Annotation of rpl/lapack/lapack/zhesvxx.f, revision 1.4
1.1 bertrand 1: SUBROUTINE ZHESVXX( 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: $ WORK( * ), X( LDX, * )
26: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
27: $ ERR_BNDS_NORM( NRHS, * ),
28: $ ERR_BNDS_COMP( NRHS, * )
29: * ..
30: *
31: * Purpose
32: * =======
33: *
34: * ZHESVXX 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. ZHESVXX 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: * ZHESVXX 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: * ZHESVXX 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 ZHESVXX 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: * N (input) INTEGER
119: * The number of linear equations, i.e., the order of the
120: * matrix A. N >= 0.
121: *
122: * NRHS (input) INTEGER
123: * The number of right hand sides, i.e., the number of columns
124: * of the matrices B and X. NRHS >= 0.
125: *
126: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
127: * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
128: * upper triangular part of A contains the upper triangular
129: * part of the matrix A, and the strictly lower triangular
130: * part of A is not referenced. If UPLO = 'L', the leading
131: * N-by-N lower triangular part of A contains the lower
132: * triangular part of the matrix A, and the strictly upper
133: * triangular part of A is not referenced.
134: *
135: * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
136: * diag(S)*A*diag(S).
137: *
138: * LDA (input) INTEGER
139: * The leading dimension of the array A. LDA >= max(1,N).
140: *
141: * AF (input or output) COMPLEX*16 array, dimension (LDAF,N)
142: * If FACT = 'F', then AF is an input argument and on entry
143: * contains the block diagonal matrix D and the multipliers
144: * used to obtain the factor U or L from the factorization A =
145: * U*D*U**T or A = L*D*L**T as computed by DSYTRF.
146: *
147: * If FACT = 'N', then AF is an output argument and on exit
148: * returns the block diagonal matrix D and the multipliers
149: * used to obtain the factor U or L from the factorization A =
150: * U*D*U**T or A = L*D*L**T.
151: *
152: * LDAF (input) INTEGER
153: * The leading dimension of the array AF. LDAF >= max(1,N).
154: *
155: * IPIV (input or output) INTEGER array, dimension (N)
156: * If FACT = 'F', then IPIV is an input argument and on entry
157: * contains details of the interchanges and the block
158: * structure of D, as determined by ZHETRF. If IPIV(k) > 0,
159: * then rows and columns k and IPIV(k) were interchanged and
160: * D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and
161: * IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
162: * -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
163: * diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
164: * then rows and columns k+1 and -IPIV(k) were interchanged
165: * and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
166: *
167: * If FACT = 'N', then IPIV is an output argument and on exit
168: * contains details of the interchanges and the block
169: * structure of D, as determined by ZHETRF.
170: *
171: * EQUED (input or output) CHARACTER*1
172: * Specifies the form of equilibration that was done.
173: * = 'N': No equilibration (always true if FACT = 'N').
174: * = 'Y': Both row and column equilibration, i.e., A has been
175: * replaced by diag(S) * A * diag(S).
176: * EQUED is an input argument if FACT = 'F'; otherwise, it is an
177: * output argument.
178: *
179: * S (input or output) DOUBLE PRECISION array, dimension (N)
180: * The scale factors for A. If EQUED = 'Y', A is multiplied on
181: * the left and right by diag(S). S is an input argument if FACT =
182: * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
183: * = 'Y', each element of S must be positive. If S is output, each
184: * element of S is a power of the radix. If S is input, each element
185: * of S should be a power of the radix to ensure a reliable solution
186: * and error estimates. Scaling by powers of the radix does not cause
187: * rounding errors unless the result underflows or overflows.
188: * Rounding errors during scaling lead to refining with a matrix that
189: * is not equivalent to the input matrix, producing error estimates
190: * that may not be reliable.
191: *
192: * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
193: * On entry, the N-by-NRHS right hand side matrix B.
194: * On exit,
195: * if EQUED = 'N', B is not modified;
196: * if EQUED = 'Y', B is overwritten by diag(S)*B;
197: *
198: * LDB (input) INTEGER
199: * The leading dimension of the array B. LDB >= max(1,N).
200: *
201: * X (output) COMPLEX*16 array, dimension (LDX,NRHS)
202: * If INFO = 0, the N-by-NRHS solution matrix X to the original
203: * system of equations. Note that A and B are modified on exit if
204: * EQUED .ne. 'N', and the solution to the equilibrated system is
205: * inv(diag(S))*X.
206: *
207: * LDX (input) INTEGER
208: * The leading dimension of the array X. LDX >= max(1,N).
209: *
210: * RCOND (output) DOUBLE PRECISION
211: * Reciprocal scaled condition number. This is an estimate of the
212: * reciprocal Skeel condition number of the matrix A after
213: * equilibration (if done). If this is less than the machine
214: * precision (in particular, if it is zero), the matrix is singular
215: * to working precision. Note that the error may still be small even
216: * if this number is very small and the matrix appears ill-
217: * conditioned.
218: *
219: * RPVGRW (output) DOUBLE PRECISION
220: * Reciprocal pivot growth. On exit, this contains the reciprocal
221: * pivot growth factor norm(A)/norm(U). The "max absolute element"
222: * norm is used. If this is much less than 1, then the stability of
223: * the LU factorization of the (equilibrated) matrix A could be poor.
224: * This also means that the solution X, estimated condition numbers,
225: * and error bounds could be unreliable. If factorization fails with
226: * 0<INFO<=N, then this contains the reciprocal pivot growth factor
227: * for the leading INFO columns of A.
228: *
229: * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
230: * Componentwise relative backward error. This is the
231: * componentwise relative backward error of each solution vector X(j)
232: * (i.e., the smallest relative change in any element of A or B that
233: * makes X(j) an exact solution).
234: *
235: * N_ERR_BNDS (input) INTEGER
236: * Number of error bounds to return for each right hand side
237: * and each type (normwise or componentwise). See ERR_BNDS_NORM and
238: * ERR_BNDS_COMP below.
239: *
240: * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
241: * For each right-hand side, this array contains information about
242: * various error bounds and condition numbers corresponding to the
243: * normwise relative error, which is defined as follows:
244: *
245: * Normwise relative error in the ith solution vector:
246: * max_j (abs(XTRUE(j,i) - X(j,i)))
247: * ------------------------------
248: * max_j abs(X(j,i))
249: *
250: * The array is indexed by the type of error information as described
251: * below. There currently are up to three pieces of information
252: * returned.
253: *
254: * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
255: * right-hand side.
256: *
257: * The second index in ERR_BNDS_NORM(:,err) contains the following
258: * three fields:
259: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
260: * reciprocal condition number is less than the threshold
261: * sqrt(n) * dlamch('Epsilon').
262: *
263: * err = 2 "Guaranteed" error bound: The estimated forward error,
264: * almost certainly within a factor of 10 of the true error
265: * so long as the next entry is greater than the threshold
266: * sqrt(n) * dlamch('Epsilon'). This error bound should only
267: * be trusted if the previous boolean is true.
268: *
269: * err = 3 Reciprocal condition number: Estimated normwise
270: * reciprocal condition number. Compared with the threshold
271: * sqrt(n) * dlamch('Epsilon') to determine if the error
272: * estimate is "guaranteed". These reciprocal condition
273: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
274: * appropriately scaled matrix Z.
275: * Let Z = S*A, where S scales each row by a power of the
276: * radix so all absolute row sums of Z are approximately 1.
277: *
278: * See Lapack Working Note 165 for further details and extra
279: * cautions.
280: *
281: * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
282: * For each right-hand side, this array contains information about
283: * various error bounds and condition numbers corresponding to the
284: * componentwise relative error, which is defined as follows:
285: *
286: * Componentwise relative error in the ith solution vector:
287: * abs(XTRUE(j,i) - X(j,i))
288: * max_j ----------------------
289: * abs(X(j,i))
290: *
291: * The array is indexed by the right-hand side i (on which the
292: * componentwise relative error depends), and the type of error
293: * information as described below. There currently are up to three
294: * pieces of information returned for each right-hand side. If
295: * componentwise accuracy is not requested (PARAMS(3) = 0.0), then
296: * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
297: * the first (:,N_ERR_BNDS) entries are returned.
298: *
299: * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
300: * right-hand side.
301: *
302: * The second index in ERR_BNDS_COMP(:,err) contains the following
303: * three fields:
304: * err = 1 "Trust/don't trust" boolean. Trust the answer if the
305: * reciprocal condition number is less than the threshold
306: * sqrt(n) * dlamch('Epsilon').
307: *
308: * err = 2 "Guaranteed" error bound: The estimated forward error,
309: * almost certainly within a factor of 10 of the true error
310: * so long as the next entry is greater than the threshold
311: * sqrt(n) * dlamch('Epsilon'). This error bound should only
312: * be trusted if the previous boolean is true.
313: *
314: * err = 3 Reciprocal condition number: Estimated componentwise
315: * reciprocal condition number. Compared with the threshold
316: * sqrt(n) * dlamch('Epsilon') to determine if the error
317: * estimate is "guaranteed". These reciprocal condition
318: * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
319: * appropriately scaled matrix Z.
320: * Let Z = S*(A*diag(x)), where x is the solution for the
321: * current right-hand side and S scales each row of
322: * A*diag(x) by a power of the radix so all absolute row
323: * sums of Z are approximately 1.
324: *
325: * See Lapack Working Note 165 for further details and extra
326: * cautions.
327: *
328: * NPARAMS (input) INTEGER
329: * Specifies the number of parameters set in PARAMS. If .LE. 0, the
330: * PARAMS array is never referenced and default values are used.
331: *
332: * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS
333: * Specifies algorithm parameters. If an entry is .LT. 0.0, then
334: * that entry will be filled with default value used for that
335: * parameter. Only positions up to NPARAMS are accessed; defaults
336: * are used for higher-numbered parameters.
337: *
338: * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
339: * refinement or not.
340: * Default: 1.0D+0
341: * = 0.0 : No refinement is performed, and no error bounds are
342: * computed.
343: * = 1.0 : Use the extra-precise refinement algorithm.
344: * (other values are reserved for future use)
345: *
346: * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
347: * computations allowed for refinement.
348: * Default: 10
349: * Aggressive: Set to 100 to permit convergence using approximate
350: * factorizations or factorizations other than LU. If
351: * the factorization uses a technique other than
352: * Gaussian elimination, the guarantees in
353: * err_bnds_norm and err_bnds_comp may no longer be
354: * trustworthy.
355: *
356: * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
357: * will attempt to find a solution with small componentwise
358: * relative error in the double-precision algorithm. Positive
359: * is true, 0.0 is false.
360: * Default: 1.0 (attempt componentwise convergence)
361: *
362: * WORK (workspace) COMPLEX*16 array, dimension (2*N)
363: *
364: * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
365: *
366: * INFO (output) INTEGER
367: * = 0: Successful exit. The solution to every right-hand side is
368: * guaranteed.
369: * < 0: If INFO = -i, the i-th argument had an illegal value
370: * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
371: * has been completed, but the factor U is exactly singular, so
372: * the solution and error bounds could not be computed. RCOND = 0
373: * is returned.
374: * = N+J: The solution corresponding to the Jth right-hand side is
375: * not guaranteed. The solutions corresponding to other right-
376: * hand sides K with K > J may not be guaranteed as well, but
377: * only the first such right-hand side is reported. If a small
378: * componentwise error is not requested (PARAMS(3) = 0.0) then
379: * the Jth right-hand side is the first with a normwise error
380: * bound that is not guaranteed (the smallest J such
381: * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
382: * the Jth right-hand side is the first with either a normwise or
383: * componentwise error bound that is not guaranteed (the smallest
384: * J such that either ERR_BNDS_NORM(J,1) = 0.0 or
385: * ERR_BNDS_COMP(J,1) = 0.0). See the definition of
386: * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
387: * about all of the right-hand sides check ERR_BNDS_NORM or
388: * ERR_BNDS_COMP.
389: *
390: * ==================================================================
391: *
392: * .. Parameters ..
393: DOUBLE PRECISION ZERO, ONE
394: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
395: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
396: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
397: INTEGER CMP_ERR_I, PIV_GROWTH_I
398: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
399: $ BERR_I = 3 )
400: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
401: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
402: $ PIV_GROWTH_I = 9 )
403: * ..
404: * .. Local Scalars ..
405: LOGICAL EQUIL, NOFACT, RCEQU
406: INTEGER INFEQU, J
407: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
408: * ..
409: * .. External Functions ..
410: EXTERNAL LSAME, DLAMCH, ZLA_HERPVGRW
411: LOGICAL LSAME
412: DOUBLE PRECISION DLAMCH, ZLA_HERPVGRW
413: * ..
414: * .. External Subroutines ..
415: EXTERNAL ZHECON, ZHEEQUB, ZHETRF, ZHETRS, ZLACPY,
416: $ ZLAQHE, XERBLA, ZLASCL2, ZHERFSX
417: * ..
418: * .. Intrinsic Functions ..
419: INTRINSIC MAX, MIN
420: * ..
421: * .. Executable Statements ..
422: *
423: INFO = 0
424: NOFACT = LSAME( FACT, 'N' )
425: EQUIL = LSAME( FACT, 'E' )
426: SMLNUM = DLAMCH( 'Safe minimum' )
427: BIGNUM = ONE / SMLNUM
428: IF( NOFACT .OR. EQUIL ) THEN
429: EQUED = 'N'
430: RCEQU = .FALSE.
431: ELSE
432: RCEQU = LSAME( EQUED, 'Y' )
433: ENDIF
434: *
435: * Default is failure. If an input parameter is wrong or
436: * factorization fails, make everything look horrible. Only the
437: * pivot growth is set here, the rest is initialized in ZHERFSX.
438: *
439: RPVGRW = ZERO
440: *
441: * Test the input parameters. PARAMS is not tested until ZHERFSX.
442: *
443: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
444: $ LSAME( FACT, 'F' ) ) THEN
445: INFO = -1
446: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
447: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
448: INFO = -2
449: ELSE IF( N.LT.0 ) THEN
450: INFO = -3
451: ELSE IF( NRHS.LT.0 ) THEN
452: INFO = -4
453: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
454: INFO = -6
455: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
456: INFO = -8
457: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
458: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
459: INFO = -9
460: ELSE
461: IF ( RCEQU ) THEN
462: SMIN = BIGNUM
463: SMAX = ZERO
464: DO 10 J = 1, N
465: SMIN = MIN( SMIN, S( J ) )
466: SMAX = MAX( SMAX, S( J ) )
467: 10 CONTINUE
468: IF( SMIN.LE.ZERO ) THEN
469: INFO = -10
470: ELSE IF( N.GT.0 ) THEN
471: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
472: ELSE
473: SCOND = ONE
474: END IF
475: END IF
476: IF( INFO.EQ.0 ) THEN
477: IF( LDB.LT.MAX( 1, N ) ) THEN
478: INFO = -12
479: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
480: INFO = -14
481: END IF
482: END IF
483: END IF
484: *
485: IF( INFO.NE.0 ) THEN
486: CALL XERBLA( 'ZHESVXX', -INFO )
487: RETURN
488: END IF
489: *
490: IF( EQUIL ) THEN
491: *
492: * Compute row and column scalings to equilibrate the matrix A.
493: *
494: CALL ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
495: IF( INFEQU.EQ.0 ) THEN
496: *
497: * Equilibrate the matrix.
498: *
499: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
500: RCEQU = LSAME( EQUED, 'Y' )
501: END IF
502: END IF
503: *
504: * Scale the right-hand side.
505: *
506: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
507: *
508: IF( NOFACT .OR. EQUIL ) THEN
509: *
510: * Compute the LDL^T or UDU^T factorization of A.
511: *
512: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
513: CALL ZHETRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
514: *
515: * Return if INFO is non-zero.
516: *
517: IF( INFO.GT.0 ) THEN
518: *
519: * Pivot in column INFO is exactly 0
520: * Compute the reciprocal pivot growth factor of the
521: * leading rank-deficient INFO columns of A.
522: *
523: IF( N.GT.0 )
524: $ RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
525: $ IPIV, RWORK )
526: RETURN
527: END IF
528: END IF
529: *
530: * Compute the reciprocal pivot growth factor RPVGRW.
531: *
532: IF( N.GT.0 )
533: $ RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV,
534: $ RWORK )
535: *
536: * Compute the solution matrix X.
537: *
538: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
539: CALL ZHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
540: *
541: * Use iterative refinement to improve the computed solution and
542: * compute error bounds and backward error estimates for it.
543: *
544: CALL ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
545: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
546: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
547: *
548: * Scale solutions.
549: *
550: IF ( RCEQU ) THEN
551: CALL ZLASCL2 ( N, NRHS, S, X, LDX )
552: END IF
553: *
554: RETURN
555: *
556: * End of ZHESVXX
557: *
558: END
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