001:       SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
002:      $                    EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
003:      $                    N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
004:      $                    NPARAMS, PARAMS, WORK, IWORK, INFO )
005: *
006: *     -- LAPACK routine (version 3.2.1)                               --
007: *     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
008: *     -- Jason Riedy of Univ. of California Berkeley.                 --
009: *     -- April 2009                                                   --
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:       INTEGER            IPIV( * ), IWORK( * )
024:       DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
025:      $                   X( LDX, * ), WORK( * )
026:       DOUBLE PRECISION   S( * ), PARAMS( * ), BERR( * ),
027:      $                   ERR_BNDS_NORM( NRHS, * ),
028:      $                   ERR_BNDS_COMP( NRHS, * )
029: *     ..
030: *
031: *     Purpose
032: *     =======
033: *
034: *     DSYSVXX uses the diagonal pivoting factorization to compute the
035: *     solution to a double precision system of linear equations A * X = B, where A
036: *     is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.
037: *
038: *     If requested, both normwise and maximum componentwise error bounds
039: *     are returned. DSYSVXX 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: *     DSYSVXX 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: *     DSYSVXX 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 DSYSVXX 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 LU decomposition is used to factor
068: *     the matrix A (after equilibration if FACT = 'E') as
069: *
070: *        A = U * D * U**T,  if UPLO = 'U', or
071: *        A = L * D * L**T,  if UPLO = 'L',
072: *
073: *     where U (or L) is a product of permutation and unit upper (lower)
074: *     triangular matrices, and D is symmetric and block diagonal with
075: *     1-by-1 and 2-by-2 diagonal blocks.
076: *
077: *     3. If some D(i,i)=0, so that D is exactly singular, then the
078: *     routine returns with INFO = i. Otherwise, the factored form of A
079: *     is used to estimate the condition number of the matrix A (see
080: *     argument RCOND).  If the reciprocal of the condition number is
081: *     less than machine precision, the routine still goes on to solve
082: *     for X and compute error bounds as described below.
083: *
084: *     4. The system of equations is solved for X using the factored form
085: *     of A.
086: *
087: *     5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
088: *     the routine will use iterative refinement to try to get a small
089: *     error and error bounds.  Refinement calculates the residual to at
090: *     least twice the working precision.
091: *
092: *     6. If equilibration was used, the matrix X is premultiplied by
093: *     diag(R) so that it solves the original system before
094: *     equilibration.
095: *
096: *     Arguments
097: *     =========
098: *
099: *     Some optional parameters are bundled in the PARAMS array.  These
100: *     settings determine how refinement is performed, but often the
101: *     defaults are acceptable.  If the defaults are acceptable, users
102: *     can pass NPARAMS = 0 which prevents the source code from accessing
103: *     the PARAMS argument.
104: *
105: *     FACT    (input) CHARACTER*1
106: *     Specifies whether or not the factored form of the matrix A is
107: *     supplied on entry, and if not, whether the matrix A should be
108: *     equilibrated before it is factored.
109: *       = 'F':  On entry, AF and IPIV contain the factored form of A.
110: *               If EQUED is not 'N', the matrix A has been
111: *               equilibrated with scaling factors given by S.
112: *               A, AF, and IPIV are not modified.
113: *       = 'N':  The matrix A will be copied to AF and factored.
114: *       = 'E':  The matrix A will be equilibrated if necessary, then
115: *               copied to AF and factored.
116: *
117: *     UPLO    (input) CHARACTER*1
118: *       = 'U':  Upper triangle of A is stored;
119: *       = 'L':  Lower triangle of A is stored.
120: *
121: *     N       (input) INTEGER
122: *     The number of linear equations, i.e., the order of the
123: *     matrix A.  N >= 0.
124: *
125: *     NRHS    (input) INTEGER
126: *     The number of right hand sides, i.e., the number of columns
127: *     of the matrices B and X.  NRHS >= 0.
128: *
129: *     A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
130: *     The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
131: *     upper triangular part of A contains the upper triangular
132: *     part of the matrix A, and the strictly lower triangular
133: *     part of A is not referenced.  If UPLO = 'L', the leading
134: *     N-by-N lower triangular part of A contains the lower
135: *     triangular part of the matrix A, and the strictly upper
136: *     triangular part of A is not referenced.
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) DOUBLE PRECISION array, dimension (LDAF,N)
145: *     If FACT = 'F', then AF is an input argument and on entry
146: *     contains the block diagonal matrix D and the multipliers
147: *     used to obtain the factor U or L from the factorization A =
148: *     U*D*U**T or A = L*D*L**T as computed by DSYTRF.
149: *
150: *     If FACT = 'N', then AF is an output argument and on exit
151: *     returns the block diagonal matrix D and the multipliers
152: *     used to obtain the factor U or L from the factorization A =
153: *     U*D*U**T or A = L*D*L**T.
154: *
155: *     LDAF    (input) INTEGER
156: *     The leading dimension of the array AF.  LDAF >= max(1,N).
157: *
158: *     IPIV    (input or output) INTEGER array, dimension (N)
159: *     If FACT = 'F', then IPIV is an input argument and on entry
160: *     contains details of the interchanges and the block
161: *     structure of D, as determined by DSYTRF.  If IPIV(k) > 0,
162: *     then rows and columns k and IPIV(k) were interchanged and
163: *     D(k,k) is a 1-by-1 diagonal block.  If UPLO = 'U' and
164: *     IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
165: *     -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
166: *     diagonal block.  If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
167: *     then rows and columns k+1 and -IPIV(k) were interchanged
168: *     and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
169: *
170: *     If FACT = 'N', then IPIV is an output argument and on exit
171: *     contains details of the interchanges and the block
172: *     structure of D, as determined by DSYTRF.
173: *
174: *     EQUED   (input or output) CHARACTER*1
175: *     Specifies the form of equilibration that was done.
176: *       = 'N':  No equilibration (always true if FACT = 'N').
177: *       = 'Y':  Both row and column equilibration, i.e., A has been
178: *               replaced by diag(S) * A * diag(S).
179: *     EQUED is an input argument if FACT = 'F'; otherwise, it is an
180: *     output argument.
181: *
182: *     S       (input or output) DOUBLE PRECISION array, dimension (N)
183: *     The scale factors for A.  If EQUED = 'Y', A is multiplied on
184: *     the left and right by diag(S).  S is an input argument if FACT =
185: *     'F'; otherwise, S is an output argument.  If FACT = 'F' and EQUED
186: *     = 'Y', each element of S must be positive.  If S is output, each
187: *     element of S is a power of the radix. If S is input, each element
188: *     of S should be a power of the radix to ensure a reliable solution
189: *     and error estimates. Scaling by powers of the radix does not cause
190: *     rounding errors unless the result underflows or overflows.
191: *     Rounding errors during scaling lead to refining with a matrix that
192: *     is not equivalent to the input matrix, producing error estimates
193: *     that may not be reliable.
194: *
195: *     B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
196: *     On entry, the N-by-NRHS right hand side matrix B.
197: *     On exit,
198: *     if EQUED = 'N', B is not modified;
199: *     if EQUED = 'Y', B is overwritten by diag(S)*B;
200: *
201: *     LDB     (input) INTEGER
202: *     The leading dimension of the array B.  LDB >= max(1,N).
203: *
204: *     X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
205: *     If INFO = 0, the N-by-NRHS solution matrix X to the original
206: *     system of equations.  Note that A and B are modified on exit if
207: *     EQUED .ne. 'N', and the solution to the equilibrated system is
208: *     inv(diag(S))*X.
209: *
210: *     LDX     (input) INTEGER
211: *     The leading dimension of the array X.  LDX >= max(1,N).
212: *
213: *     RCOND   (output) DOUBLE PRECISION
214: *     Reciprocal scaled condition number.  This is an estimate of the
215: *     reciprocal Skeel condition number of the matrix A after
216: *     equilibration (if done).  If this is less than the machine
217: *     precision (in particular, if it is zero), the matrix is singular
218: *     to working precision.  Note that the error may still be small even
219: *     if this number is very small and the matrix appears ill-
220: *     conditioned.
221: *
222: *     RPVGRW  (output) DOUBLE PRECISION
223: *     Reciprocal pivot growth.  On exit, this contains the reciprocal
224: *     pivot growth factor norm(A)/norm(U). The "max absolute element"
225: *     norm is used.  If this is much less than 1, then the stability of
226: *     the LU factorization of the (equilibrated) matrix A could be poor.
227: *     This also means that the solution X, estimated condition numbers,
228: *     and error bounds could be unreliable. If factorization fails with
229: *     0<INFO<=N, then this contains the reciprocal pivot growth factor
230: *     for the leading INFO columns of A.
