1: *> \brief <b> ZHESVXX computes the solution to system of linear equations A * X = B for HE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHESVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
22: * EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
23: * N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
24: * NPARAMS, PARAMS, WORK, RWORK, INFO )
25: *
26: * .. Scalar Arguments ..
27: * CHARACTER EQUED, FACT, UPLO
28: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
29: * $ N_ERR_BNDS
30: * DOUBLE PRECISION RCOND, RPVGRW
31: * ..
32: * .. Array Arguments ..
33: * INTEGER IPIV( * )
34: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35: * $ WORK( * ), X( LDX, * )
36: * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
37: * $ ERR_BNDS_NORM( NRHS, * ),
38: * $ ERR_BNDS_COMP( NRHS, * )
39: * ..
40: *
41: *
42: *> \par Purpose:
43: * =============
44: *>
45: *> \verbatim
46: *>
47: *> ZHESVXX uses the diagonal pivoting factorization to compute the
48: *> solution to a complex*16 system of linear equations A * X = B, where
49: *> A is an N-by-N Hermitian matrix and X and B are N-by-NRHS
50: *> matrices.
51: *>
52: *> If requested, both normwise and maximum componentwise error bounds
53: *> are returned. ZHESVXX will return a solution with a tiny
54: *> guaranteed error (O(eps) where eps is the working machine
55: *> precision) unless the matrix is very ill-conditioned, in which
56: *> case a warning is returned. Relevant condition numbers also are
57: *> calculated and returned.
58: *>
59: *> ZHESVXX accepts user-provided factorizations and equilibration
60: *> factors; see the definitions of the FACT and EQUED options.
61: *> Solving with refinement and using a factorization from a previous
62: *> ZHESVXX call will also produce a solution with either O(eps)
63: *> errors or warnings, but we cannot make that claim for general
64: *> user-provided factorizations and equilibration factors if they
65: *> differ from what ZHESVXX would itself produce.
66: *> \endverbatim
67: *
68: *> \par Description:
69: * =================
70: *>
71: *> \verbatim
72: *>
73: *> The following steps are performed:
74: *>
75: *> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
76: *> the system:
77: *>
78: *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
79: *>
80: *> Whether or not the system will be equilibrated depends on the
81: *> scaling of the matrix A, but if equilibration is used, A is
82: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
83: *>
84: *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
85: *> the matrix A (after equilibration if FACT = 'E') as
86: *>
87: *> A = U * D * U**T, if UPLO = 'U', or
88: *> A = L * D * L**T, if UPLO = 'L',
89: *>
90: *> where U (or L) is a product of permutation and unit upper (lower)
91: *> triangular matrices, and D is Hermitian and block diagonal with
92: *> 1-by-1 and 2-by-2 diagonal blocks.
93: *>
94: *> 3. If some D(i,i)=0, so that D is exactly singular, then the
95: *> routine returns with INFO = i. Otherwise, the factored form of A
96: *> is used to estimate the condition number of the matrix A (see
97: *> argument RCOND). If the reciprocal of the condition number is
98: *> less than machine precision, the routine still goes on to solve
99: *> for X and compute error bounds as described below.
100: *>
101: *> 4. The system of equations is solved for X using the factored form
102: *> of A.
103: *>
104: *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
105: *> the routine will use iterative refinement to try to get a small
106: *> error and error bounds. Refinement calculates the residual to at
107: *> least twice the working precision.
108: *>
109: *> 6. If equilibration was used, the matrix X is premultiplied by
110: *> diag(R) so that it solves the original system before
111: *> equilibration.
112: *> \endverbatim
113: *
114: * Arguments:
115: * ==========
116: *
117: *> \verbatim
118: *> Some optional parameters are bundled in the PARAMS array. These
119: *> settings determine how refinement is performed, but often the
120: *> defaults are acceptable. If the defaults are acceptable, users
121: *> can pass NPARAMS = 0 which prevents the source code from accessing
122: *> the PARAMS argument.
123: *> \endverbatim
124: *>
125: *> \param[in] FACT
126: *> \verbatim
127: *> FACT is CHARACTER*1
128: *> Specifies whether or not the factored form of the matrix A is
129: *> supplied on entry, and if not, whether the matrix A should be
130: *> equilibrated before it is factored.
131: *> = 'F': On entry, AF and IPIV contain the factored form of A.
132: *> If EQUED is not 'N', the matrix A has been
133: *> equilibrated with scaling factors given by S.
134: *> A, AF, and IPIV are not modified.
135: *> = 'N': The matrix A will be copied to AF and factored.
136: *> = 'E': The matrix A will be equilibrated if necessary, then
137: *> copied to AF and factored.
138: *> \endverbatim
139: *>
140: *> \param[in] UPLO
141: *> \verbatim
142: *> UPLO is CHARACTER*1
143: *> = 'U': Upper triangle of A is stored;
144: *> = 'L': Lower triangle of A is stored.
145: *> \endverbatim
146: *>
147: *> \param[in] N
148: *> \verbatim
149: *> N is INTEGER
150: *> The number of linear equations, i.e., the order of the
151: *> matrix A. N >= 0.
152: *> \endverbatim
153: *>
154: *> \param[in] NRHS
155: *> \verbatim
156: *> NRHS is INTEGER
157: *> The number of right hand sides, i.e., the number of columns
158: *> of the matrices B and X. NRHS >= 0.
159: *> \endverbatim
160: *>
161: *> \param[in,out] A
162: *> \verbatim
163: *> A is COMPLEX*16 array, dimension (LDA,N)
164: *> The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
165: *> upper triangular part of A contains the upper triangular
166: *> part of the matrix A, and the strictly lower triangular
167: *> part of A is not referenced. If UPLO = 'L', the leading
168: *> N-by-N lower triangular part of A contains the lower
169: *> triangular part of the matrix A, and the strictly upper
170: *> triangular part of A is not referenced.
171: *>
172: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
173: *> diag(S)*A*diag(S).
174: *> \endverbatim
175: *>
176: *> \param[in] LDA
177: *> \verbatim
178: *> LDA is INTEGER
179: *> The leading dimension of the array A. LDA >= max(1,N).
180: *> \endverbatim
181: *>
182: *> \param[in,out] AF
183: *> \verbatim
184: *> AF is COMPLEX*16 array, dimension (LDAF,N)
185: *> If FACT = 'F', then AF is an input argument and on entry
186: *> contains the block diagonal matrix D and the multipliers
187: *> used to obtain the factor U or L from the factorization A =
188: *> U*D*U**H or A = L*D*L**H as computed by ZHETRF.
189: *>
190: *> If FACT = 'N', then AF is an output argument and on exit
191: *> returns the block diagonal matrix D and the multipliers
192: *> used to obtain the factor U or L from the factorization A =
193: *> U*D*U**H or A = L*D*L**H.
194: *> \endverbatim
195: *>
196: *> \param[in] LDAF
197: *> \verbatim
198: *> LDAF is INTEGER
199: *> The leading dimension of the array AF. LDAF >= max(1,N).
200: *> \endverbatim
201: *>
202: *> \param[in,out] IPIV
203: *> \verbatim
204: *> IPIV is INTEGER array, dimension (N)
205: *> If FACT = 'F', then IPIV is an input argument and on entry
206: *> contains details of the interchanges and the block
207: *> structure of D, as determined by ZHETRF. If IPIV(k) > 0,
208: *> then rows and columns k and IPIV(k) were interchanged and
209: *> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and
210: *> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
211: *> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
212: *> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
213: *> then rows and columns k+1 and -IPIV(k) were interchanged
214: *> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
215: *>
216: *> If FACT = 'N', then IPIV is an output argument and on exit
217: *> contains details of the interchanges and the block
218: *> structure of D, as determined by ZHETRF.
219: *> \endverbatim
220: *>
221: *> \param[in,out] EQUED
222: *> \verbatim
223: *> EQUED is CHARACTER*1
224: *> Specifies the form of equilibration that was done.
225: *> = 'N': No equilibration (always true if FACT = 'N').
226: *> = 'Y': Both row and column equilibration, i.e., A has been
227: *> replaced by diag(S) * A * diag(S).
228: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
229: *> output argument.
