Annotation of rpl/lapack/lapack/zposvxx.f, revision 1.2
1.1 bertrand 1: SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
2: $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
3: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
4: $ NPARAMS, PARAMS, WORK, RWORK, INFO )
5: *
6: * -- LAPACK driver routine (version 3.2.2) --
7: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
8: * -- Jason Riedy of Univ. of California Berkeley. --
9: * -- June 2010 --
10: *
11: * -- LAPACK is a software package provided by Univ. of Tennessee, --
12: * -- Univ. of California Berkeley and NAG Ltd. --
13: *
14: IMPLICIT NONE
15: * ..
16: * .. Scalar Arguments ..
17: CHARACTER EQUED, FACT, UPLO
18: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
19: $ N_ERR_BNDS
20: DOUBLE PRECISION RCOND, RPVGRW
21: * ..
22: * .. Array Arguments ..
23: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
24: $ WORK( * ), X( LDX, * )
25: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
26: $ ERR_BNDS_NORM( NRHS, * ),
27: $ ERR_BNDS_COMP( NRHS, * )
28: * ..
29: *
30: * Purpose
31: * =======
32: *
33: * ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
34: * to compute the solution to a complex*16 system of linear equations
35: * A * X = B, where A is an N-by-N symmetric positive definite matrix
36: * and X and B are N-by-NRHS matrices.
37: *
38: * If requested, both normwise and maximum componentwise error bounds
39: * are returned. ZPOSVXX will return a solution with a tiny
40: * guaranteed error (O(eps) where eps is the working machine
41: * precision) unless the matrix is very ill-conditioned, in which
42: * case a warning is returned. Relevant condition numbers also are
43: * calculated and returned.
44: *
45: * ZPOSVXX accepts user-provided factorizations and equilibration
46: * factors; see the definitions of the FACT and EQUED options.
47: * Solving with refinement and using a factorization from a previous
48: * ZPOSVXX call will also produce a solution with either O(eps)
49: * errors or warnings, but we cannot make that claim for general
50: * user-provided factorizations and equilibration factors if they
51: * differ from what ZPOSVXX would itself produce.
52: *
53: * Description
54: * ===========
55: *
56: * The following steps are performed:
57: *
58: * 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
59: * the system:
60: *
61: * diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
62: *
63: * Whether or not the system will be equilibrated depends on the
64: * scaling of the matrix A, but if equilibration is used, A is
65: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
66: *
67: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
68: * factor the matrix A (after equilibration if FACT = 'E') as
69: * A = U**T* U, if UPLO = 'U', or
70: * A = L * L**T, if UPLO = 'L',
71: * where U is an upper triangular matrix and L is a lower triangular
72: * matrix.
73: *
74: * 3. If the leading i-by-i principal minor is not positive definite,
75: * then the routine returns with INFO = i. Otherwise, the factored
76: * form of A is used to estimate the condition number of the matrix
77: * A (see argument RCOND). If the reciprocal of the condition number
78: * is less than machine precision, the routine still goes on to solve
79: * for X and compute error bounds as described below.
80: *
81: * 4. The system of equations is solved for X using the factored form
82: * of A.
83: *
84: * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
85: * the routine will use iterative refinement to try to get a small
86: * error and error bounds. Refinement calculates the residual to at
87: * least twice the working precision.
88: *
89: * 6. If equilibration was used, the matrix X is premultiplied by
90: * diag(S) so that it solves the original system before
91: * equilibration.
92: *
93: * Arguments
94: * =========
95: *
96: * Some optional parameters are bundled in the PARAMS array. These
97: * settings determine how refinement is performed, but often the
98: * defaults are acceptable. If the defaults are acceptable, users
99: * can pass NPARAMS = 0 which prevents the source code from accessing
100: * the PARAMS argument.
101: *
102: * FACT (input) CHARACTER*1
103: * Specifies whether or not the factored form of the matrix A is
104: * supplied on entry, and if not, whether the matrix A should be
105: * equilibrated before it is factored.
