001:       SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
002:      $                    S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
003:      $                    N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
004:      $                    NPARAMS, PARAMS, WORK, RWORK, INFO )
005: *
006: *     -- LAPACK driver routine (version 3.2)                          --
007: *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
008: *     -- Jason Riedy of Univ. of California Berkeley.                 --
009: *     -- November 2008                                                --
010: *
011: *     -- LAPACK is a software package provided by Univ. of Tennessee, --
012: *     -- Univ. of California Berkeley and NAG Ltd.                    --
013: *
014:       IMPLICIT NONE
015: *     ..
016: *     .. Scalar Arguments ..
017:       CHARACTER          EQUED, FACT, UPLO
018:       INTEGER            INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
019:      $                   N_ERR_BNDS
020:       DOUBLE PRECISION   RCOND, RPVGRW
021: *     ..
022: *     .. Array Arguments ..
023:       COMPLEX*16         A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
024:      $                   WORK( * ), X( LDX, * )
025:       DOUBLE PRECISION   S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
026:      $                   ERR_BNDS_NORM( NRHS, * ),
027:      $                   ERR_BNDS_COMP( NRHS, * )
028: *     ..
029: *
030: *     Purpose
031: *     =======
032: *
033: *     ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
034: *     to compute the solution to a complex*16 system of linear equations
035: *     A * X = B, where A is an N-by-N symmetric positive definite matrix
036: *     and X and B are N-by-NRHS matrices.
037: *
038: *     If requested, both normwise and maximum componentwise error bounds
039: *     are returned. ZPOSVXX will return a solution with a tiny
040: *     guaranteed error (O(eps) where eps is the working machine
041: *     precision) unless the matrix is very ill-conditioned, in which
042: *     case a warning is returned. Relevant condition numbers also are
043: *     calculated and returned.
044: *
045: *     ZPOSVXX accepts user-provided factorizations and equilibration
046: *     factors; see the definitions of the FACT and EQUED options.
047: *     Solving with refinement and using a factorization from a previous
048: *     ZPOSVXX call will also produce a solution with either O(eps)
049: *     errors or warnings, but we cannot make that claim for general
050: *     user-provided factorizations and equilibration factors if they
051: *     differ from what ZPOSVXX would itself produce.
052: *
053: *     Description
054: *     ===========
055: *
056: *     The following steps are performed:
057: *
058: *     1. If FACT = 'E', double precision scaling factors are computed to equilibrate
059: *     the system:
060: *
061: *       diag(S)*A*diag(S)     *inv(diag(S))*X = diag(S)*B
062: *
063: *     Whether or not the system will be equilibrated depends on the
064: *     scaling of the matrix A, but if equilibration is used, A is
065: *     overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
066: *
067: *     2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
068: *     factor the matrix A (after equilibration if FACT = 'E') as
069: *        A = U**T* U,  if UPLO = 'U', or
070: *        A = L * L**T,  if UPLO = 'L',
071: *     where U is an upper triangular matrix and L is a lower triangular
072: *     matrix.
073: *
074: *     3. If the leading i-by-i principal minor is not positive definite,
075: *     then the routine returns with INFO = i. Otherwise, the factored
076: *     form of A is used to estimate the condition number of the matrix
077: *     A (see argument RCOND).  If the reciprocal of the condition number
078: *     is less than machine precision, the routine still goes on to solve
079: *     for X and compute error bounds as described below.
080: *
081: *     4. The system of equations is solved for X using the factored form
082: *     of A.
083: *
084: *     5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
085: *     the routine will use iterative refinement to try to get a small
086: *     error and error bounds.  Refinement calculates the residual to at
087: *     least twice the working precision.
088: *
089: *     6. If equilibration was used, the matrix X is premultiplied by
090: *     diag(S) so that it solves the original system before
091: *     equilibration.
092: *
093: *     Arguments
094: *     =========
095: *
096: *     Some optional parameters are bundled in the PARAMS array.  These
097: *     settings determine how refinement is performed, but often the
098: *     defaults are acceptable.  If the defaults are acceptable, users
099: *     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) DOUBLE PRECISION array, dimension (4*N)
357: *
358: *     IWORK   (workspace) INTEGER array, dimension (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 LU 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, WORK )
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, WORK )
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
550: