Annotation of rpl/lapack/lapack/dposvxx.f, revision 1.2
1.1 bertrand 1: SUBROUTINE DPOSVXX( 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, IWORK, 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: INTEGER 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: * DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
35: * to compute the solution to a double precision system of linear equations
36: * A * X = B, where A is an N-by-N symmetric positive definite matrix
37: * and X and B are N-by-NRHS matrices.
38: *
39: * If requested, both normwise and maximum componentwise error bounds
40: * are returned. DPOSVXX will return a solution with a tiny
41: * guaranteed error (O(eps) where eps is the working machine
42: * precision) unless the matrix is very ill-conditioned, in which
43: * case a warning is returned. Relevant condition numbers also are
44: * calculated and returned.
45: *
46: * DPOSVXX accepts user-provided factorizations and equilibration
47: * factors; see the definitions of the FACT and EQUED options.
48: * Solving with refinement and using a factorization from a previous
49: * DPOSVXX call will also produce a solution with either O(eps)
50: * errors or warnings, but we cannot make that claim for general
51: * user-provided factorizations and equilibration factors if they
52: * differ from what DPOSVXX would itself produce.
53: *
54: * Description
55: * ===========
56: *
57: * The following steps are performed:
58: *
59: * 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
60: * the system:
61: *
62: * diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
63: *
64: * Whether or not the system will be equilibrated depends on the
65: * scaling of the matrix A, but if equilibration is used, A is
66: * overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
67: *
68: * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
69: * factor the matrix A (after equilibration if FACT = 'E') as
70: * A = U**T* U, if UPLO = 'U', or
71: * A = L * L**T, if UPLO = 'L',
72: * where U is an upper triangular matrix and L is a lower triangular
73: * matrix.
74: *
75: * 3. If the leading i-by-i principal minor is not positive definite,
76: * then the routine returns with INFO = i. Otherwise, the factored
77: * form of A is used to estimate the condition number of the matrix
78: * A (see argument RCOND). If the reciprocal of the condition number
79: * is less than machine precision, the routine still goes on to solve
80: * for X and compute error bounds as described below.
81: *
82: * 4. The system of equations is solved for X using the factored form
83: * of A.
84: *
85: * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
86: * the routine will use iterative refinement to try to get a small
87: * error and error bounds. Refinement calculates the residual to at
88: * least twice the working precision.
89: *
90: * 6. If equilibration was used, the matrix X is premultiplied by
91: * diag(S) so that it solves the original system before
92: * equilibration.
93: *
94: * Arguments
95: * =========
96: *
97: * Some optional parameters are bundled in the PARAMS array. These
98: * settings determine how refinement is performed, but often the
99: * defaults are acceptable. If the defaults are acceptable, users
100: * can pass NPARAMS = 0 which prevents the source code from accessing
101: * the PARAMS argument.
102: *
103: * FACT (input) CHARACTER*1
104: * Specifies whether or not the factored form of the matrix A is
105: * supplied on entry, and if not, whether the matrix A should be
106: * equilibrated before it is factored.
107: * = 'F': On entry, AF contains the factored form of A.
108: * If EQUED is not 'N', the matrix A has been
109: * equilibrated with scaling factors given by S.
110: * A and AF are not modified.
111: * = 'N': The matrix A will be copied to AF and factored.
112: * = 'E': The matrix A will be equilibrated if necessary, then
113: * copied to AF and factored.
114: *
115: * UPLO (input) CHARACTER*1
116: * = 'U': Upper triangle of A is stored;
117: * = 'L': Lower triangle of A is stored.
118: *
119: * N (input) INTEGER
120: * The number of linear equations, i.e., the order of the
121: * matrix A. N >= 0.
122: *
123: * NRHS (input) INTEGER
124: * The number of right hand sides, i.e., the number of columns
125: * of the matrices B and X. NRHS >= 0.
