Annotation of rpl/lapack/lapack/dsysvxx.f, revision 1.4
1.1 bertrand 1: SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
2: $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
3: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
4: $ NPARAMS, PARAMS, WORK, IWORK, INFO )
5: *
6: * -- LAPACK 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: INTEGER IPIV( * ), IWORK( * )
24: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
25: $ X( LDX, * ), WORK( * )
26: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
27: $ ERR_BNDS_NORM( NRHS, * ),
28: $ ERR_BNDS_COMP( NRHS, * )
29: * ..
30: *
31: * Purpose
32: * =======
33: *
34: * DSYSVXX uses the diagonal pivoting factorization to compute the
35: * solution to a double precision system of linear equations A * X = B, where A
36: * is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.
37: *
38: * If requested, both normwise and maximum componentwise error bounds
39: * are returned. DSYSVXX 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: * DSYSVXX 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: * DSYSVXX 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 DSYSVXX 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 LU decomposition is used to factor
68: * the matrix A (after equilibration if FACT = 'E') as
69: *
70: * A = U * D * U**T, if UPLO = 'U', or
71: * A = L * D * L**T, if UPLO = 'L',
72: *
73: * where U (or L) is a product of permutation and unit upper (lower)
74: * triangular matrices, and D is symmetric and block diagonal with
75: * 1-by-1 and 2-by-2 diagonal blocks.
76: *
77: * 3. If some D(i,i)=0, so that D is exactly singular, then the
78: * routine returns with INFO = i. Otherwise, the factored form of A
79: * is used to estimate the condition number of the matrix A (see
80: * argument RCOND). If the reciprocal of the condition number is
81: * less than machine precision, the routine still goes on to solve
82: * for X and compute error bounds as described below.
83: *
84: * 4. The system of equations is solved for X using the factored form
85: * of A.
86: *
87: * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
88: * the routine will use iterative refinement to try to get a small
89: * error and error bounds. Refinement calculates the residual to at
90: * least twice the working precision.
91: *
92: * 6. If equilibration was used, the matrix X is premultiplied by
93: * diag(R) so that it solves the original system before
94: * equilibration.
95: *
96: * Arguments
97: * =========
98: *
99: * 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 LDL^T or UDU^T 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
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