106: * = 'F': On entry, AF contains the factored form of A.
107: * If EQUED is not 'N', the matrix A has been
108: * equilibrated with scaling factors given by S.
109: * A and AF are not modified.
110: * = 'N': The matrix A will be copied to AF and factored.
111: * = 'E': The matrix A will be equilibrated if necessary, then
112: * copied to AF and factored.
113: *
114: * UPLO (input) CHARACTER*1
115: * = 'U': Upper triangle of A is stored;
116: * = 'L': Lower triangle of A is stored.
117: *
118: * N (input) INTEGER
119: * The number of linear equations, i.e., the order of the
120: * matrix A. N >= 0.
121: *
122: * NRHS (input) INTEGER
123: * The number of right hand sides, i.e., the number of columns
124: * of the matrices B and X. NRHS >= 0.
125: *
126: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
127: * On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
128: * 'Y', then A must contain the equilibrated matrix
129: * diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
130: * triangular part of A contains the upper triangular part of the
131: * matrix A, and the strictly lower triangular part of A is not
132: * referenced. If UPLO = 'L', the leading N-by-N lower triangular
133: * part of A contains the lower triangular part of the matrix A, and
134: * the strictly upper triangular part of A is not referenced. A is
135: * not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
136: * 'N' on exit.
137: *
138: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
139: * diag(S)*A*diag(S).
140: *
141: * LDA (input) INTEGER
142: * The leading dimension of the array A. LDA >= max(1,N).
143: *
144: * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
145: * If FACT = 'F', then AF is an input argument and on entry
146: * contains the triangular factor U or L from the Cholesky
147: * factorization A = U**T*U or A = L*L**T, in the same storage
148: * format as A. If EQUED .ne. 'N', then AF is the factored
149: * form of the equilibrated matrix diag(S)*A*diag(S).
150: *
151: * If FACT = 'N', then AF is an output argument and on exit
152: * returns the triangular factor U or L from the Cholesky
153: * factorization A = U**T*U or A = L*L**T of the original
154: * matrix A.
155: *
156: * If FACT = 'E', then AF is an output argument and on exit
157: * returns the triangular factor U or L from the Cholesky
158: * factorization A = U**T*U or A = L*L**T of the equilibrated
159: * matrix A (see the description of A for the form of the
160: * equilibrated matrix).
161: *
162: * LDAF (input) INTEGER
163: * The leading dimension of the array AF. LDAF >= max(1,N).
164: *
165: * EQUED (input or output) CHARACTER*1
166: * Specifies the form of equilibration that was done.
167: * = 'N': No equilibration (always true if FACT = 'N').
168: * = 'Y': Both row and column equilibration, i.e., A has been
169: * replaced by diag(S) * A * diag(S).
170: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
171: * output argument.
172: *
173: * S (input or output) DOUBLE PRECISION array, dimension (N)
174: * The row scale factors for A. If EQUED = 'Y', A is multiplied on
175: * the left and right by diag(S). S is an input argument if FACT =
176: * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
177: * = 'Y', each element of S must be positive. If S is output, each
178: * element of S is a power of the radix. If S is input, each element
179: * of S should be a power of the radix to ensure a reliable solution
180: * and error estimates. Scaling by powers of the radix does not cause
181: * rounding errors unless the result underflows or overflows.
182: * Rounding errors during scaling lead to refining with a matrix that
183: * is not equivalent to the input matrix, producing error estimates
184: * that may not be reliable.
185: *
186: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
187: * On entry, the N-by-NRHS right hand side matrix B.
188: * On exit,
189: * if EQUED = 'N', B is not modified;
190: * if EQUED = 'Y', B is overwritten by diag(S)*B;
191: *
192: * LDB (input) INTEGER
193: * The leading dimension of the array B. LDB >= max(1,N).
194: *
195: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
196: * If INFO = 0, the N-by-NRHS solution matrix X to the original
197: * system of equations. Note that A and B are modified on exit if
198: * EQUED .ne. 'N', and the solution to the equilibrated system is
199: * inv(diag(S))*X.
