L) is a product of permutation and unit upper (lower)
90: *> triangular matrices, and D is symmetric and block diagonal with
91: *> 1-by-1 and 2-by-2 diagonal blocks.
92: *>
93: *> 3. If some D(i,i)=0, so that D is exactly singular, then the
94: *> routine returns with INFO = i. Otherwise, the factored form of A
95: *> is used to estimate the condition number of the matrix A (see
96: *> argument RCOND). If the reciprocal of the condition number is
97: *> less than machine precision, the routine still goes on to solve
98: *> for X and compute error bounds as described below.
99: *>
100: *> 4. The system of equations is solved for X using the factored form
101: *> of A.
102: *>
103: *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
104: *> the routine will use iterative refinement to try to get a small
105: *> error and error bounds. Refinement calculates the residual to at
106: *> least twice the working precision.
107: *>
108: *> 6. If equilibration was used, the matrix X is premultiplied by
109: *> diag(R) so that it solves the original system before
110: *> equilibration.
111: *> \endverbatim
112: *
113: * Arguments:
114: * ==========
115: *
116: *> \verbatim
117: *> Some optional parameters are bundled in the PARAMS array. These
118: *> settings determine how refinement is performed, but often the
119: *> defaults are acceptable. If the defaults are acceptable, users
120: *> can pass NPARAMS = 0 which prevents the source code from accessing
121: *> the PARAMS argument.
122: *> \endverbatim
123: *>
124: *> \param[in] FACT
125: *> \verbatim
126: *> FACT is CHARACTER*1
127: *> Specifies whether or not the factored form of the matrix A is
128: *> supplied on entry, and if not, whether the matrix A should be
129: *> equilibrated before it is factored.
130: *> = 'F': On entry, AF and IPIV contain the factored form of A.
131: *> If EQUED is not 'N', the matrix A has been
132: *> equilibrated with scaling factors given by S.
133: *> A, AF, and IPIV are not modified.
134: *> = 'N': The matrix A will be copied to AF and factored.
135: *> = 'E': The matrix A will be equilibrated if necessary, then
136: *> copied to AF and factored.
137: *> \endverbatim
138: *>
139: *> \param[in] UPLO
140: *> \verbatim
141: *> UPLO is CHARACTER*1
142: *> = 'U': Upper triangle of A is stored;
143: *> = 'L': Lower triangle of A is stored.
144: *> \endverbatim
145: *>
146: *> \param[in] N
147: *> \verbatim
148: *> N is INTEGER
149: *> The number of linear equations, i.e., the order of the
150: *> matrix A. N >= 0.
151: *> \endverbatim
152: *>
153: *> \param[in] NRHS
154: *> \verbatim
155: *> NRHS is INTEGER
156: *> The number of right hand sides, i.e., the number of columns
157: *> of the matrices B and X. NRHS >= 0.
158: *> \endverbatim
159: *>
160: *> \param[in,out] A
161: *> \verbatim
162: *> A is DOUBLE PRECISION array, dimension (LDA,N)
163: *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
164: *> upper triangular part of A contains the upper triangular
165: *> part of the matrix A, and the strictly lower triangular
166: *> part of A is not referenced. If UPLO = 'L', the leading
167: *> N-by-N lower triangular part of A contains the lower
168: *> triangular part of the matrix A, and the strictly upper
169: *> triangular part of A is not referenced.
170: *>
171: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
172: *> diag(S)*A*diag(S).
173: *> \endverbatim
174: *>
175: *> \param[in] LDA
176: *> \verbatim
177: *> LDA is INTEGER
178: *> The leading dimension of the array A. LDA >= max(1,N).
179: *> \endverbatim
180: *>
181: *> \param[in,out] AF
182: *> \verbatim
183: *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
184: *> If FACT = 'F', then AF is an input argument and on entry
185: *> contains the block diagonal matrix D and the multipliers
186: *> used to obtain the factor U or L from the factorization A =
187: *> U*D*U**T or A = L*D*L**T as computed by DSYTRF.
188: *>
189: *> If FACT = 'N', then AF is an output argument and on exit
190: *> returns the block diagonal matrix D and the multipliers
191: *> used to obtain the factor U or L from the factorization A =
192: *> U*D*U**T or A = L*D*L**T.
193: *> \endverbatim
194: *>
195: *> \param[in] LDAF
196: *> \verbatim
197: *> LDAF is INTEGER
198: *> The leading dimension of the array AF. LDAF >= max(1,N).