126: *
127: * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
128: * On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
129: * 'Y', then A must contain the equilibrated matrix
130: * diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
131: * triangular part of A contains the upper triangular part of the
132: * matrix A, and the strictly lower triangular part of A is not
133: * referenced. If UPLO = 'L', the leading N-by-N lower triangular
134: * part of A contains the lower triangular part of the matrix A, and
135: * the strictly upper triangular part of A is not referenced. A is
136: * not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
137: * 'N' on exit.
138: *
139: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
140: * diag(S)*A*diag(S).
141: *
142: * LDA (input) INTEGER
143: * The leading dimension of the array A. LDA >= max(1,N).
144: *
145: * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
146: * If FACT = 'F', then AF is an input argument and on entry
147: * contains the triangular factor U or L from the Cholesky
148: * factorization A = U**T*U or A = L*L**T, in the same storage
149: * format as A. If EQUED .ne. 'N', then AF is the factored
150: * form of the equilibrated matrix diag(S)*A*diag(S).
151: *
152: * If FACT = 'N', then AF is an output argument and on exit
153: * returns the triangular factor U or L from the Cholesky
154: * factorization A = U**T*U or A = L*L**T of the original
155: * matrix A.
156: *
157: * If FACT = 'E', then AF is an output argument and on exit
158: * returns the triangular factor U or L from the Cholesky
159: * factorization A = U**T*U or A = L*L**T of the equilibrated
160: * matrix A (see the description of A for the form of the
161: * equilibrated matrix).
162: *
163: * LDAF (input) INTEGER
164: * The leading dimension of the array AF. LDAF >= max(1,N).
165: *
166: * EQUED (input or output) CHARACTER*1
167: * Specifies the form of equilibration that was done.
168: * = 'N': No equilibration (always true if FACT = 'N').
169: * = 'Y': Both row and column equilibration, i.e., A has been
170: * replaced by diag(S) * A * diag(S).
171: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
172: * output argument.
173: *
174: * S (input or output) DOUBLE PRECISION array, dimension (N)
175: * The row scale factors for A. If EQUED = 'Y', A is multiplied on
176: * the left and right by diag(S). S is an input argument if FACT =
177: * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
178: * = 'Y', each element of S must be positive. If S is output, each
179: * element of S is a power of the radix. If S is input, each element
180: * of S should be a power of the radix to ensure a reliable solution
181: * and error estimates. Scaling by powers of the radix does not cause
182: * rounding errors unless the result underflows or overflows.
183: * Rounding errors during scaling lead to refining with a matrix that
184: * is not equivalent to the input matrix, producing error estimates
185: * that may not be reliable.
186: *
187: * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
188: * On entry, the N-by-NRHS right hand side matrix B.
189: * On exit,
190: * if EQUED = 'N', B is not modified;
191: * if EQUED = 'Y', B is overwritten by diag(S)*B;
192: *
193: * LDB (input) INTEGER
194: * The leading dimension of the array B. LDB >= max(1,N).
195: *
196: * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
197: * If INFO = 0, the N-by-NRHS solution matrix X to the original
198: * system of equations. Note that A and B are modified on exit if
199: * EQUED .ne. 'N', and the solution to the equilibrated system is
200: * inv(diag(S))*X.
201: *
202: * LDX (input) INTEGER
203: * The leading dimension of the array X. LDX >= max(1,N).
204: *
205: * RCOND (output) DOUBLE PRECISION
206: * Reciprocal scaled condition number. This is an estimate of the
207: * reciprocal Skeel condition number of the matrix A after
208: * equilibration (if done). If this is less than the machine
209: * precision (in particular, if it is zero), the matrix is singular
210: * to working precision. Note that the error may still be small even
211: * if this number is very small and the matrix appears ill-
212: * conditioned.
213: *
214: * RPVGRW (output) DOUBLE PRECISION
215: * Reciprocal pivot growth. On exit, this contains the reciprocal
216: * pivot growth factor norm(A)/norm(U). The "max absolute element"
217: * norm is used. If this is much less than 1, then the stability of
218: * the LU factorization of the (equilibrated) matrix A could be poor.
219: * This also means that the solution X, estimated condition numbers,
220: * and error bounds could be unreliable. If factorization fails with
221: * 0<INFO<=N, then this contains the reciprocal pivot growth factor
222: * for the leading INFO columns of A.
223: *
224: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
225: * Componentwise relative backward error. This is the
226: * componentwise relative backward error of each solution vector X(j)
227: * (i.e., the smallest relative change in any element of A or B that
228: * makes X(j) an exact solution).
229: *
230: * N_ERR_BNDS (input) INTEGER
231: * Number of error bounds to return for each right hand side
232: * and each type (normwise or componentwise). See ERR_BNDS_NORM and
233: * ERR_BNDS_COMP below.
234: *
235: * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
236: * For each right-hand side, this array contains information about
237: * various error bounds and condition numbers corresponding to the
238: * normwise relative error, which is defined as follows:
239: *
240: * Normwise relative error in the ith solution vector:
241: * max_j (abs(XTRUE(j,i) - X(j,i)))
242: * ------------------------------
243: * max_j abs(X(j,i))
244: *
245: * The array is indexed by the type of error information as described
246: * below. There currently are up to three pieces of information
247: * returned.
248: *
249: * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
250: * right-hand side.
251: *
252: * The second index in ERR_BNDS_NORM(:,err) contains the following
253: * three fields:
254: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
255: * reciprocal condition number is less than the threshold
256: * sqrt(n) * dlamch('Epsilon').
257: *
258: * err = 2 "Guaranteed" error bound: The estimated forward error,
259: * almost certainly within a factor of 10 of the true error
260: * so long as the next entry is greater than the threshold
261: * sqrt(n) * dlamch('Epsilon'). This error bound should only
262: * be trusted if the previous boolean is true.
263: *
264: * err = 3 Reciprocal condition number: Estimated normwise
265: * reciprocal condition number. Compared with the threshold
266: * sqrt(n) * dlamch('Epsilon') to determine if the error
267: * estimate is "guaranteed". These reciprocal condition
268: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
269: * appropriately scaled matrix Z.
270: * Let Z = S*A, where S scales each row by a power of the
271: * radix so all absolute row sums of Z are approximately 1.
272: *
273: * See Lapack Working Note 165 for further details and extra
274: * cautions.
275: *
276: * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
277: * For each right-hand side, this array contains information about
278: * various error bounds and condition numbers corresponding to the
279: * componentwise relative error, which is defined as follows:
280: *
281: * Componentwise relative error in the ith solution vector:
282: * abs(XTRUE(j,i) - X(j,i))
283: * max_j ----------------------
284: * abs(X(j,i))
285: *
286: * The array is indexed by the right-hand side i (on which the
287: * componentwise relative error depends), and the type of error
288: * information as described below. There currently are up to three
289: * pieces of information returned for each right-hand side. If
290: * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
291: * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
292: * the first (:,N_ERR_BNDS) entries are returned.
293: *
294: * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
295: * right-hand side.
296: *
297: * The second index in ERR_BNDS_COMP(:,err) contains the following
298: * three fields:
299: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
300: * reciprocal condition number is less than the threshold
301: * sqrt(n) * dlamch('Epsilon').
302: *
303: * err = 2 "Guaranteed" error bound: The estimated forward error,
304: * almost certainly within a factor of 10 of the true error
305: * so long as the next entry is greater than the threshold
306: * sqrt(n) * dlamch('Epsilon'). This error bound should only
307: * be trusted if the previous boolean is true.
308: *
309: * err = 3 Reciprocal condition number: Estimated componentwise
310: * reciprocal condition number. Compared with the threshold
311: * sqrt(n) * dlamch('Epsilon') to determine if the error
312: * estimate is "guaranteed". These reciprocal condition
313: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
314: * appropriately scaled matrix Z.
315: * Let Z = S*(A*diag(x)), where x is the solution for the
316: * current right-hand side and S scales each row of
317: * A*diag(x) by a power of the radix so all absolute row
318: * sums of Z are approximately 1.
319: *
320: * See Lapack Working Note 165 for further details and extra
321: * cautions.
322: *
323: * NPARAMS (input) INTEGER
324: * Specifies the number of parameters set in PARAMS. If .LE. 0, the
325: * PARAMS array is never referenced and default values are used.
326: *
327: * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
328: * Specifies algorithm parameters. If an entry is .LT. 0.0, then
329: * that entry will be filled with default value used for that
330: * parameter. Only positions up to NPARAMS are accessed; defaults
331: * are used for higher-numbered parameters.
332: *
333: * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
334: * refinement or not.
335: * Default: 1.0D+0
336: * = 0.0 : No refinement is performed, and no error bounds are
337: * computed.
338: * = 1.0 : Use the extra-precise refinement algorithm.
339: * (other values are reserved for future use)
340: *
341: * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
342: * computations allowed for refinement.
343: * Default: 10
344: * Aggressive: Set to 100 to permit convergence using approximate
345: * factorizations or factorizations other than LU. If
346: * the factorization uses a technique other than
347: * Gaussian elimination, the guarantees in
348: * err_bnds_norm and err_bnds_comp may no longer be
349: * trustworthy.
350: *
351: * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
352: * will attempt to find a solution with small componentwise
353: * relative error in the double-precision algorithm. Positive
354: * is true, 0.0 is false.
355: * Default: 1.0 (attempt componentwise convergence)
356: *
357: * WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
358: *
359: * IWORK (workspace) INTEGER array, dimension (N)
360: *
361: * INFO (output) INTEGER
362: * = 0: Successful exit. The solution to every right-hand side is
363: * guaranteed.
364: * < 0: If INFO = -i, the i-th argument had an illegal value
365: * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
366: * has been completed, but the factor U is exactly singular, so
367: * the solution and error bounds could not be computed. RCOND = 0
368: * is returned.
369: * = N+J: The solution corresponding to the Jth right-hand side is
370: * not guaranteed. The solutions corresponding to other right-
371: * hand sides K with K > J may not be guaranteed as well, but
372: * only the first such right-hand side is reported. If a small
373: * componentwise error is not requested (PARAMS(3) = 0.0) then
374: * the Jth right-hand side is the first with a normwise error
375: * bound that is not guaranteed (the smallest J such
376: * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
377: * the Jth right-hand side is the first with either a normwise or
378: * componentwise error bound that is not guaranteed (the smallest
379: * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
380: * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
381: * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
382: * about all of the right-hand sides check ERR_BNDS_NORM or
383: * ERR_BNDS_COMP.
384: *
385: * ==================================================================
386: *
387: * .. Parameters ..
388: DOUBLE PRECISION ZERO, ONE
389: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
390: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
391: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
392: INTEGER CMP_ERR_I, PIV_GROWTH_I
393: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
394: $ BERR_I = 3 )
395: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
396: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
397: $ PIV_GROWTH_I = 9 )
398: * ..
399: * .. Local Scalars ..
400: LOGICAL EQUIL, NOFACT, RCEQU
401: INTEGER INFEQU, J
402: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX,
403: $ SCOND, SMLNUM
404: * ..
405: * .. External Functions ..
406: EXTERNAL LSAME, DLAMCH, DLA_PORPVGRW
407: LOGICAL LSAME
408: DOUBLE PRECISION DLAMCH, DLA_PORPVGRW
409: * ..
410: * .. External Subroutines ..
411: EXTERNAL DPOEQUB, DPOTRF, DPOTRS, DLACPY, DLAQSY,
412: $ XERBLA, DLASCL2, DPORFSX
413: * ..
414: * .. Intrinsic Functions ..
415: INTRINSIC MAX, MIN
416: * ..
417: * .. Executable Statements ..
418: *
419: INFO = 0
420: NOFACT = LSAME( FACT, 'N' )
421: EQUIL = LSAME( FACT, 'E' )
422: SMLNUM = DLAMCH( 'Safe minimum' )
423: BIGNUM = ONE / SMLNUM
424: IF( NOFACT .OR. EQUIL ) THEN
425: EQUED = 'N'
426: RCEQU = .FALSE.
427: ELSE
428: RCEQU = LSAME( EQUED, 'Y' )
429: ENDIF
430: *
431: * Default is failure. If an input parameter is wrong or
432: * factorization fails, make everything look horrible. Only the
433: * pivot growth is set here, the rest is initialized in DPORFSX.
434: *
435: RPVGRW = ZERO
436: *
437: * Test the input parameters. PARAMS is not tested until DPORFSX.
438: *
439: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
440: $ LSAME( FACT, 'F' ) ) THEN
441: INFO = -1
442: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
443: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
444: INFO = -2
445: ELSE IF( N.LT.0 ) THEN
446: INFO = -3
447: ELSE IF( NRHS.LT.0 ) THEN
448: INFO = -4
449: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
450: INFO = -6
451: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
452: INFO = -8
453: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
454: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
455: INFO = -9
456: ELSE
457: IF ( RCEQU ) THEN
458: SMIN = BIGNUM
459: SMAX = ZERO
460: DO 10 J = 1, N
461: SMIN = MIN( SMIN, S( J ) )
462: SMAX = MAX( SMAX, S( J ) )
463: 10 CONTINUE
464: IF( SMIN.LE.ZERO ) THEN
465: INFO = -10
466: ELSE IF( N.GT.0 ) THEN
467: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
468: ELSE
469: SCOND = ONE
470: END IF
471: END IF
472: IF( INFO.EQ.0 ) THEN
473: IF( LDB.LT.MAX( 1, N ) ) THEN
474: INFO = -12
475: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
476: INFO = -14
477: END IF
478: END IF
479: END IF
480: *
481: IF( INFO.NE.0 ) THEN
482: CALL XERBLA( 'DPOSVXX', -INFO )
483: RETURN
484: END IF
485: *
486: IF( EQUIL ) THEN
487: *
488: * Compute row and column scalings to equilibrate the matrix A.
489: *
490: CALL DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU )
491: IF( INFEQU.EQ.0 ) THEN
492: *
493: * Equilibrate the matrix.
494: *
495: CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
496: RCEQU = LSAME( EQUED, 'Y' )
497: END IF
498: END IF
499: *
500: * Scale the right-hand side.
501: *
502: IF( RCEQU ) CALL DLASCL2( N, NRHS, S, B, LDB )
503: *
504: IF( NOFACT .OR. EQUIL ) THEN
505: *
506: * Compute the Cholesky factorization of A.
507: *
508: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
509: CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
510: *
511: * Return if INFO is non-zero.
512: *
513: IF( INFO.NE.0 ) THEN
514: *
515: * Pivot in column INFO is exactly 0
516: * Compute the reciprocal pivot growth factor of the
517: * leading rank-deficient INFO columns of A.
518: *
519: RPVGRW = DLA_PORPVGRW( UPLO, INFO, A, LDA, AF, LDAF, WORK )
520: RETURN
521: ENDIF
522: END IF
523: *
524: * Compute the reciprocal growth factor RPVGRW.
525: *
526: RPVGRW = DLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK )
527: *
528: * Compute the solution matrix X.
529: *
530: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
531: CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
532: *
533: * Use iterative refinement to improve the computed solution and
534: * compute error bounds and backward error estimates for it.
535: *
536: CALL DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF,
537: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
538: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO )
539:
540: *
541: * Scale solutions.
542: *
543: IF ( RCEQU ) THEN
544: CALL DLASCL2 ( N, NRHS, S, X, LDX )
545: END IF
546: *
547: RETURN
548: *
549: * End of DPOSVXX
550: *
551: END
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