200: *
201: * LDX (input) INTEGER
202: * The leading dimension of the array X. LDX >= max(1,N).
203: *
204: * RCOND (output) DOUBLE PRECISION
205: * Reciprocal scaled condition number. This is an estimate of the
206: * reciprocal Skeel condition number of the matrix A after
207: * equilibration (if done). If this is less than the machine
208: * precision (in particular, if it is zero), the matrix is singular
209: * to working precision. Note that the error may still be small even
210: * if this number is very small and the matrix appears ill-
211: * conditioned.
212: *
213: * RPVGRW (output) DOUBLE PRECISION
214: * Reciprocal pivot growth. On exit, this contains the reciprocal
215: * pivot growth factor norm(A)/norm(U). The "max absolute element"
216: * norm is used. If this is much less than 1, then the stability of
217: * the LU factorization of the (equilibrated) matrix A could be poor.
218: * This also means that the solution X, estimated condition numbers,
219: * and error bounds could be unreliable. If factorization fails with
220: * 0<INFO<=N, then this contains the reciprocal pivot growth factor
221: * for the leading INFO columns of A.
222: *
223: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
224: * Componentwise relative backward error. This is the
225: * componentwise relative backward error of each solution vector X(j)
226: * (i.e., the smallest relative change in any element of A or B that
227: * makes X(j) an exact solution).
228: *
229: * N_ERR_BNDS (input) INTEGER
230: * Number of error bounds to return for each right hand side
231: * and each type (normwise or componentwise). See ERR_BNDS_NORM and
232: * ERR_BNDS_COMP below.
233: *
234: * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
235: * For each right-hand side, this array contains information about
236: * various error bounds and condition numbers corresponding to the
237: * normwise relative error, which is defined as follows:
238: *
239: * Normwise relative error in the ith solution vector:
240: * max_j (abs(XTRUE(j,i) - X(j,i)))
241: * ------------------------------
242: * max_j abs(X(j,i))
243: *
244: * The array is indexed by the type of error information as described
245: * below. There currently are up to three pieces of information
246: * returned.
247: *
248: * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
249: * right-hand side.
250: *
251: * The second index in ERR_BNDS_NORM(:,err) contains the following
252: * three fields:
253: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
254: * reciprocal condition number is less than the threshold
255: * sqrt(n) * dlamch('Epsilon').
256: *
257: * err = 2 "Guaranteed" error bound: The estimated forward error,
258: * almost certainly within a factor of 10 of the true error
259: * so long as the next entry is greater than the threshold
260: * sqrt(n) * dlamch('Epsilon'). This error bound should only
261: * be trusted if the previous boolean is true.
262: *
263: * err = 3 Reciprocal condition number: Estimated normwise
264: * reciprocal condition number. Compared with the threshold
265: * sqrt(n) * dlamch('Epsilon') to determine if the error
266: * estimate is "guaranteed". These reciprocal condition
267: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
268: * appropriately scaled matrix Z.
269: * Let Z = S*A, where S scales each row by a power of the
270: * radix so all absolute row sums of Z are approximately 1.
271: *
272: * See Lapack Working Note 165 for further details and extra
273: * cautions.
274: *
275: * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
276: * For each right-hand side, this array contains information about
277: * various error bounds and condition numbers corresponding to the
278: * componentwise relative error, which is defined as follows:
279: *
280: * Componentwise relative error in the ith solution vector:
281: * abs(XTRUE(j,i) - X(j,i))
282: * max_j ----------------------
283: * abs(X(j,i))
284: *
285: * The array is indexed by the right-hand side i (on which the
286: * componentwise relative error depends), and the type of error
287: * information as described below. There currently are up to three
288: * pieces of information returned for each right-hand side. If
289: * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
290: * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
291: * the first (:,N_ERR_BNDS) entries are returned.
292: *
293: * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
294: * right-hand side.
295: *
296: * The second index in ERR_BNDS_COMP(:,err) contains the following
297: * three fields:
298: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
299: * reciprocal condition number is less than the threshold
300: * sqrt(n) * dlamch('Epsilon').