230: *> \endverbatim
231: *>
232: *> \param[in,out] S
233: *> \verbatim
234: *> S is DOUBLE PRECISION array, dimension (N)
235: *> The scale factors for A. If EQUED = 'Y', A is multiplied on
236: *> the left and right by diag(S). S is an input argument if FACT =
237: *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
238: *> = 'Y', each element of S must be positive. If S is output, each
239: *> element of S is a power of the radix. If S is input, each element
240: *> of S should be a power of the radix to ensure a reliable solution
241: *> and error estimates. Scaling by powers of the radix does not cause
242: *> rounding errors unless the result underflows or overflows.
243: *> Rounding errors during scaling lead to refining with a matrix that
244: *> is not equivalent to the input matrix, producing error estimates
245: *> that may not be reliable.
246: *> \endverbatim
247: *>
248: *> \param[in,out] B
249: *> \verbatim
250: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
251: *> On entry, the N-by-NRHS right hand side matrix B.
252: *> On exit,
253: *> if EQUED = 'N', B is not modified;
254: *> if EQUED = 'Y', B is overwritten by diag(S)*B;
255: *> \endverbatim
256: *>
257: *> \param[in] LDB
258: *> \verbatim
259: *> LDB is INTEGER
260: *> The leading dimension of the array B. LDB >= max(1,N).
261: *> \endverbatim
262: *>
263: *> \param[out] X
264: *> \verbatim
265: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
266: *> If INFO = 0, the N-by-NRHS solution matrix X to the original
267: *> system of equations. Note that A and B are modified on exit if
268: *> EQUED .ne. 'N', and the solution to the equilibrated system is
269: *> inv(diag(S))*X.
270: *> \endverbatim
271: *>
272: *> \param[in] LDX
273: *> \verbatim
274: *> LDX is INTEGER
275: *> The leading dimension of the array X. LDX >= max(1,N).
276: *> \endverbatim
277: *>
278: *> \param[out] RCOND
279: *> \verbatim
280: *> RCOND is DOUBLE PRECISION
281: *> Reciprocal scaled condition number. This is an estimate of the
282: *> reciprocal Skeel condition number of the matrix A after
283: *> equilibration (if done). If this is less than the machine
284: *> precision (in particular, if it is zero), the matrix is singular
285: *> to working precision. Note that the error may still be small even
286: *> if this number is very small and the matrix appears ill-
287: *> conditioned.
288: *> \endverbatim
289: *>
290: *> \param[out] RPVGRW
291: *> \verbatim
292: *> RPVGRW is DOUBLE PRECISION
293: *> Reciprocal pivot growth. On exit, this contains the reciprocal
294: *> pivot growth factor norm(A)/norm(U). The "max absolute element"
295: *> norm is used. If this is much less than 1, then the stability of
296: *> the LU factorization of the (equilibrated) matrix A could be poor.
297: *> This also means that the solution X, estimated condition numbers,
298: *> and error bounds could be unreliable. If factorization fails with
299: *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
300: *> for the leading INFO columns of A.
301: *> \endverbatim
302: *>
303: *> \param[out] BERR
304: *> \verbatim
305: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
306: *> Componentwise relative backward error. This is the
307: *> componentwise relative backward error of each solution vector X(j)
308: *> (i.e., the smallest relative change in any element of A or B that
309: *> makes X(j) an exact solution).
310: *> \endverbatim
311: *>
312: *> \param[in] N_ERR_BNDS
313: *> \verbatim
314: *> N_ERR_BNDS is INTEGER
315: *> Number of error bounds to return for each right hand side
316: *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
317: *> ERR_BNDS_COMP below.
318: *> \endverbatim
319: *>
320: *> \param[out] ERR_BNDS_NORM
321: *> \verbatim
322: *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
323: *> For each right-hand side, this array contains information about
324: *> various error bounds and condition numbers corresponding to the
325: *> normwise relative error, which is defined as follows:
326: *>
327: *> Normwise relative error in the ith solution vector:
328: *> max_j (abs(XTRUE(j,i) - X(j,i)))
329: *> ------------------------------
330: *> max_j abs(X(j,i))
331: *>
332: *> The array is indexed by the type of error information as described
333: *> below. There currently are up to three pieces of information
334: *> returned.
335: *>
336: *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
337: *> right-hand side.
338: *>
339: *> The second index in ERR_BNDS_NORM(:,err) contains the following
340: *> three fields:
341: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
342: *> reciprocal condition number is less than the threshold
343: *> sqrt(n) * dlamch('Epsilon').
344: *>
345: *> err = 2 "Guaranteed" error bound: The estimated forward error,
346: *> almost certainly within a factor of 10 of the true error
347: *> so long as the next entry is greater than the threshold
348: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
349: *> be trusted if the previous boolean is true.
350: *>
351: *> err = 3 Reciprocal condition number: Estimated normwise
352: *> reciprocal condition number. Compared with the threshold
353: *> sqrt(n) * dlamch('Epsilon') to determine if the error
354: *> estimate is "guaranteed". These reciprocal condition
355: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
356: *> appropriately scaled matrix Z.
357: *> Let Z = S*A, where S scales each row by a power of the
358: *> radix so all absolute row sums of Z are approximately 1.
359: *>
360: *> See Lapack Working Note 165 for further details and extra
361: *> cautions.
362: *> \endverbatim
363: *>
364: *> \param[out] ERR_BNDS_COMP
365: *> \verbatim
366: *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
367: *> For each right-hand side, this array contains information about
368: *> various error bounds and condition numbers corresponding to the
369: *> componentwise relative error, which is defined as follows:
370: *>
371: *> Componentwise relative error in the ith solution vector:
372: *> abs(XTRUE(j,i) - X(j,i))
373: *> max_j ----------------------
374: *> abs(X(j,i))
375: *>
376: *> The array is indexed by the right-hand side i (on which the
377: *> componentwise relative error depends), and the type of error
378: *> information as described below. There currently are up to three
379: *> pieces of information returned for each right-hand side. If
380: *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
381: *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
382: *> the first (:,N_ERR_BNDS) entries are returned.
383: *>
384: *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
385: *> right-hand side.
386: *>
387: *> The second index in ERR_BNDS_COMP(:,err) contains the following
388: *> three fields:
389: *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
390: *> reciprocal condition number is less than the threshold
391: *> sqrt(n) * dlamch('Epsilon').
392: *>
393: *> err = 2 "Guaranteed" error bound: The estimated forward error,
394: *> almost certainly within a factor of 10 of the true error
395: *> so long as the next entry is greater than the threshold
396: *> sqrt(n) * dlamch('Epsilon'). This error bound should only
397: *> be trusted if the previous boolean is true.
398: *>
399: *> err = 3 Reciprocal condition number: Estimated componentwise
400: *> reciprocal condition number. Compared with the threshold
401: *> sqrt(n) * dlamch('Epsilon') to determine if the error
402: *> estimate is "guaranteed". These reciprocal condition
403: *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
404: *> appropriately scaled matrix Z.
405: *> Let Z = S*(A*diag(x)), where x is the solution for the
406: *> current right-hand side and S scales each row of
407: *> A*diag(x) by a power of the radix so all absolute row
408: *> sums of Z are approximately 1.
409: *>
410: *> See Lapack Working Note 165 for further details and extra
411: *> cautions.
412: *> \endverbatim
413: *>
414: *> \param[in] NPARAMS
415: *> \verbatim
416: *> NPARAMS is INTEGER
417: *> Specifies the number of parameters set in PARAMS. If <= 0, the
418: *> PARAMS array is never referenced and default values are used.
419: *> \endverbatim
420: *>
421: *> \param[in,out] PARAMS
422: *> \verbatim
423: *> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
424: *> Specifies algorithm parameters. If an entry is < 0.0, then
425: *> that entry will be filled with default value used for that
426: *> parameter. Only positions up to NPARAMS are accessed; defaults
427: *> are used for higher-numbered parameters.
428: *>
429: *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
430: *> refinement or not.
431: *> Default: 1.0D+0
432: *> = 0.0: No refinement is performed, and no error bounds are
433: *> computed.
434: *> = 1.0: Use the extra-precise refinement algorithm.
435: *> (other values are reserved for future use)
436: *>
437: *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
438: *> computations allowed for refinement.
439: *> Default: 10
440: *> Aggressive: Set to 100 to permit convergence using approximate
441: *> factorizations or factorizations other than LU. If
442: *> the factorization uses a technique other than
443: *> Gaussian elimination, the guarantees in
444: *> err_bnds_norm and err_bnds_comp may no longer be
445: *> trustworthy.
446: *>
447: *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
448: *> will attempt to find a solution with small componentwise
449: *> relative error in the double-precision algorithm. Positive
450: *> is true, 0.0 is false.
451: *> Default: 1.0 (attempt componentwise convergence)
452: *> \endverbatim
453: *>
454: *> \param[out] WORK
455: *> \verbatim
456: *> WORK is COMPLEX*16 array, dimension (5*N)
457: *> \endverbatim
458: *>
459: *> \param[out] RWORK
460: *> \verbatim
461: *> RWORK is DOUBLE PRECISION array, dimension (2*N)
462: *> \endverbatim
463: *>
464: *> \param[out] INFO
465: *> \verbatim
466: *> INFO is INTEGER
467: *> = 0: Successful exit. The solution to every right-hand side is
468: *> guaranteed.
469: *> < 0: If INFO = -i, the i-th argument had an illegal value
470: *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
471: *> has been completed, but the factor U is exactly singular, so
472: *> the solution and error bounds could not be computed. RCOND = 0
473: *> is returned.
474: *> = N+J: The solution corresponding to the Jth right-hand side is
475: *> not guaranteed. The solutions corresponding to other right-
476: *> hand sides K with K > J may not be guaranteed as well, but
477: *> only the first such right-hand side is reported. If a small
478: *> componentwise error is not requested (PARAMS(3) = 0.0) then
479: *> the Jth right-hand side is the first with a normwise error
480: *> bound that is not guaranteed (the smallest J such
481: *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
482: *> the Jth right-hand side is the first with either a normwise or
483: *> componentwise error bound that is not guaranteed (the smallest
484: *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
485: *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
486: *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
487: *> about all of the right-hand sides check ERR_BNDS_NORM or
488: *> ERR_BNDS_COMP.
489: *> \endverbatim
490: *
491: * Authors:
492: * ========
493: *
494: *> \author Univ. of Tennessee
495: *> \author Univ. of California Berkeley
496: *> \author Univ. of Colorado Denver
497: *> \author NAG Ltd.
498: *
499: *> \ingroup complex16HEsolve
500: *
501: * =====================================================================
502: SUBROUTINE ZHESVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
503: $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
504: $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
505: $ NPARAMS, PARAMS, WORK, RWORK, INFO )
506: *
507: * -- LAPACK driver routine --
508: * -- LAPACK is a software package provided by Univ. of Tennessee, --
509: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
510: *
511: * .. Scalar Arguments ..
512: CHARACTER EQUED, FACT, UPLO
513: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
514: $ N_ERR_BNDS
515: DOUBLE PRECISION RCOND, RPVGRW
516: * ..
517: * .. Array Arguments ..
518: INTEGER IPIV( * )
519: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
520: $ WORK( * ), X( LDX, * )
521: DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
522: $ ERR_BNDS_NORM( NRHS, * ),
523: $ ERR_BNDS_COMP( NRHS, * )
524: * ..
525: *
526: * ==================================================================
527: *
528: * .. Parameters ..
529: DOUBLE PRECISION ZERO, ONE
530: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
531: INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
532: INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
533: INTEGER CMP_ERR_I, PIV_GROWTH_I
534: PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
535: $ BERR_I = 3 )
536: PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
537: PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
538: $ PIV_GROWTH_I = 9 )
539: * ..
540: * .. Local Scalars ..
541: LOGICAL EQUIL, NOFACT, RCEQU
542: INTEGER INFEQU, J
543: DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
544: * ..
545: * .. External Functions ..
546: EXTERNAL LSAME, DLAMCH, ZLA_HERPVGRW
547: LOGICAL LSAME
548: DOUBLE PRECISION DLAMCH, ZLA_HERPVGRW
549: * ..
550: * .. External Subroutines ..
551: EXTERNAL ZHEEQUB, ZHETRF, ZHETRS, ZLACPY,
552: $ ZLAQHE, XERBLA, ZLASCL2, ZHERFSX
553: * ..