199: *> \endverbatim
200: *>
201: *> \param[in,out] IPIV
202: *> \verbatim
203: *> IPIV is INTEGER array, dimension (N)
204: *> If FACT = 'F', then IPIV is an input argument and on entry
205: *> contains details of the interchanges and the block
206: *> structure of D, as determined by DSYTRF. If IPIV(k) > 0,
207: *> then rows and columns k and IPIV(k) were interchanged and
208: *> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and
209: *> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
210: *> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
211: *> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
212: *> then rows and columns k+1 and -IPIV(k) were interchanged
213: *> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
214: *>
215: *> If FACT = 'N', then IPIV is an output argument and on exit
216: *> contains details of the interchanges and the block
217: *> structure of D, as determined by DSYTRF.
218: *> \endverbatim
219: *>
220: *> \param[in,out] EQUED
221: *> \verbatim
222: *> EQUED is CHARACTER*1
223: *> Specifies the form of equilibration that was done.
224: *> = 'N': No equilibration (always true if FACT = 'N').
225: *> = 'Y': Both row and column equilibration, i.e., A has been
226: *> replaced by diag(S) * A * diag(S).
227: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
228: *> output argument.
229: *> \endverbatim
230: *>
231: *> \param[in,out] S
232: *> \verbatim
233: *> S is DOUBLE PRECISION array, dimension (N)
234: *> The scale factors for A. If EQUED = 'Y', A is multiplied on
235: *> the left and right by diag(S). S is an input argument if FACT =
236: *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
237: *> = 'Y', each element of S must be positive. If S is output, each
238: *> element of S is a power of the radix. If S is input, each element
239: *> of S should be a power of the radix to ensure a reliable solution
240: *> and error estimates. Scaling by powers of the radix does not cause
241: *> rounding errors unless the result underflows or overflows.
242: *> Rounding errors during scaling lead to refining with a matrix that
243: *> is not equivalent to the input matrix, producing error estimates
244: *> that may not be reliable.
245: *> \endverbatim
246: *>
247: *> \param[in,out] B
248: *> \verbatim
249: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
250: *> On entry, the N-by-NRHS right hand side matrix B.
251: *> On exit,
252: *> if EQUED = 'N', B is not modified;
253: *> if EQUED = 'Y', B is overwritten by diag(S)*B;
254: *> \endverbatim
255: *>
256: *> \param[in] LDB
257: *> \verbatim
258: *> LDB is INTEGER
259: *> The leading dimension of the array B. LDB >= max(1,N).
260: *> \endverbatim
261: *>
262: *> \param[out] X
263: *> \verbatim
264: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
265: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
266: *> system of equations. Note that A and B are modified on exit if
267: *> EQUED .ne. 'N', and the solution to the equilibrated system is
268: *> inv(diag(S))*X.
269: *> \endverbatim
270: *>
271: *> \param[in] LDX
272: *> \verbatim
273: *> LDX is INTEGER
274: *> The leading dimension of the array X. LDX >= max(1,N).
275: *> \endverbatim
276: *>
277: *> \param[out] RCOND
278: *> \verbatim
279: *> RCOND is DOUBLE PRECISION
280: *> Reciprocal scaled condition number. This is an estimate of the
281: *> reciprocal Skeel condition number of the matrix A after
282: *> equilibration (if done). If this is less than the machine
283: *> precision (in particular, if it is zero), the matrix is singular
284: *> to working precision. Note that the error may still be small even
285: *> if this number is very small and the matrix appears ill-
286: *> conditioned.
287: *> \endverbatim
288: *>
289: *> \param[out] RPVGRW
290: *> \verbatim
291: *> RPVGRW is DOUBLE PRECISION
292: *> Reciprocal pivot growth. On exit, this contains the reciprocal
293: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
294: *> norm is used. If this is much less than 1, then the stability of
295: *> the LU factorization of the (equilibrated) matrix A could be poor.
296: *> This also means that the solution X, estimated condition numbers,
297: *> and error bounds could be unreliable. If factorization fails with
298: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
299: *> for the leading INFO columns of A.
300: *> \endverbatim
301: *>
302: *> \param[out] BERR
303: *> \verbatim
304: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
305: *> Componentwise relative backward error. This is the
306: *> componentwise relative backward error of each solution vector X(j)
307: *> (i.e., the smallest relative change in any element of A or B that
308: *> makes X(j) an exact solution).