301: *
302: * err = 2 "Guaranteed" error bound: The estimated forward error,
303: * almost certainly within a factor of 10 of the true error
304: * so long as the next entry is greater than the threshold
305: * sqrt(n) * dlamch('Epsilon'). This error bound should only
306: * be trusted if the previous boolean is true.
307: *
308: * err = 3 Reciprocal condition number: Estimated componentwise
309: * reciprocal condition number. Compared with the threshold
310: * sqrt(n) * dlamch('Epsilon') to determine if the error
311: * estimate is "guaranteed". These reciprocal condition
312: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
313: * appropriately scaled matrix Z.
314: * Let Z = S*(A*diag(x)), where x is the solution for the
315: * current right-hand side and S scales each row of
316: * A*diag(x) by a power of the radix so all absolute row
317: * sums of Z are approximately 1.
318: *
319: * See Lapack Working Note 165 for further details and extra
320: * cautions.
321: *
322: * NPARAMS (input) INTEGER
323: * Specifies the number of parameters set in PARAMS. If .LE. 0, the
324: * PARAMS array is never referenced and default values are used.
325: *
326: * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
327: * Specifies algorithm parameters. If an entry is .LT. 0.0, then
328: * that entry will be filled with default value used for that
329: * parameter. Only positions up to NPARAMS are accessed; defaults
330: * are used for higher-numbered parameters.
331: *
332: * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
333: * refinement or not.
334: * Default: 1.0D+0
335: * = 0.0 : No refinement is performed, and no error bounds are
336: * computed.
337: * = 1.0 : Use the extra-precise refinement algorithm.
338: * (other values are reserved for future use)
339: *
340: * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
341: * computations allowed for refinement.
342: * Default: 10
343: * Aggressive: Set to 100 to permit convergence using approximate
344: * factorizations or factorizations other than LU. If
345: * the factorization uses a technique other than
346: * Gaussian elimination, the guarantees in
347: * err_bnds_norm and err_bnds_comp may no longer be
348: * trustworthy.
349: *
350: * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
351: * will attempt to find a solution with small componentwise
352: * relative error in the double-precision algorithm. Positive
353: * is true, 0.0 is false.
354: * Default: 1.0 (attempt componentwise convergence)
355: *
356: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
357: *
358: * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
359: *
360: * INFO (output) INTEGER
361: * = 0: Successful exit. The solution to every right-hand side is
362: * guaranteed.
363: * < 0: If INFO = -i, the i-th argument had an illegal value
364: * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
365: * has been completed, but the factor U is exactly singular, so
366: * the solution and error bounds could not be computed. RCOND = 0
367: * is returned.
368: * = N+J: The solution corresponding to the Jth right-hand side is
369: * not guaranteed. The solutions corresponding to other right-
370: * hand sides K with K > J may not be guaranteed as well, but
371: * only the first such right-hand side is reported. If a small
372: * componentwise error is not requested (PARAMS(3) = 0.0) then
373: * the Jth right-hand side is the first with a normwise error
374: * bound that is not guaranteed (the smallest J such
375: * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
376: * the Jth right-hand side is the first with either a normwise or
377: * componentwise error bound that is not guaranteed (the smallest
378: * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
379: * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
380: * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
381: * about all of the right-hand sides check ERR_BNDS_NORM or
382: * ERR_BNDS_COMP.
383: *
384: * ==================================================================
385: *
386: * .. Parameters ..
387: DOUBLE PRECISION ZERO, ONE
388: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
389: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
390: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
391: INTEGER CMP_ERR_I, PIV_GROWTH_I
392: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
393: $ BERR_I = 3 )
394: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
395: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
396: $ PIV_GROWTH_I = 9 )
397: * ..
398: * .. Local Scalars ..
399: LOGICAL EQUIL, NOFACT, RCEQU
400: INTEGER INFEQU, J
401: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
402: * ..
403: * .. External Functions ..
404: EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW
405: LOGICAL LSAME
406: DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
407: * ..