231: *
232: *     BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
233: *     Componentwise relative backward error.  This is the
234: *     componentwise relative backward error of each solution vector X(j)
235: *     (i.e., the smallest relative change in any element of A or B that
236: *     makes X(j) an exact solution).
237: *
238: *     N_ERR_BNDS (input) INTEGER
239: *     Number of error bounds to return for each right hand side
240: *     and each type (normwise or componentwise).  See ERR_BNDS_NORM and
241: *     ERR_BNDS_COMP below.
242: *
243: *     ERR_BNDS_NORM  (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
244: *     For each right-hand side, this array contains information about
245: *     various error bounds and condition numbers corresponding to the
246: *     normwise relative error, which is defined as follows:
247: *
248: *     Normwise relative error in the ith solution vector:
249: *             max_j (abs(XTRUE(j,i) - X(j,i)))
250: *            ------------------------------
251: *                  max_j abs(X(j,i))
252: *
253: *     The array is indexed by the type of error information as described
254: *     below. There currently are up to three pieces of information
255: *     returned.
256: *
257: *     The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
258: *     right-hand side.
259: *
260: *     The second index in ERR_BNDS_NORM(:,err) contains the following
261: *     three fields:
262: *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
263: *              reciprocal condition number is less than the threshold
264: *              sqrt(n) * dlamch('Epsilon').
265: *
266: *     err = 2 "Guaranteed" error bound: The estimated forward error,
267: *              almost certainly within a factor of 10 of the true error
268: *              so long as the next entry is greater than the threshold
269: *              sqrt(n) * dlamch('Epsilon'). This error bound should only
270: *              be trusted if the previous boolean is true.
271: *
272: *     err = 3  Reciprocal condition number: Estimated normwise
273: *              reciprocal condition number.  Compared with the threshold
274: *              sqrt(n) * dlamch('Epsilon') to determine if the error
275: *              estimate is "guaranteed". These reciprocal condition
276: *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
277: *              appropriately scaled matrix Z.
278: *              Let Z = S*A, where S scales each row by a power of the
279: *              radix so all absolute row sums of Z are approximately 1.
280: *
281: *     See Lapack Working Note 165 for further details and extra
282: *     cautions.
283: *
284: *     ERR_BNDS_COMP  (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
285: *     For each right-hand side, this array contains information about
286: *     various error bounds and condition numbers corresponding to the
287: *     componentwise relative error, which is defined as follows:
288: *
289: *     Componentwise relative error in the ith solution vector:
290: *                    abs(XTRUE(j,i) - X(j,i))
291: *             max_j ----------------------
292: *                         abs(X(j,i))
293: *
294: *     The array is indexed by the right-hand side i (on which the
295: *     componentwise relative error depends), and the type of error
296: *     information as described below. There currently are up to three
297: *     pieces of information returned for each right-hand side. If
298: *     componentwise accuracy is not requested (PARAMS(3) = 0.0), then
299: *     ERR_BNDS_COMP is not accessed.  If N_ERR_BNDS .LT. 3, then at most
300: *     the first (:,N_ERR_BNDS) entries are returned.
301: *
302: *     The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
303: *     right-hand side.
304: *
305: *     The second index in ERR_BNDS_COMP(:,err) contains the following
306: *     three fields:
307: *     err = 1 "Trust/don't trust" boolean. Trust the answer if the
308: *              reciprocal condition number is less than the threshold
309: *              sqrt(n) * dlamch('Epsilon').
310: *
311: *     err = 2 "Guaranteed" error bound: The estimated forward error,
312: *              almost certainly within a factor of 10 of the true error
313: *              so long as the next entry is greater than the threshold
314: *              sqrt(n) * dlamch('Epsilon'). This error bound should only
315: *              be trusted if the previous boolean is true.
316: *
317: *     err = 3  Reciprocal condition number: Estimated componentwise
318: *              reciprocal condition number.  Compared with the threshold
319: *              sqrt(n) * dlamch('Epsilon') to determine if the error
320: *              estimate is "guaranteed". These reciprocal condition
321: *              numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
322: *              appropriately scaled matrix Z.
323: *              Let Z = S*(A*diag(x)), where x is the solution for the
324: *              current right-hand side and S scales each row of
325: *              A*diag(x) by a power of the radix so all absolute row
326: *              sums of Z are approximately 1.
327: *
328: *     See Lapack Working Note 165 for further details and extra
329: *     cautions.