309: *> \endverbatim
310: *>
311: *> \param[in] N_ERR_BNDS
312: *> \verbatim
313: *> N_ERR_BNDS is INTEGER
314: *> Number of error bounds to return for each right hand side
315: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
316: *> ERR_BNDS_COMP below.
317: *> \endverbatim
318: *>
319: *> \param[out] ERR_BNDS_NORM
320: *> \verbatim
321: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
322: *> For each right-hand side, this array contains information about
323: *> various error bounds and condition numbers corresponding to the
324: *> normwise relative error, which is defined as follows:
325: *>
326: *> Normwise relative error in the ith solution vector:
327: *> max_j (abs(XTRUE(j,i) - X(j,i)))
328: *> ------------------------------
329: *> max_j abs(X(j,i))
330: *>
331: *> The array is indexed by the type of error information as described
332: *> below. There currently are up to three pieces of information
333: *> returned.
334: *>
335: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
336: *> right-hand side.
337: *>
338: *> The second index in ERR_BNDS_NORM(:,err) contains the following
339: *> three fields:
340: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
341: *> reciprocal condition number is less than the threshold
342: *> sqrt(n) * dlamch('Epsilon').
343: *>
344: *> err = 2 "Guaranteed" error bound: The estimated forward error,
345: *> almost certainly within a factor of 10 of the true error
346: *> so long as the next entry is greater than the threshold
347: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
348: *> be trusted if the previous boolean is true.
349: *>
350: *> err = 3 Reciprocal condition number: Estimated normwise
351: *> reciprocal condition number. Compared with the threshold
352: *> sqrt(n) * dlamch('Epsilon') to determine if the error
353: *> estimate is "guaranteed". These reciprocal condition
354: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
355: *> appropriately scaled matrix Z.
356: *> Let Z = S*A, where S scales each row by a power of the
357: *> radix so all absolute row sums of Z are approximately 1.
358: *>
359: *> See Lapack Working Note 165 for further details and extra
360: *> cautions.
361: *> \endverbatim
362: *>
363: *> \param[out] ERR_BNDS_COMP
364: *> \verbatim
365: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
366: *> For each right-hand side, this array contains information about
367: *> various error bounds and condition numbers corresponding to the
368: *> componentwise relative error, which is defined as follows:
369: *>
370: *> Componentwise relative error in the ith solution vector:
371: *> abs(XTRUE(j,i) - X(j,i))
372: *> max_j ----------------------
373: *> abs(X(j,i))
374: *>
375: *> The array is indexed by the right-hand side i (on which the
376: *> componentwise relative error depends), and the type of error
377: *> information as described below. There currently are up to three
378: *> pieces of information returned for each right-hand side. If
379: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
380: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
381: *> the first (:,N_ERR_BNDS) entries are returned.
382: *>
383: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
384: *> right-hand side.
385: *>
386: *> The second index in ERR_BNDS_COMP(:,err) contains the following
387: *> three fields:
388: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
389: *> reciprocal condition number is less than the threshold
390: *> sqrt(n) * dlamch('Epsilon').
391: *>
392: *> err = 2 "Guaranteed" error bound: The estimated forward error,
393: *> almost certainly within a factor of 10 of the true error
394: *> so long as the next entry is greater than the threshold
395: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
396: *> be trusted if the previous boolean is true.
397: *>
398: *> err = 3 Reciprocal condition number: Estimated componentwise
399: *> reciprocal condition number. Compared with the threshold
400: *> sqrt(n) * dlamch('Epsilon') to determine if the error
401: *> estimate is "guaranteed". These reciprocal condition
402: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
403: *> appropriately scaled matrix Z.
404: *> Let Z = S*(A*diag(x)), where x is the solution for the
405: *> current right-hand side and S scales each row of
406: *> A*diag(x) by a power of the radix so all absolute row
407: *> sums of Z are approximately 1.
408: *>
409: *> See Lapack Working Note 165 for further details and extra
410: *> cautions.
411: *> \endverbatim
412: *>
413: *> \param[in] NPARAMS
414: *> \verbatim
415: *> NPARAMS is INTEGER
416: *> Specifies the number of parameters set in PARAMS. If <= 0, the
417: *> PARAMS array is never referenced and default values are used.
418: *> \endverbatim
419: *>
420: *> \param[in,out] PARAMS
421: *> \verbatim
422: *> PARAMS is DOUBLE PRECISION array, dimension (NPARAMS)
423: *> Specifies algorithm parameters. If an entry is < 0.0, then
424: *> that entry will be filled with default value used for that
425: *> parameter. Only positions up to NPARAMS are accessed; defaults
426: *> are used for higher-numbered parameters.
427: *>
428: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
429: *> refinement or not.
430: *> Default: 1.0D+0
431: *> = 0.0: No refinement is performed, and no error bounds are
432: *> computed.
433: *> = 1.0: Use the extra-precise refinement algorithm.
434: *> (other values are reserved for future use)
435: *>
436: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
437: *> computations allowed for refinement.
438: *> Default: 10
439: *> Aggressive: Set to 100 to permit convergence using approximate
440: *> factorizations or factorizations other than LU. If
441: *> the factorization uses a technique other than
442: *> Gaussian elimination, the guarantees in
443: *> err_bnds_norm and err_bnds_comp may no longer be
444: *> trustworthy.
445: *>
446: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
447: *> will attempt to find a solution with small componentwise
448: *> relative error in the double-precision algorithm. Positive
449: *> is true, 0.0 is false.
450: *> Default: 1.0 (attempt componentwise convergence)
451: *> \endverbatim
452: *>
453: *> \param[out] WORK
454: *> \verbatim
455: *> WORK is DOUBLE PRECISION array, dimension (4*N)
456: *> \endverbatim
457: *>
458: *> \param[out] IWORK
459: *> \verbatim
460: *> IWORK is INTEGER array, dimension (N)
461: *> \endverbatim
462: *>
463: *> \param[out] INFO
464: *> \verbatim
465: *> INFO is INTEGER
466: *> = 0: Successful exit. The solution to every right-hand side is
467: *> guaranteed.
468: *> < 0: If INFO = -i, the i-th argument had an illegal value
469: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
470: *> has been completed, but the factor U is exactly singular, so
471: *> the solution and error bounds could not be computed. RCOND = 0
472: *> is returned.
473: *> = N+J: The solution corresponding to the Jth right-hand side is
474: *> not guaranteed. The solutions corresponding to other right-
475: *> hand sides K with K > J may not be guaranteed as well, but
476: *> only the first such right-hand side is reported. If a small
477: *> componentwise error is not requested (PARAMS(3) = 0.0) then
478: *> the Jth right-hand side is the first with a normwise error
479: *> bound that is not guaranteed (the smallest J such
480: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
481: *> the Jth right-hand side is the first with either a normwise or
482: *> componentwise error bound that is not guaranteed (the smallest
483: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
484: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
485: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
486: *> about all of the right-hand sides check ERR_BNDS_NORM or
487: *> ERR_BNDS_COMP.
488: *> \endverbatim
489: *
490: * Authors:
491: * ========
492: *
493: *> \author Univ. of Tennessee
494: *> \author Univ. of California Berkeley
495: *> \author Univ. of Colorado Denver
496: *> \author NAG Ltd.
497: *
498: *> \ingroup doubleSYsolve
499: *
500: * =====================================================================
501: SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
502: $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
503: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
504: $ NPARAMS, PARAMS, WORK, IWORK, INFO )
505: *
506: * -- LAPACK driver routine --
507: * -- LAPACK is a software package provided by Univ. of Tennessee, --
508: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
509: *
510: * .. Scalar Arguments ..
511: CHARACTER EQUED, FACT, UPLO
512: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
513: $ N_ERR_BNDS
514: DOUBLE PRECISION RCOND, RPVGRW
515: * ..
516: * .. Array Arguments ..
517: INTEGER IPIV( * ), IWORK( * )
518: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
519: $ X( LDX, * ), WORK( * )
520: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
521: $ ERR_BNDS_NORM( NRHS, * ),
522: $ ERR_BNDS_COMP( NRHS, * )
523: * ..
524: *
525: * ==================================================================
526: *
527: * .. Parameters ..
528: DOUBLE PRECISION ZERO, ONE
529: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
530: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
531: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
532: INTEGER CMP_ERR_I, PIV_GROWTH_I
533: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
534: $ BERR_I = 3 )
535: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
536: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
537: $ PIV_GROWTH_I = 9 )
538: * ..
539: * .. Local Scalars ..
540: LOGICAL EQUIL, NOFACT, RCEQU
541: INTEGER INFEQU, J
542: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
543: * ..