554: * .. Intrinsic Functions ..
555: INTRINSIC MAX, MIN
556: * ..
557: * .. Executable Statements ..
558: *
559: INFO = 0
560: NOFACT = LSAME( FACT, 'N' )
561: EQUIL = LSAME( FACT, 'E' )
562: SMLNUM = DLAMCH( 'Safe minimum' )
563: BIGNUM = ONE / SMLNUM
564: IF( NOFACT .OR. EQUIL ) THEN
565: EQUED = 'N'
566: RCEQU = .FALSE.
567: ELSE
568: RCEQU = LSAME( EQUED, 'Y' )
569: ENDIF
570: *
571: * Default is failure. If an input parameter is wrong or
572: * factorization fails, make everything look horrible. Only the
573: * pivot growth is set here, the rest is initialized in ZHERFSX.
574: *
575: RPVGRW = ZERO
576: *
577: * Test the input parameters. PARAMS is not tested until ZHERFSX.
578: *
579: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.
580: $ LSAME( FACT, 'F' ) ) THEN
581: INFO = -1
582: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND.
583: $ .NOT.LSAME( UPLO, 'L' ) ) THEN
584: INFO = -2
585: ELSE IF( N.LT.0 ) THEN
586: INFO = -3
587: ELSE IF( NRHS.LT.0 ) THEN
588: INFO = -4
589: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
590: INFO = -6
591: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
592: INFO = -8
593: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
594: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
595: INFO = -9
596: ELSE
597: IF ( RCEQU ) THEN
598: SMIN = BIGNUM
599: SMAX = ZERO
600: DO 10 J = 1, N
601: SMIN = MIN( SMIN, S( J ) )
602: SMAX = MAX( SMAX, S( J ) )
603: 10 CONTINUE
604: IF( SMIN.LE.ZERO ) THEN
605: INFO = -10
606: ELSE IF( N.GT.0 ) THEN
607: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
608: ELSE
609: SCOND = ONE
610: END IF
611: END IF
612: IF( INFO.EQ.0 ) THEN
613: IF( LDB.LT.MAX( 1, N ) ) THEN
614: INFO = -12
615: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
616: INFO = -14
617: END IF
618: END IF
619: END IF
620: *
621: IF( INFO.NE.0 ) THEN
622: CALL XERBLA( 'ZHESVXX', -INFO )
623: RETURN
624: END IF
625: *
626: IF( EQUIL ) THEN
627: *
628: * Compute row and column scalings to equilibrate the matrix A.
629: *
630: CALL ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
631: IF( INFEQU.EQ.0 ) THEN
632: *
633: * Equilibrate the matrix.
634: *
635: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
636: RCEQU = LSAME( EQUED, 'Y' )
637: END IF
638: END IF
639: *
640: * Scale the right-hand side.
641: *
642: IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB )
643: *
644: IF( NOFACT .OR. EQUIL ) THEN
645: *
646: * Compute the LDL^H or UDU^H factorization of A.
647: *
648: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
649: CALL ZHETRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
650: *
651: * Return if INFO is non-zero.
652: *
653: IF( INFO.GT.0 ) THEN
654: *
655: * Pivot in column INFO is exactly 0
656: * Compute the reciprocal pivot growth factor of the
657: * leading rank-deficient INFO columns of A.
658: *
659: IF( N.GT.0 )
660: $ RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
661: $ IPIV, RWORK )
662: RETURN
663: END IF
664: END IF
665: *
666: * Compute the reciprocal pivot growth factor RPVGRW.
667: *
668: IF( N.GT.0 )
669: $ RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV,
670: $ RWORK )
671: *
672: * Compute the solution matrix X.
673: *
674: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
675: CALL ZHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
676: *
677: * Use iterative refinement to improve the computed solution and
678: * compute error bounds and backward error estimates for it.
679: *
680: CALL ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
681: $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
682: $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
683: *
684: * Scale solutions.
685: *
686: IF ( RCEQU ) THEN
687: CALL ZLASCL2 ( N, NRHS, S, X, LDX )
688: END IF
689: *
690: RETURN
691: *
692: * End of ZHESVXX
693: *
694: END
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