Annotation of rpl/lapack/lapack/zsysvxx.f, revision 1.2
1.1 bertrand 1: SUBROUTINE ZSYSVXX( 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, RWORK, INFO )
5: *
6: * -- LAPACK driver routine (version 3.2.2) --
7: * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and --
8: * -- Jason Riedy of Univ. of California Berkeley. --
9: * -- June 2010 --
10: *
11: * -- LAPACK is a software package provided by Univ. of Tennessee, --
12: * -- Univ. of California Berkeley and NAG Ltd. --
13: *
14: IMPLICIT NONE
15: * ..
16: * .. Scalar Arguments ..
17: CHARACTER EQUED, FACT, UPLO
18: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
19: $ N_ERR_BNDS
20: DOUBLE PRECISION RCOND, RPVGRW
21: * ..
22: * .. Array Arguments ..
23: INTEGER IPIV( * )
24: COMPLEX*16 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, * ), RWORK( * )
29: * ..
30: *
31: * Purpose
32: * =======
33: *
34: * ZSYSVXX uses the diagonal pivoting factorization to compute the
35: * solution to a complex*16 system of linear equations A * X = B, where
36: * A is an N-by-N symmetric matrix and X and B are N-by-NRHS
37: * matrices.
38: *
39: * If requested, both normwise and maximum componentwise error bounds
40: * are returned. ZSYSVXX 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: * ZSYSVXX 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: * ZSYSVXX 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 ZSYSVXX 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 LU decomposition is used to factor
69: * the matrix A (after equilibration if FACT = 'E') as
70: *
71: * A = U * D * U**T, if UPLO = 'U', or
72: * A = L * D * L**T, if UPLO = 'L',
73: *
74: * where U (or L) is a product of permutation and unit upper (lower)
75: * triangular matrices, and D is symmetric and block diagonal with
76: * 1-by-1 and 2-by-2 diagonal blocks.
77: *
78: * 3. If some D(i,i)=0, so that D is exactly singular, then the
79: * routine returns with INFO = i. Otherwise, the factored form of A
80: * is used to estimate the condition number of the matrix A (see
81: * argument RCOND). If the reciprocal of the condition number is
82: * less than machine precision, the routine still goes on to solve
83: * for X and compute error bounds as described below.
84: *
85: * 4. The system of equations is solved for X using the factored form
86: * of A.
87: *
88: * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
89: * the routine will use iterative refinement to try to get a small
90: * error and error bounds. Refinement calculates the residual to at
91: * least twice the working precision.
92: *
93: * 6. If equilibration was used, the matrix X is premultiplied by
94: * diag(R) so that it solves the original system before
95: * equilibration.
96: *
97: * Arguments
98: * =========
99: *
100: * Some optional parameters are bundled in the PARAMS array. These
101: * settings determine how refinement is performed, but often the
102: * defaults are acceptable. If the defaults are acceptable, users
103: * can pass NPARAMS = 0 which prevents the source code from accessing
104: * the PARAMS argument.
105: *
106: * FACT (input) CHARACTER*1
107: * Specifies whether or not the factored form of the matrix A is
108: * supplied on entry, and if not, whether the matrix A should be
109: * equilibrated before it is factored.
110: * = 'F': On entry, AF and IPIV contain the factored form of A.
111: * If EQUED is not 'N', the matrix A has been
112: * equilibrated with scaling factors given by S.
113: * A, AF, and IPIV are not modified.
114: * = 'N': The matrix A will be copied to AF and factored.
115: * = 'E': The matrix A will be equilibrated if necessary, then
116: * copied to AF and factored.
117: *
118: * UPLO (input) CHARACTER*1
119: * = 'U': Upper triangle of A is stored;
120: * = 'L': Lower triangle of A is stored.
121: *
122: * N (input) INTEGER
123: * The number of linear equations, i.e., the order of the
124: * matrix A. N >= 0.
125: *
126: * NRHS (input) INTEGER
127: * The number of right hand sides, i.e., the number of columns
128: * of the matrices B and X. NRHS >= 0.