544: * .. External Functions ..
545: EXTERNAL LSAME, DLAMCH, DLA_SYRPVGRW
546: LOGICAL LSAME
547: DOUBLE PRECISION DLAMCH, DLA_SYRPVGRW
548: * ..
549: * .. External Subroutines ..
550: EXTERNAL DSYEQUB, DSYTRF, DSYTRS,
551: $ DLACPY, DLAQSY, XERBLA, DLASCL2, DSYRFSX
552: * ..
553: * .. Intrinsic Functions ..
554: INTRINSIC MAX, MIN
555: * ..
556: * .. Executable Statements ..
557: *
558: INFO = 0
559: NOFACT = LSAME( FACT, 'N' )
560: EQUIL = LSAME( FACT, 'E' )
561: SMLNUM = DLAMCH( 'Safe minimum' )
562: BIGNUM = ONE / SMLNUM
563: IF( NOFACT .OR. EQUIL ) THEN
564: EQUED = 'N'
565: RCEQU = .FALSE.
566: ELSE
567: RCEQU = LSAME( EQUED, 'Y' )
568: ENDIF
569: *
570: * Default is failure. If an input parameter is wrong or
571: * factorization fails, make everything look horrible. Only the
572: * pivot growth is set here, the rest is initialized in DSYRFSX.
573: *
574: RPVGRW = ZERO
575: *
576: * Test the input parameters. PARAMS is not tested until DSYRFSX.
577: *
578: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
579: $ LSAME( FACT, 'F' ) ) THEN
580: INFO = -1
581: ELSE IF( .NOT.LSAME(UPLO, 'U') .AND.
582: $ .NOT.LSAME(UPLO, 'L') ) THEN
583: INFO = -2
584: ELSE IF( N.LT.0 ) THEN
585: INFO = -3
586: ELSE IF( NRHS.LT.0 ) THEN
587: INFO = -4
588: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
589: INFO = -6
590: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
591: INFO = -8
592: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
593: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
594: INFO = -10
595: ELSE
596: IF ( RCEQU ) THEN
597: SMIN = BIGNUM
598: SMAX = ZERO
599: DO 10 J = 1, N
600: SMIN = MIN( SMIN, S( J ) )
601: SMAX = MAX( SMAX, S( J ) )
602: 10 CONTINUE
603: IF( SMIN.LE.ZERO ) THEN
604: INFO = -11
605: ELSE IF( N.GT.0 ) THEN
606: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
607: ELSE
608: SCOND = ONE
609: END IF
610: END IF
611: IF( INFO.EQ.0 ) THEN
612: IF( LDB.LT.MAX( 1, N ) ) THEN
613: INFO = -13
614: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
615: INFO = -15
616: END IF
617: END IF
618: END IF
619: *
620: IF( INFO.NE.0 ) THEN
621: CALL XERBLA( 'DSYSVXX', -INFO )
622: RETURN
623: END IF
624: *
625: IF( EQUIL ) THEN
626: *
627: * Compute row and column scalings to equilibrate the matrix A.
628: *
629: CALL DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, 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), INFO )
649: *
650: * Return if INFO is non-zero.
651: *
652: IF( INFO.GT.0 ) THEN
653: *
654: * Pivot in column INFO is exactly 0
655: * Compute the reciprocal pivot growth factor of the
656: * leading rank-deficient INFO columns of A.
657: *
658: IF ( N.GT.0 )
659: $ RPVGRW = DLA_SYRPVGRW(UPLO, N, INFO, A, LDA, AF,
660: $ LDAF, IPIV, WORK )
661: RETURN
662: END IF
663: END IF
664: *
665: * Compute the reciprocal pivot growth factor RPVGRW.
666: *
667: IF ( N.GT.0 )
668: $ RPVGRW = DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
669: $ IPIV, WORK )
670: *
671: * Compute the solution matrix X.
672: *
673: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
674: CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
675: *
676: * Use iterative refinement to improve the computed solution and
677: * compute error bounds and backward error estimates for it.
678: *
679: CALL DSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
680: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
681: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO )
682: *
683: * Scale solutions.
684: *
685: IF ( RCEQU ) THEN
686: CALL DLASCL2 ( N, NRHS, S, X, LDX )
687: END IF
688: *
689: RETURN
690: *
691: * End of DSYSVXX
692: *
693: END
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