408: * .. External Subroutines ..
409: EXTERNAL ZPOCON, ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY,
410: $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX
411: * ..
412: * .. Intrinsic Functions ..
413: INTRINSIC MAX, MIN
414: * ..
415: * .. Executable Statements ..
416: *
417: INFO = 0
418: NOFACT = LSAME( FACT, 'N' )
419: EQUIL = LSAME( FACT, 'E' )
420: SMLNUM = DLAMCH( 'Safe minimum' )
421: BIGNUM = ONE / SMLNUM
422: IF( NOFACT .OR. EQUIL ) THEN
423: EQUED = 'N'
424: RCEQU = .FALSE.
425: ELSE
426: RCEQU = LSAME( EQUED, 'Y' )
427: ENDIF
428: *
429: * Default is failure. If an input parameter is wrong or
430: * factorization fails, make everything look horrible. Only the
431: * pivot growth is set here, the rest is initialized in ZPORFSX.
432: *
433: RPVGRW = ZERO
434: *
435: * Test the input parameters. PARAMS is not tested until ZPORFSX.
436: *
437: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
438: $ LSAME( FACT, 'F' ) ) THEN
439: INFO = -1
440: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
441: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
442: INFO = -2
443: ELSE IF( N.LT.0 ) THEN
444: INFO = -3
445: ELSE IF( NRHS.LT.0 ) THEN
446: INFO = -4
447: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
448: INFO = -6
449: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
450: INFO = -8
451: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
452: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
453: INFO = -9
454: ELSE
455: IF ( RCEQU ) THEN
456: SMIN = BIGNUM
457: SMAX = ZERO
458: DO 10 J = 1, N
459: SMIN = MIN( SMIN, S( J ) )
460: SMAX = MAX( SMAX, S( J ) )
461: 10 CONTINUE
462: IF( SMIN.LE.ZERO ) THEN
463: INFO = -10
464: ELSE IF( N.GT.0 ) THEN
465: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
466: ELSE
467: SCOND = ONE
468: END IF
469: END IF
470: IF( INFO.EQ.0 ) THEN
471: IF( LDB.LT.MAX( 1, N ) ) THEN
472: INFO = -12
473: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
474: INFO = -14
475: END IF
476: END IF
477: END IF
478: *
479: IF( INFO.NE.0 ) THEN
480: CALL XERBLA( 'ZPOSVXX', -INFO )
481: RETURN
482: END IF
483: *
484: IF( EQUIL ) THEN
485: *
486: * Compute row and column scalings to equilibrate the matrix A.
487: *
488: CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
489: IF( INFEQU.EQ.0 ) THEN
490: *
491: * Equilibrate the matrix.
492: *
493: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
494: RCEQU = LSAME( EQUED, 'Y' )
495: END IF
496: END IF
497: *
498: * Scale the right-hand side.
499: *
500: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
501: *
502: IF( NOFACT .OR. EQUIL ) THEN
503: *
504: * Compute the Cholesky factorization of A.
505: *
506: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
507: CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
508: *
509: * Return if INFO is non-zero.
510: *
511: IF( INFO.GT.0 ) THEN
512: *
513: * Pivot in column INFO is exactly 0
514: * Compute the reciprocal pivot growth factor of the
515: * leading rank-deficient INFO columns of A.
516: *
517: RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
518: RETURN
519: END IF
520: END IF
521: *
522: * Compute the reciprocal pivot growth factor RPVGRW.
523: *
524: RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK )
525: *
526: * Compute the solution matrix X.
527: *
528: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
529: CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
530: *
531: * Use iterative refinement to improve the computed solution and
532: * compute error bounds and backward error estimates for it.
533: *
534: CALL ZPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
535: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
536: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
537:
538: *
539: * Scale solutions.
540: *
541: IF ( RCEQU ) THEN
542: CALL ZLASCL2( N, NRHS, S, X, LDX )
543: END IF
544: *
545: RETURN
546: *
547: * End of ZPOSVXX
548: *
549: END
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