330: *
331: *     NPARAMS (input) INTEGER
332: *     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
333: *     PARAMS array is never referenced and default values are used.
334: *
335: *     PARAMS  (input / output) DOUBLE PRECISION array, dimension NPARAMS
336: *     Specifies algorithm parameters.  If an entry is .LT. 0.0, then
337: *     that entry will be filled with default value used for that
338: *     parameter.  Only positions up to NPARAMS are accessed; defaults
339: *     are used for higher-numbered parameters.
340: *
341: *       PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
342: *            refinement or not.
343: *         Default: 1.0D+0
344: *            = 0.0 : No refinement is performed, and no error bounds are
345: *                    computed.
346: *            = 1.0 : Use the extra-precise refinement algorithm.
347: *              (other values are reserved for future use)
348: *
349: *       PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
350: *            computations allowed for refinement.
351: *         Default: 10
352: *         Aggressive: Set to 100 to permit convergence using approximate
353: *                     factorizations or factorizations other than LU. If
354: *                     the factorization uses a technique other than
355: *                     Gaussian elimination, the guarantees in
356: *                     err_bnds_norm and err_bnds_comp may no longer be
357: *                     trustworthy.
358: *
359: *       PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
360: *            will attempt to find a solution with small componentwise
361: *            relative error in the double-precision algorithm.  Positive
362: *            is true, 0.0 is false.
363: *         Default: 1.0 (attempt componentwise convergence)
364: *
365: *     WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)
366: *
367: *     IWORK   (workspace) INTEGER array, dimension (N)
368: *
369: *     INFO    (output) INTEGER
370: *       = 0:  Successful exit. The solution to every right-hand side is
371: *         guaranteed.
372: *       < 0:  If INFO = -i, the i-th argument had an illegal value
373: *       > 0 and <= N:  U(INFO,INFO) is exactly zero.  The factorization
374: *         has been completed, but the factor U is exactly singular, so
375: *         the solution and error bounds could not be computed. RCOND = 0
376: *         is returned.
377: *       = N+J: The solution corresponding to the Jth right-hand side is
378: *         not guaranteed. The solutions corresponding to other right-
379: *         hand sides K with K > J may not be guaranteed as well, but
380: *         only the first such right-hand side is reported. If a small
381: *         componentwise error is not requested (PARAMS(3) = 0.0) then
382: *         the Jth right-hand side is the first with a normwise error
383: *         bound that is not guaranteed (the smallest J such
384: *         that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
385: *         the Jth right-hand side is the first with either a normwise or
386: *         componentwise error bound that is not guaranteed (the smallest
387: *         J such that either ERR_BNDS_NORM(J,1) = 0.0 or
388: *         ERR_BNDS_COMP(J,1) = 0.0). See the definition of
389: *         ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
390: *         about all of the right-hand sides check ERR_BNDS_NORM or
391: *         ERR_BNDS_COMP.
392: *
393: *     ==================================================================
394: *
395: *     .. Parameters ..
396:       DOUBLE PRECISION   ZERO, ONE
397:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
398:       INTEGER            FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
399:       INTEGER            RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
400:       INTEGER            CMP_ERR_I, PIV_GROWTH_I
401:       PARAMETER          ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
402:      $                   BERR_I = 3 )
403:       PARAMETER          ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
404:       PARAMETER          ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
405:      $                   PIV_GROWTH_I = 9 )
406: *     ..
407: *     .. Local Scalars ..
408:       LOGICAL            EQUIL, NOFACT, RCEQU
409:       INTEGER            INFEQU, J
410:       DOUBLE PRECISION   AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
411: *     ..
412: *     .. External Functions ..
413:       EXTERNAL           LSAME, DLAMCH, DLA_SYRPVGRW
414:       LOGICAL            LSAME
415:       DOUBLE PRECISION   DLAMCH, DLA_SYRPVGRW
416: *     ..
417: *     .. External Subroutines ..
418:       EXTERNAL           DSYCON, DSYEQUB, DSYTRF, DSYTRS,
419:      $                   DLACPY, DLAQSY, XERBLA, DLASCL2, DSYRFSX
420: *     ..
421: *     .. Intrinsic Functions ..
422:       INTRINSIC          MAX, MIN
423: *     ..
424: *     .. Executable Statements ..