129: *
130: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
131: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
132: * upper triangular part of A contains the upper triangular
133: * part of the matrix A, and the strictly lower triangular
134: * part of A is not referenced. If UPLO = 'L', the leading
135: * N-by-N lower triangular part of A contains the lower
136: * triangular part of the matrix A, and the strictly upper
137: * triangular part of A is not referenced.
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) COMPLEX*16 array, dimension (LDAF,N)
146: * If FACT = 'F', then AF is an input argument and on entry
147: * contains the block diagonal matrix D and the multipliers
148: * used to obtain the factor U or L from the factorization A =
149: * U*D*U**T or A = L*D*L**T as computed by DSYTRF.
150: *
151: * If FACT = 'N', then AF is an output argument and on exit
152: * returns the block diagonal matrix D and the multipliers
153: * used to obtain the factor U or L from the factorization A =
154: * U*D*U**T or A = L*D*L**T.
155: *
156: * LDAF (input) INTEGER
157: * The leading dimension of the array AF. LDAF >= max(1,N).
158: *
159: * IPIV (input or output) INTEGER array, dimension (N)
160: * If FACT = 'F', then IPIV is an input argument and on entry
161: * contains details of the interchanges and the block
162: * structure of D, as determined by DSYTRF. If IPIV(k) > 0,
163: * then rows and columns k and IPIV(k) were interchanged and
164: * D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and
165: * IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
166: * -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
167: * diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
168: * then rows and columns k+1 and -IPIV(k) were interchanged
169: * and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
170: *
171: * If FACT = 'N', then IPIV is an output argument and on exit
172: * contains details of the interchanges and the block
173: * structure of D, as determined by DSYTRF.
174: *
175: * EQUED (input or output) CHARACTER*1
176: * Specifies the form of equilibration that was done.
177: * = 'N': No equilibration (always true if FACT = 'N').
178: * = 'Y': Both row and column equilibration, i.e., A has been
179: * replaced by diag(S) * A * diag(S).
180: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
181: * output argument.
182: *
183: * S (input or output) DOUBLE PRECISION array, dimension (N)
184: * The scale factors for A. If EQUED = 'Y', A is multiplied on
185: * the left and right by diag(S). S is an input argument if FACT =
186: * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
187: * = 'Y', each element of S must be positive. If S is output, each
188: * element of S is a power of the radix. If S is input, each element
189: * of S should be a power of the radix to ensure a reliable solution
190: * and error estimates. Scaling by powers of the radix does not cause
191: * rounding errors unless the result underflows or overflows.
192: * Rounding errors during scaling lead to refining with a matrix that
193: * is not equivalent to the input matrix, producing error estimates
194: * that may not be reliable.
195: *
196: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
197: * On entry, the N-by-NRHS right hand side matrix B.
198: * On exit,
199: * if EQUED = 'N', B is not modified;
200: * if EQUED = 'Y', B is overwritten by diag(S)*B;
201: *
202: * LDB (input) INTEGER
203: * The leading dimension of the array B. LDB >= max(1,N).
204: *
205: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
206: * If INFO = 0, the N-by-NRHS solution matrix X to the original
207: * system of equations. Note that A and B are modified on exit if
208: * EQUED .ne. 'N', and the solution to the equilibrated system is
209: * inv(diag(S))*X.
210: *
211: * LDX (input) INTEGER
212: * The leading dimension of the array X. LDX >= max(1,N).
213: *
214: * RCOND (output) DOUBLE PRECISION
215: * Reciprocal scaled condition number. This is an estimate of the
216: * reciprocal Skeel condition number of the matrix A after
217: * equilibration (if done). If this is less than the machine
218: * precision (in particular, if it is zero), the matrix is singular
219: * to working precision. Note that the error may still be small even
220: * if this number is very small and the matrix appears ill-
221: * conditioned.
222: *
223: * RPVGRW (output) DOUBLE PRECISION
224: * Reciprocal pivot growth. On exit, this contains the reciprocal
225: * pivot growth factor norm(A)/norm(U). The "max absolute element"
226: * norm is used. If this is much less than 1, then the stability of
227: * the LU factorization of the (equilibrated) matrix A could be poor.
228: * This also means that the solution X, estimated condition numbers,
229: * and error bounds could be unreliable. If factorization fails with
230: * 0<INFO<=N, then this contains the reciprocal pivot growth factor
231: * for the leading INFO columns of A.
232: *
233: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
234: * Componentwise relative backward error. This is the
235: * componentwise relative backward error of each solution vector X(j)
236: * (i.e., the smallest relative change in any element of A or B that
237: * makes X(j) an exact solution).
238: *
239: * N_ERR_BNDS (input) INTEGER
240: * Number of error bounds to return for each right hand side
241: * and each type (normwise or componentwise). See ERR_BNDS_NORM and
242: * ERR_BNDS_COMP below.
243: *
244: * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
245: * For each right-hand side, this array contains information about
246: * various error bounds and condition numbers corresponding to the
247: * normwise relative error, which is defined as follows:
248: *
249: * Normwise relative error in the ith solution vector:
250: * max_j (abs(XTRUE(j,i) - X(j,i)))
251: * ------------------------------
252: * max_j abs(X(j,i))
253: *
254: * The array is indexed by the type of error information as described
255: * below. There currently are up to three pieces of information
256: * returned.
257: *
258: * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
259: * right-hand side.
260: *
261: * The second index in ERR_BNDS_NORM(:,err) contains the following
262: * three fields:
263: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
264: * reciprocal condition number is less than the threshold
265: * sqrt(n) * dlamch('Epsilon').
266: *
267: * err = 2 "Guaranteed" error bound: The estimated forward error,
268: * almost certainly within a factor of 10 of the true error
269: * so long as the next entry is greater than the threshold
270: * sqrt(n) * dlamch('Epsilon'). This error bound should only
271: * be trusted if the previous boolean is true.
272: *
273: * err = 3 Reciprocal condition number: Estimated normwise
274: * reciprocal condition number. Compared with the threshold
275: * sqrt(n) * dlamch('Epsilon') to determine if the error
276: * estimate is "guaranteed". These reciprocal condition
277: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
278: * appropriately scaled matrix Z.
279: * Let Z = S*A, where S scales each row by a power of the
280: * radix so all absolute row sums of Z are approximately 1.
281: *
282: * See Lapack Working Note 165 for further details and extra
283: * cautions.
284: *
285: * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
286: * For each right-hand side, this array contains information about
287: * various error bounds and condition numbers corresponding to the
288: * componentwise relative error, which is defined as follows:
289: *
290: * Componentwise relative error in the ith solution vector:
291: * abs(XTRUE(j,i) - X(j,i))
292: * max_j ----------------------
293: * abs(X(j,i))
294: *
295: * The array is indexed by the right-hand side i (on which the
296: * componentwise relative error depends), and the type of error
297: * information as described below. There currently are up to three
298: * pieces of information returned for each right-hand side. If
299: * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
300: * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
301: * the first (:,N_ERR_BNDS) entries are returned.
302: *
303: * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
304: * right-hand side.
305: *
306: * The second index in ERR_BNDS_COMP(:,err) contains the following
307: * three fields:
308: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
309: * reciprocal condition number is less than the threshold
310: * sqrt(n) * dlamch('Epsilon').
311: *
312: * err = 2 "Guaranteed" error bound: The estimated forward error,
313: * almost certainly within a factor of 10 of the true error
314: * so long as the next entry is greater than the threshold
315: * sqrt(n) * dlamch('Epsilon'). This error bound should only
316: * be trusted if the previous boolean is true.
317: *
318: * err = 3 Reciprocal condition number: Estimated componentwise
319: * reciprocal condition number. Compared with the threshold
320: * sqrt(n) * dlamch('Epsilon') to determine if the error
321: * estimate is "guaranteed". These reciprocal condition
322: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
323: * appropriately scaled matrix Z.
324: * Let Z = S*(A*diag(x)), where x is the solution for the
325: * current right-hand side and S scales each row of
326: * A*diag(x) by a power of the radix so all absolute row
327: * sums of Z are approximately 1.
328: *
329: * See Lapack Working Note 165 for further details and extra
330: * cautions.
331: *
332: * NPARAMS (input) INTEGER
333: * Specifies the number of parameters set in PARAMS. If .LE. 0, the
334: * PARAMS array is never referenced and default values are used.
335: *
336: * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
337: * Specifies algorithm parameters. If an entry is .LT. 0.0, then
338: * that entry will be filled with default value used for that
339: * parameter. Only positions up to NPARAMS are accessed; defaults
340: * are used for higher-numbered parameters.
341: *
342: * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
343: * refinement or not.
344: * Default: 1.0D+0
345: * = 0.0 : No refinement is performed, and no error bounds are
346: * computed.
347: * = 1.0 : Use the extra-precise refinement algorithm.
348: * (other values are reserved for future use)
349: *
350: * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
351: * computations allowed for refinement.
352: * Default: 10
353: * Aggressive: Set to 100 to permit convergence using approximate
354: * factorizations or factorizations other than LU. If
355: * the factorization uses a technique other than
356: * Gaussian elimination, the guarantees in
357: * err_bnds_norm and err_bnds_comp may no longer be
358: * trustworthy.
359: *
360: * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
361: * will attempt to find a solution with small componentwise
362: * relative error in the double-precision algorithm. Positive
363: * is true, 0.0 is false.
364: * Default: 1.0 (attempt componentwise convergence)
365: *
366: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
367: *
368: * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
369: *
370: * INFO (output) INTEGER
371: * = 0: Successful exit. The solution to every right-hand side is
372: * guaranteed.
373: * < 0: If INFO = -i, the i-th argument had an illegal value
374: * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
375: * has been completed, but the factor U is exactly singular, so
376: * the solution and error bounds could not be computed. RCOND = 0
377: * is returned.
378: * = N+J: The solution corresponding to the Jth right-hand side is
379: * not guaranteed. The solutions corresponding to other right-
380: * hand sides K with K > J may not be guaranteed as well, but
381: * only the first such right-hand side is reported. If a small
382: * componentwise error is not requested (PARAMS(3) = 0.0) then
383: * the Jth right-hand side is the first with a normwise error
384: * bound that is not guaranteed (the smallest J such
385: * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
386: * the Jth right-hand side is the first with either a normwise or
387: * componentwise error bound that is not guaranteed (the smallest
388: * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
389: * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
390: * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
391: * about all of the right-hand sides check ERR_BNDS_NORM or
392: * ERR_BNDS_COMP.
393: *
394: * ==================================================================
395: *
396: * .. Parameters ..
397: DOUBLE PRECISION ZERO, ONE
398: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
399: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
400: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
401: INTEGER CMP_ERR_I, PIV_GROWTH_I
402: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
403: $ BERR_I = 3 )
404: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
405: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
406: $ PIV_GROWTH_I = 9 )
407: * ..
408: * .. Local Scalars ..
409: LOGICAL EQUIL, NOFACT, RCEQU
410: INTEGER INFEQU, J
411: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
412: * ..
413: * .. External Functions ..
414: EXTERNAL LSAME, DLAMCH, ZLA_SYRPVGRW
415: LOGICAL LSAME
416: DOUBLE PRECISION DLAMCH, ZLA_SYRPVGRW
417: * ..
418: * .. External Subroutines ..
419: EXTERNAL ZSYCON, ZSYEQUB, ZSYTRF, ZSYTRS, ZLACPY,
420: $ ZLAQSY, XERBLA, ZLASCL2, ZSYRFSX
421: * ..
422: * .. Intrinsic Functions ..
423: INTRINSIC MAX, MIN
424: * ..
425: * .. Executable Statements ..
426: *
427: INFO = 0
428: NOFACT = LSAME( FACT, 'N' )
429: EQUIL = LSAME( FACT, 'E' )
430: SMLNUM = DLAMCH( 'Safe minimum' )
431: BIGNUM = ONE / SMLNUM
432: IF( NOFACT .OR. EQUIL ) THEN
433: EQUED = 'N'
434: RCEQU = .FALSE.
435: ELSE
436: RCEQU = LSAME( EQUED, 'Y' )
437: ENDIF
438: *
439: * Default is failure. If an input parameter is wrong or
440: * factorization fails, make everything look horrible. Only the
441: * pivot growth is set here, the rest is initialized in ZSYRFSX.
442: *
443: RPVGRW = ZERO
444: *
445: * Test the input parameters. PARAMS is not tested until ZSYRFSX.
446: *
447: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
448: $ LSAME( FACT, 'F' ) ) THEN
449: INFO = -1
450: ELSE IF( .NOT.LSAME(UPLO, 'U') .AND.
451: $ .NOT.LSAME(UPLO, 'L') ) THEN
452: INFO = -2
453: ELSE IF( N.LT.0 ) THEN
454: INFO = -3
455: ELSE IF( NRHS.LT.0 ) THEN
456: INFO = -4
457: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
458: INFO = -6
459: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
460: INFO = -8
461: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
462: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
463: INFO = -9
464: ELSE
465: IF ( RCEQU ) THEN
466: SMIN = BIGNUM
467: SMAX = ZERO
468: DO 10 J = 1, N
469: SMIN = MIN( SMIN, S( J ) )
470: SMAX = MAX( SMAX, S( J ) )
471: 10 CONTINUE
472: IF( SMIN.LE.ZERO ) THEN
473: INFO = -10
474: ELSE IF( N.GT.0 ) THEN
475: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
476: ELSE
477: SCOND = ONE
478: END IF
479: END IF
480: IF( INFO.EQ.0 ) THEN
481: IF( LDB.LT.MAX( 1, N ) ) THEN
482: INFO = -12
483: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
484: INFO = -14
485: END IF
486: END IF
487: END IF
488: *
489: IF( INFO.NE.0 ) THEN
490: CALL XERBLA( 'ZSYSVXX', -INFO )
491: RETURN
492: END IF
493: *
494: IF( EQUIL ) THEN
495: *
496: * Compute row and column scalings to equilibrate the matrix A.
497: *
498: CALL ZSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
499: IF( INFEQU.EQ.0 ) THEN
500: *
501: * Equilibrate the matrix.
502: *
503: CALL ZLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
504: RCEQU = LSAME( EQUED, 'Y' )
505: END IF
506:
507: END IF
508: *
509: * Scale the right hand-side.
510: *
511: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
512: *
513: IF( NOFACT .OR. EQUIL ) THEN
514: *
515: * Compute the LDL^T or UDU^T factorization of A.
516: *
517: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
518: CALL ZSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
519: *
520: * Return if INFO is non-zero.
521: *
522: IF( INFO.GT.0 ) THEN
523: *
524: * Pivot in column INFO is exactly 0
525: * Compute the reciprocal pivot growth factor of the
526: * leading rank-deficient INFO columns of A.
527: *
528: IF ( N.GT.0 )
529: $ RPVGRW = ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
530: $ LDAF, IPIV, RWORK )
531: RETURN
532: END IF
533: END IF
534: *
535: * Compute the reciprocal pivot growth factor RPVGRW.
536: *
537: IF ( N.GT.0 )
538: $ RPVGRW = ZLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
539: $ IPIV, RWORK )
540: *
541: * Compute the solution matrix X.
542: *
543: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
544: CALL ZSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
545: *
546: * Use iterative refinement to improve the computed solution and
547: * compute error bounds and backward error estimates for it.
548: *
549: CALL ZSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
550: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
551: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
552: *
553: * Scale solutions.
554: *
555: IF ( RCEQU ) THEN
556: CALL ZLASCL2 (N, NRHS, S, X, LDX )
557: END IF
558: *
559: RETURN
560: *
561: * End of ZSYSVXX
562: *
563: END
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