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1: *> \brief <b> DPOSVXX 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
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPOSVXX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvxx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvxx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvxx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPOSVXX( 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: * DOUBLE PRECISION RCOND, RPVGRW
31: * ..
32: * .. Array Arguments ..
33: * INTEGER 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: *> DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
48: *> to compute the solution to a double precision 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. DPOSVXX 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: *> DPOSVXX 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: *> DPOSVXX 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 DPOSVXX 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', double precision 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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 .LT. 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) * dlamch('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) * dlamch('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) * dlamch('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 .LE. 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 DOUBLE PRECISION array, dimension NPARAMS
412: *> Specifies algorithm parameters. If an entry is .LT. 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: 1.0D+0
420: *> = 0.0 : No refinement is performed, and no error bounds are
421: *> computed.
422: *> = 1.0 : Use the extra-precise refinement algorithm.
423: *> (other values are reserved for future use)
424: *>
425: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
426: *> computations allowed for refinement.
427: *> Default: 10
428: *> Aggressive: Set to 100 to permit convergence using approximate
429: *> factorizations or factorizations other than LU. If
430: *> the factorization uses a technique other than
431: *> Gaussian elimination, the guarantees in
432: *> err_bnds_norm and err_bnds_comp may no longer be
433: *> trustworthy.
434: *>
435: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
436: *> will attempt to find a solution with small componentwise
437: *> relative error in the double-precision algorithm. Positive
438: *> is true, 0.0 is false.
439: *> Default: 1.0 (attempt componentwise convergence)
440: *> \endverbatim
441: *>
442: *> \param[out] WORK
443: *> \verbatim
444: *> WORK is DOUBLE PRECISION array, dimension (4*N)
445: *> \endverbatim
446: *>
447: *> \param[out] IWORK
448: *> \verbatim
449: *> IWORK is INTEGER array, dimension (N)
450: *> \endverbatim
451: *>
452: *> \param[out] INFO
453: *> \verbatim
454: *> INFO is INTEGER
455: *> = 0: Successful exit. The solution to every right-hand side is
456: *> guaranteed.
457: *> < 0: If INFO = -i, the i-th argument had an illegal value
458: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
459: *> has been completed, but the factor U is exactly singular, so
460: *> the solution and error bounds could not be computed. RCOND = 0
461: *> is returned.
462: *> = N+J: The solution corresponding to the Jth right-hand side is
463: *> not guaranteed. The solutions corresponding to other right-
464: *> hand sides K with K > J may not be guaranteed as well, but
465: *> only the first such right-hand side is reported. If a small
466: *> componentwise error is not requested (PARAMS(3) = 0.0) then
467: *> the Jth right-hand side is the first with a normwise error
468: *> bound that is not guaranteed (the smallest J such
469: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
470: *> the Jth right-hand side is the first with either a normwise or
471: *> componentwise error bound that is not guaranteed (the smallest
472: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
473: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
474: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
475: *> about all of the right-hand sides check ERR_BNDS_NORM or
476: *> ERR_BNDS_COMP.
477: *> \endverbatim
478: *
479: * Authors:
480: * ========
481: *
482: *> \author Univ. of Tennessee
483: *> \author Univ. of California Berkeley
484: *> \author Univ. of Colorado Denver
485: *> \author NAG Ltd.
486: *
487: *> \date April 2012
488: *
489: *> \ingroup doublePOsolve
490: *
491: * =====================================================================
492: SUBROUTINE DPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
493: $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
494: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
495: $ NPARAMS, PARAMS, WORK, IWORK, INFO )
496: *
497: * -- LAPACK driver routine (version 3.4.1) --
498: * -- LAPACK is a software package provided by Univ. of Tennessee, --
499: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
500: * April 2012
501: *
502: * .. Scalar Arguments ..
503: CHARACTER EQUED, FACT, UPLO
504: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
505: $ N_ERR_BNDS
506: DOUBLE PRECISION RCOND, RPVGRW
507: * ..
508: * .. Array Arguments ..
509: INTEGER IWORK( * )
510: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
511: $ X( LDX, * ), WORK( * )
512: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
513: $ ERR_BNDS_NORM( NRHS, * ),
514: $ ERR_BNDS_COMP( NRHS, * )
515: * ..
516: *
517: * ==================================================================
518: *
519: * .. Parameters ..
520: DOUBLE PRECISION ZERO, ONE
521: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+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: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX,
535: $ SCOND, SMLNUM
536: * ..
537: * .. External Functions ..
538: EXTERNAL LSAME, DLAMCH, DLA_PORPVGRW
539: LOGICAL LSAME
540: DOUBLE PRECISION DLAMCH, DLA_PORPVGRW
541: * ..
542: * .. External Subroutines ..
543: EXTERNAL DPOEQUB, DPOTRF, DPOTRS, DLACPY, DLAQSY,
544: $ XERBLA, DLASCL2, DPORFSX
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 = DLAMCH( '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 DPORFSX.
566: *
567: RPVGRW = ZERO
568: *
569: * Test the input parameters. PARAMS is not tested until DPORFSX.
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( 'DPOSVXX', -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 DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
623: IF( INFEQU.EQ.0 ) THEN
624: *
625: * Equilibrate the matrix.
626: *
627: CALL DLAQSY( 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 DLASCL2( N, NRHS, S, B, LDB )
635: *
636: IF( NOFACT .OR. EQUIL ) THEN
637: *
638: * Compute the Cholesky factorization of A.
639: *
640: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
641: CALL DPOTRF( 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 = DLA_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 = DLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK )
659: *
660: * Compute the solution matrix X.
661: *
662: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
663: CALL DPOTRS( 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 DPORFSX( 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 DLASCL2 ( N, NRHS, S, X, LDX )
677: END IF
678: *
679: RETURN
680: *
681: * End of DPOSVXX
682: *
683: END
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