425: *
426:       INFO = 0
427:       NOFACT = LSAME( FACT, 'N' )
428:       EQUIL = LSAME( FACT, 'E' )
429:       SMLNUM = DLAMCH( 'Safe minimum' )
430:       BIGNUM = ONE / SMLNUM
431:       IF( NOFACT .OR. EQUIL ) THEN
432:          EQUED = 'N'
433:          RCEQU = .FALSE.
434:       ELSE
435:          RCEQU = LSAME( EQUED, 'Y' )
436:       ENDIF
437: *
438: *     Default is failure.  If an input parameter is wrong or
439: *     factorization fails, make everything look horrible.  Only the
440: *     pivot growth is set here, the rest is initialized in DSYRFSX.
441: *
442:       RPVGRW = ZERO
443: *
444: *     Test the input parameters.  PARAMS is not tested until DSYRFSX.
445: *
446:       IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
447:      $     LSAME( FACT, 'F' ) ) THEN
448:          INFO = -1
449:       ELSE IF( .NOT.LSAME(UPLO, 'U') .AND.
450:      $         .NOT.LSAME(UPLO, 'L') ) THEN
451:          INFO = -2
452:       ELSE IF( N.LT.0 ) THEN
453:          INFO = -3
454:       ELSE IF( NRHS.LT.0 ) THEN
455:          INFO = -4
456:       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
457:          INFO = -6
458:       ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
459:          INFO = -8
460:       ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
461:      $        ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
462:          INFO = -9
463:       ELSE
464:          IF ( RCEQU ) THEN
465:             SMIN = BIGNUM
466:             SMAX = ZERO
467:             DO 10 J = 1, N
468:                SMIN = MIN( SMIN, S( J ) )
469:                SMAX = MAX( SMAX, S( J ) )
470:  10         CONTINUE
471:             IF( SMIN.LE.ZERO ) THEN
472:                INFO = -10
473:             ELSE IF( N.GT.0 ) THEN
474:                SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
475:             ELSE
476:                SCOND = ONE
477:             END IF
478:          END IF
479:          IF( INFO.EQ.0 ) THEN
480:             IF( LDB.LT.MAX( 1, N ) ) THEN
481:                INFO = -12
482:             ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
483:                INFO = -14
484:             END IF
485:          END IF
486:       END IF
487: *
488:       IF( INFO.NE.0 ) THEN
489:          CALL XERBLA( 'DSYSVXX', -INFO )
490:          RETURN
491:       END IF
492: *
493:       IF( EQUIL ) THEN
494: *
495: *     Compute row and column scalings to equilibrate the matrix A.
496: *
497:          CALL DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
498:          IF( INFEQU.EQ.0 ) THEN
499: *
500: *     Equilibrate the matrix.
501: *
502:             CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
503:             RCEQU = LSAME( EQUED, 'Y' )
504:          END IF
505:       END IF
506: *
507: *     Scale the right-hand side.
508: *
509:       IF( RCEQU ) CALL DLASCL2( N, NRHS, S, B, LDB )
510: *
511:       IF( NOFACT .OR. EQUIL ) THEN
512: *
513: *        Compute the LU factorization of A.
514: *
515:          CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
516:          CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
517: *
518: *        Return if INFO is non-zero.
519: *
520:          IF( INFO.GT.0 ) THEN
521: *
522: *           Pivot in column INFO is exactly 0
523: *           Compute the reciprocal pivot growth factor of the
524: *           leading rank-deficient INFO columns of A.
525: *
526:             IF ( N.GT.0 )
527:      $           RPVGRW = DLA_SYRPVGRW(UPLO, N, INFO, A, LDA, AF,
528:      $           LDAF, IPIV, WORK )
529:             RETURN
530:          END IF
531:       END IF
532: *
533: *     Compute the reciprocal pivot growth factor RPVGRW.
534: *
535:       IF ( N.GT.0 )
536:      $     RPVGRW = DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
537:      $     IPIV, WORK )
538: *
539: *     Compute the solution matrix X.
540: *
541:       CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
542:       CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
543: *
544: *     Use iterative refinement to improve the computed solution and
545: *     compute error bounds and backward error estimates for it.
546: *
547:       CALL DSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
548:      $     S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
549:      $     ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO )
550: *
551: *     Scale solutions.
552: *
553:       IF ( RCEQU ) THEN
554:          CALL DLASCL2 ( N, NRHS, S, X, LDX )
555:       END IF
556: *
557:       RETURN
558: *
559: *     End of DSYSVXX
560: *
561:       END
562: