Annotation of rpl/lapack/lapack/zposvxx.f, revision 1.9
1.5 bertrand 1: *> \brief <b> ZPOSVXX 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 ZPOSVXX + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zposvxx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zposvxx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPOSVXX( 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, RWORK, 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: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
34: * $ WORK( * ), X( LDX, * )
35: * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
36: * $ ERR_BNDS_NORM( NRHS, * ),
37: * $ ERR_BNDS_COMP( NRHS, * )
38: * ..
39: *
40: *
41: *> \par Purpose:
42: * =============
43: *>
44: *> \verbatim
45: *>
46: *> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
47: *> to compute the solution to a complex*16 system of linear equations
48: *> A * X = B, where A is an N-by-N symmetric positive definite matrix
49: *> and X and B are N-by-NRHS matrices.
50: *>
51: *> If requested, both normwise and maximum componentwise error bounds
52: *> are returned. ZPOSVXX 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: *> ZPOSVXX 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: *> ZPOSVXX 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 ZPOSVXX 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 Cholesky decomposition is used to
84: *> factor the matrix A (after equilibration if FACT = 'E') as
85: *> A = U**T* U, if UPLO = 'U', or
86: *> A = L * L**T, if UPLO = 'L',
87: *> where U is an upper triangular matrix and L is a lower triangular
88: *> matrix.
89: *>
90: *> 3. If the leading i-by-i principal minor is not positive definite,
91: *> then the routine returns with INFO = i. Otherwise, the factored
92: *> form of A is used to estimate the condition number of the matrix
93: *> A (see argument RCOND). If the reciprocal of the condition number
94: *> is less than machine precision, the routine still goes on to solve
95: *> for X and compute error bounds as described below.
96: *>
97: *> 4. The system of equations is solved for X using the factored form
98: *> of A.
99: *>
100: *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
101: *> the routine will use iterative refinement to try to get a small
102: *> error and error bounds. Refinement calculates the residual to at
103: *> least twice the working precision.
104: *>
105: *> 6. If equilibration was used, the matrix X is premultiplied by
106: *> diag(S) so that it solves the original system before
107: *> equilibration.
108: *> \endverbatim
109: *
110: * Arguments:
111: * ==========
112: *
113: *> \verbatim
114: *> Some optional parameters are bundled in the PARAMS array. These
115: *> settings determine how refinement is performed, but often the
116: *> defaults are acceptable. If the defaults are acceptable, users
117: *> can pass NPARAMS = 0 which prevents the source code from accessing
118: *> the PARAMS argument.
119: *> \endverbatim
120: *>
121: *> \param[in] FACT
122: *> \verbatim
123: *> FACT is CHARACTER*1
124: *> Specifies whether or not the factored form of the matrix A is
125: *> supplied on entry, and if not, whether the matrix A should be
126: *> equilibrated before it is factored.
127: *> = 'F': On entry, AF contains the factored form of A.
128: *> If EQUED is not 'N', the matrix A has been
129: *> equilibrated with scaling factors given by S.
130: *> A and AF are not modified.
131: *> = 'N': The matrix A will be copied to AF and factored.
132: *> = 'E': The matrix A will be equilibrated if necessary, then
133: *> copied to AF and factored.
134: *> \endverbatim
135: *>
136: *> \param[in] UPLO
137: *> \verbatim
138: *> UPLO is CHARACTER*1
139: *> = 'U': Upper triangle of A is stored;
140: *> = 'L': Lower triangle of A is stored.
141: *> \endverbatim
142: *>
143: *> \param[in] N
144: *> \verbatim
145: *> N is INTEGER
146: *> The number of linear equations, i.e., the order of the
147: *> matrix A. N >= 0.
148: *> \endverbatim
149: *>
150: *> \param[in] NRHS
151: *> \verbatim
152: *> NRHS is INTEGER
153: *> The number of right hand sides, i.e., the number of columns
154: *> of the matrices B and X. NRHS >= 0.
155: *> \endverbatim
156: *>
157: *> \param[in,out] A
158: *> \verbatim
159: *> A is COMPLEX*16 array, dimension (LDA,N)
160: *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
161: *> 'Y', then A must contain the equilibrated matrix
162: *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
163: *> triangular part of A contains the upper triangular part of the
164: *> matrix A, and the strictly lower triangular part of A is not
165: *> referenced. If UPLO = 'L', the leading N-by-N lower triangular
166: *> part of A contains the lower triangular part of the matrix A, and
167: *> the strictly upper triangular part of A is not referenced. A is
168: *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
169: *> 'N' on exit.
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
1.7 bertrand 183: *> AF is COMPLEX*16 array, dimension (LDAF,N)
1.5 bertrand 184: *> If FACT = 'F', then AF is an input argument and on entry
185: *> contains the triangular factor U or L from the Cholesky
186: *> factorization A = U**T*U or A = L*L**T, in the same storage
187: *> format as A. If EQUED .ne. 'N', then AF is the factored
188: *> form of the equilibrated matrix diag(S)*A*diag(S).
189: *>
190: *> If FACT = 'N', then AF is an output argument and on exit
191: *> returns the triangular factor U or L from the Cholesky
192: *> factorization A = U**T*U or A = L*L**T of the original
193: *> matrix A.
194: *>
195: *> If FACT = 'E', then AF is an output argument and on exit
196: *> returns the triangular factor U or L from the Cholesky
197: *> factorization A = U**T*U or A = L*L**T of the equilibrated
198: *> matrix A (see the description of A for the form of the
199: *> equilibrated matrix).
200: *> \endverbatim
201: *>
202: *> \param[in] LDAF
203: *> \verbatim
204: *> LDAF is INTEGER
205: *> The leading dimension of the array AF. LDAF >= max(1,N).
206: *> \endverbatim
207: *>
208: *> \param[in,out] EQUED
209: *> \verbatim
1.7 bertrand 210: *> EQUED is CHARACTER*1
1.5 bertrand 211: *> Specifies the form of equilibration that was done.
212: *> = 'N': No equilibration (always true if FACT = 'N').
213: *> = 'Y': Both row and column equilibration, i.e., A has been
214: *> replaced by diag(S) * A * diag(S).
215: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
216: *> output argument.
217: *> \endverbatim
218: *>
219: *> \param[in,out] S
220: *> \verbatim
1.7 bertrand 221: *> S is DOUBLE PRECISION array, dimension (N)
1.5 bertrand 222: *> The row scale factors for A. If EQUED = 'Y', A is multiplied on
223: *> the left and right by diag(S). S is an input argument if FACT =
224: *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
225: *> = 'Y', each element of S must be positive. If S is output, each
226: *> element of S is a power of the radix. If S is input, each element
227: *> of S should be a power of the radix to ensure a reliable solution
228: *> and error estimates. Scaling by powers of the radix does not cause
229: *> rounding errors unless the result underflows or overflows.
230: *> Rounding errors during scaling lead to refining with a matrix that
231: *> is not equivalent to the input matrix, producing error estimates
232: *> that may not be reliable.
233: *> \endverbatim
234: *>
235: *> \param[in,out] B
236: *> \verbatim
237: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
238: *> On entry, the N-by-NRHS right hand side matrix B.
239: *> On exit,
240: *> if EQUED = 'N', B is not modified;
241: *> if EQUED = 'Y', B is overwritten by diag(S)*B;
242: *> \endverbatim
243: *>
244: *> \param[in] LDB
245: *> \verbatim
246: *> LDB is INTEGER
247: *> The leading dimension of the array B. LDB >= max(1,N).
248: *> \endverbatim
249: *>
250: *> \param[out] X
251: *> \verbatim
252: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
253: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
254: *> system of equations. Note that A and B are modified on exit if
255: *> EQUED .ne. 'N', and the solution to the equilibrated system is
256: *> inv(diag(S))*X.
257: *> \endverbatim
258: *>
259: *> \param[in] LDX
260: *> \verbatim
261: *> LDX is INTEGER
262: *> The leading dimension of the array X. LDX >= max(1,N).
263: *> \endverbatim
264: *>
265: *> \param[out] RCOND
266: *> \verbatim
267: *> RCOND is DOUBLE PRECISION
268: *> Reciprocal scaled condition number. This is an estimate of the
269: *> reciprocal Skeel condition number of the matrix A after
270: *> equilibration (if done). If this is less than the machine
271: *> precision (in particular, if it is zero), the matrix is singular
272: *> to working precision. Note that the error may still be small even
273: *> if this number is very small and the matrix appears ill-
274: *> conditioned.
275: *> \endverbatim
276: *>
277: *> \param[out] RPVGRW
278: *> \verbatim
279: *> RPVGRW is DOUBLE PRECISION
280: *> Reciprocal pivot growth. On exit, this contains the reciprocal
281: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
282: *> norm is used. If this is much less than 1, then the stability of
283: *> the LU factorization of the (equilibrated) matrix A could be poor.
284: *> This also means that the solution X, estimated condition numbers,
285: *> and error bounds could be unreliable. If factorization fails with
286: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
287: *> for the leading INFO columns of A.
288: *> \endverbatim
289: *>
290: *> \param[out] BERR
291: *> \verbatim
292: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
293: *> Componentwise relative backward error. This is the
294: *> componentwise relative backward error of each solution vector X(j)
295: *> (i.e., the smallest relative change in any element of A or B that
296: *> makes X(j) an exact solution).
297: *> \endverbatim
298: *>
299: *> \param[in] N_ERR_BNDS
300: *> \verbatim
301: *> N_ERR_BNDS is INTEGER
302: *> Number of error bounds to return for each right hand side
303: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
304: *> ERR_BNDS_COMP below.
305: *> \endverbatim
306: *>
307: *> \param[out] ERR_BNDS_NORM
308: *> \verbatim
309: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
310: *> For each right-hand side, this array contains information about
311: *> various error bounds and condition numbers corresponding to the
312: *> normwise relative error, which is defined as follows:
313: *>
314: *> Normwise relative error in the ith solution vector:
315: *> max_j (abs(XTRUE(j,i) - X(j,i)))
316: *> ------------------------------
317: *> max_j abs(X(j,i))
318: *>
319: *> The array is indexed by the type of error information as described
320: *> below. There currently are up to three pieces of information
321: *> returned.
322: *>
323: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
324: *> right-hand side.
325: *>
326: *> The second index in ERR_BNDS_NORM(:,err) contains the following
327: *> three fields:
328: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
329: *> reciprocal condition number is less than the threshold
330: *> sqrt(n) * dlamch('Epsilon').
331: *>
332: *> err = 2 "Guaranteed" error bound: The estimated forward error,
333: *> almost certainly within a factor of 10 of the true error
334: *> so long as the next entry is greater than the threshold
335: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
336: *> be trusted if the previous boolean is true.
337: *>
338: *> err = 3 Reciprocal condition number: Estimated normwise
339: *> reciprocal condition number. Compared with the threshold
340: *> sqrt(n) * dlamch('Epsilon') to determine if the error
341: *> estimate is "guaranteed". These reciprocal condition
342: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
343: *> appropriately scaled matrix Z.
344: *> Let Z = S*A, where S scales each row by a power of the
345: *> radix so all absolute row sums of Z are approximately 1.
346: *>
347: *> See Lapack Working Note 165 for further details and extra
348: *> cautions.
349: *> \endverbatim
350: *>
351: *> \param[out] ERR_BNDS_COMP
352: *> \verbatim
353: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
354: *> For each right-hand side, this array contains information about
355: *> various error bounds and condition numbers corresponding to the
356: *> componentwise relative error, which is defined as follows:
357: *>
358: *> Componentwise relative error in the ith solution vector:
359: *> abs(XTRUE(j,i) - X(j,i))
360: *> max_j ----------------------
361: *> abs(X(j,i))
362: *>
363: *> The array is indexed by the right-hand side i (on which the
364: *> componentwise relative error depends), and the type of error
365: *> information as described below. There currently are up to three
366: *> pieces of information returned for each right-hand side. If
367: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
368: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
369: *> the first (:,N_ERR_BNDS) entries are returned.
370: *>
371: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
372: *> right-hand side.
373: *>
374: *> The second index in ERR_BNDS_COMP(:,err) contains the following
375: *> three fields:
376: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
377: *> reciprocal condition number is less than the threshold
378: *> sqrt(n) * dlamch('Epsilon').
379: *>
380: *> err = 2 "Guaranteed" error bound: The estimated forward error,
381: *> almost certainly within a factor of 10 of the true error
382: *> so long as the next entry is greater than the threshold
383: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
384: *> be trusted if the previous boolean is true.
385: *>
386: *> err = 3 Reciprocal condition number: Estimated componentwise
387: *> reciprocal condition number. Compared with the threshold
388: *> sqrt(n) * dlamch('Epsilon') to determine if the error
389: *> estimate is "guaranteed". These reciprocal condition
390: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
391: *> appropriately scaled matrix Z.
392: *> Let Z = S*(A*diag(x)), where x is the solution for the
393: *> current right-hand side and S scales each row of
394: *> A*diag(x) by a power of the radix so all absolute row
395: *> sums of Z are approximately 1.
396: *>
397: *> See Lapack Working Note 165 for further details and extra
398: *> cautions.
399: *> \endverbatim
400: *>
401: *> \param[in] NPARAMS
402: *> \verbatim
403: *> NPARAMS is INTEGER
404: *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
405: *> PARAMS array is never referenced and default values are used.
406: *> \endverbatim
407: *>
408: *> \param[in,out] PARAMS
409: *> \verbatim
1.7 bertrand 410: *> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
1.5 bertrand 411: *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
412: *> that entry will be filled with default value used for that
413: *> parameter. Only positions up to NPARAMS are accessed; defaults
414: *> are used for higher-numbered parameters.
415: *>
416: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
417: *> refinement or not.
418: *> Default: 1.0D+0
419: *> = 0.0 : No refinement is performed, and no error bounds are
420: *> computed.
421: *> = 1.0 : Use the extra-precise refinement algorithm.
422: *> (other values are reserved for future use)
423: *>
424: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
425: *> computations allowed for refinement.
426: *> Default: 10
427: *> Aggressive: Set to 100 to permit convergence using approximate
428: *> factorizations or factorizations other than LU. If
429: *> the factorization uses a technique other than
430: *> Gaussian elimination, the guarantees in
431: *> err_bnds_norm and err_bnds_comp may no longer be
432: *> trustworthy.
433: *>
434: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
435: *> will attempt to find a solution with small componentwise
436: *> relative error in the double-precision algorithm. Positive
437: *> is true, 0.0 is false.
438: *> Default: 1.0 (attempt componentwise convergence)
439: *> \endverbatim
440: *>
441: *> \param[out] WORK
442: *> \verbatim
443: *> WORK is COMPLEX*16 array, dimension (2*N)
444: *> \endverbatim
445: *>
446: *> \param[out] RWORK
447: *> \verbatim
448: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
449: *> \endverbatim
450: *>
451: *> \param[out] INFO
452: *> \verbatim
453: *> INFO is INTEGER
454: *> = 0: Successful exit. The solution to every right-hand side is
455: *> guaranteed.
456: *> < 0: If INFO = -i, the i-th argument had an illegal value
457: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
458: *> has been completed, but the factor U is exactly singular, so
459: *> the solution and error bounds could not be computed. RCOND = 0
460: *> is returned.
461: *> = N+J: The solution corresponding to the Jth right-hand side is
462: *> not guaranteed. The solutions corresponding to other right-
463: *> hand sides K with K > J may not be guaranteed as well, but
464: *> only the first such right-hand side is reported. If a small
465: *> componentwise error is not requested (PARAMS(3) = 0.0) then
466: *> the Jth right-hand side is the first with a normwise error
467: *> bound that is not guaranteed (the smallest J such
468: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
469: *> the Jth right-hand side is the first with either a normwise or
470: *> componentwise error bound that is not guaranteed (the smallest
471: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
472: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
473: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
474: *> about all of the right-hand sides check ERR_BNDS_NORM or
475: *> ERR_BNDS_COMP.
476: *> \endverbatim
477: *
478: * Authors:
479: * ========
480: *
481: *> \author Univ. of Tennessee
482: *> \author Univ. of California Berkeley
483: *> \author Univ. of Colorado Denver
484: *> \author NAG Ltd.
485: *
1.7 bertrand 486: *> \date April 2012
1.5 bertrand 487: *
488: *> \ingroup complex16POsolve
489: *
490: * =====================================================================
1.1 bertrand 491: SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
492: $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
493: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
494: $ NPARAMS, PARAMS, WORK, RWORK, INFO )
495: *
1.7 bertrand 496: * -- LAPACK driver routine (version 3.4.1) --
1.5 bertrand 497: * -- LAPACK is a software package provided by Univ. of Tennessee, --
498: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.7 bertrand 499: * April 2012
1.1 bertrand 500: *
501: * .. Scalar Arguments ..
502: CHARACTER EQUED, FACT, UPLO
503: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
504: $ N_ERR_BNDS
505: DOUBLE PRECISION RCOND, RPVGRW
506: * ..
507: * .. Array Arguments ..
508: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
509: $ WORK( * ), X( LDX, * )
510: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
511: $ ERR_BNDS_NORM( NRHS, * ),
512: $ ERR_BNDS_COMP( NRHS, * )
513: * ..
514: *
1.5 bertrand 515: * ==================================================================
1.1 bertrand 516: *
517: * .. Parameters ..
518: DOUBLE PRECISION ZERO, ONE
519: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
520: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
521: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
522: INTEGER CMP_ERR_I, PIV_GROWTH_I
523: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
524: $ BERR_I = 3 )
525: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
526: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
527: $ PIV_GROWTH_I = 9 )
528: * ..
529: * .. Local Scalars ..
530: LOGICAL EQUIL, NOFACT, RCEQU
531: INTEGER INFEQU, J
532: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
533: * ..
534: * .. External Functions ..
535: EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW
536: LOGICAL LSAME
537: DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
538: * ..
539: * .. External Subroutines ..
540: EXTERNAL ZPOCON, ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY,
541: $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX
542: * ..
543: * .. Intrinsic Functions ..
544: INTRINSIC MAX, MIN
545: * ..
546: * .. Executable Statements ..
547: *
548: INFO = 0
549: NOFACT = LSAME( FACT, 'N' )
550: EQUIL = LSAME( FACT, 'E' )
551: SMLNUM = DLAMCH( 'Safe minimum' )
552: BIGNUM = ONE / SMLNUM
553: IF( NOFACT .OR. EQUIL ) THEN
554: EQUED = 'N'
555: RCEQU = .FALSE.
556: ELSE
557: RCEQU = LSAME( EQUED, 'Y' )
558: ENDIF
559: *
560: * Default is failure. If an input parameter is wrong or
561: * factorization fails, make everything look horrible. Only the
562: * pivot growth is set here, the rest is initialized in ZPORFSX.
563: *
564: RPVGRW = ZERO
565: *
566: * Test the input parameters. PARAMS is not tested until ZPORFSX.
567: *
568: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
569: $ LSAME( FACT, 'F' ) ) THEN
570: INFO = -1
571: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
572: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
573: INFO = -2
574: ELSE IF( N.LT.0 ) THEN
575: INFO = -3
576: ELSE IF( NRHS.LT.0 ) THEN
577: INFO = -4
578: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
579: INFO = -6
580: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
581: INFO = -8
582: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
583: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
584: INFO = -9
585: ELSE
586: IF ( RCEQU ) THEN
587: SMIN = BIGNUM
588: SMAX = ZERO
589: DO 10 J = 1, N
590: SMIN = MIN( SMIN, S( J ) )
591: SMAX = MAX( SMAX, S( J ) )
592: 10 CONTINUE
593: IF( SMIN.LE.ZERO ) THEN
594: INFO = -10
595: ELSE IF( N.GT.0 ) THEN
596: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
597: ELSE
598: SCOND = ONE
599: END IF
600: END IF
601: IF( INFO.EQ.0 ) THEN
602: IF( LDB.LT.MAX( 1, N ) ) THEN
603: INFO = -12
604: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
605: INFO = -14
606: END IF
607: END IF
608: END IF
609: *
610: IF( INFO.NE.0 ) THEN
611: CALL XERBLA( 'ZPOSVXX', -INFO )
612: RETURN
613: END IF
614: *
615: IF( EQUIL ) THEN
616: *
617: * Compute row and column scalings to equilibrate the matrix A.
618: *
619: CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
620: IF( INFEQU.EQ.0 ) THEN
621: *
622: * Equilibrate the matrix.
623: *
624: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
625: RCEQU = LSAME( EQUED, 'Y' )
626: END IF
627: END IF
628: *
629: * Scale the right-hand side.
630: *
631: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
632: *
633: IF( NOFACT .OR. EQUIL ) THEN
634: *
635: * Compute the Cholesky factorization of A.
636: *
637: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
638: CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
639: *
640: * Return if INFO is non-zero.
641: *
642: IF( INFO.GT.0 ) THEN
643: *
644: * Pivot in column INFO is exactly 0
645: * Compute the reciprocal pivot growth factor of the
646: * leading rank-deficient INFO columns of A.
647: *
648: RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
649: RETURN
650: END IF
651: END IF
652: *
653: * Compute the reciprocal pivot growth factor RPVGRW.
654: *
655: RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
656: *
657: * Compute the solution matrix X.
658: *
659: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
660: CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
661: *
662: * Use iterative refinement to improve the computed solution and
663: * compute error bounds and backward error estimates for it.
664: *
665: CALL ZPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
666: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
667: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
668:
669: *
670: * Scale solutions.
671: *
672: IF ( RCEQU ) THEN
673: CALL ZLASCL2( N, NRHS, S, X, LDX )
674: END IF
675: *
676: RETURN
677: *
678: * End of ZPOSVXX
679: *
680: END
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