1: *> \brief <b> ZPOSVX computes the solution to system of linear equations A * X = B for PO matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZPOSVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zposvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zposvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zposvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
22: * S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
23: * RWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER EQUED, FACT, UPLO
27: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
28: * DOUBLE PRECISION RCOND
29: * ..
30: * .. Array Arguments ..
31: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
32: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
33: * $ WORK( * ), X( LDX, * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
43: *> compute the solution to a complex system of linear equations
44: *> A * X = B,
45: *> where A is an N-by-N Hermitian positive definite matrix and X and B
46: *> are N-by-NRHS matrices.
47: *>
48: *> Error bounds on the solution and a condition estimate are also
49: *> provided.
50: *> \endverbatim
51: *
52: *> \par Description:
53: * =================
54: *>
55: *> \verbatim
56: *>
57: *> The following steps are performed:
58: *>
59: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
60: *> the system:
61: *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
62: *> Whether or not the system will be equilibrated depends on the
63: *> scaling of the matrix A, but if equilibration is used, A is
64: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
65: *>
66: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
67: *> factor the matrix A (after equilibration if FACT = 'E') as
68: *> A = U**H* U, if UPLO = 'U', or
69: *> A = L * L**H, if UPLO = 'L',
70: *> where U is an upper triangular matrix and L is a lower triangular
71: *> matrix.
72: *>
73: *> 3. If the leading i-by-i principal minor is not positive definite,
74: *> then the routine returns with INFO = i. Otherwise, the factored
75: *> form of A is used to estimate the condition number of the matrix
76: *> A. If the reciprocal of the condition number is less than machine
77: *> precision, INFO = N+1 is returned as a warning, but the routine
78: *> still goes on to solve for X and compute error bounds as
79: *> described below.
80: *>
81: *> 4. The system of equations is solved for X using the factored form
82: *> of A.
83: *>
84: *> 5. Iterative refinement is applied to improve the computed solution
85: *> matrix and calculate error bounds and backward error estimates
86: *> for it.
87: *>
88: *> 6. If equilibration was used, the matrix X is premultiplied by
89: *> diag(S) so that it solves the original system before
90: *> equilibration.
91: *> \endverbatim
92: *
93: * Arguments:
94: * ==========
95: *
96: *> \param[in] FACT
97: *> \verbatim
98: *> FACT is CHARACTER*1
99: *> Specifies whether or not the factored form of the matrix A is
100: *> supplied on entry, and if not, whether the matrix A should be
101: *> equilibrated before it is factored.
102: *> = 'F': On entry, AF contains the factored form of A.
103: *> If EQUED = 'Y', the matrix A has been equilibrated
104: *> with scaling factors given by S. A and AF will not
105: *> be modified.
106: *> = 'N': The matrix A will be copied to AF and factored.
107: *> = 'E': The matrix A will be equilibrated if necessary, then
108: *> copied to AF and factored.
109: *> \endverbatim
110: *>
111: *> \param[in] UPLO
112: *> \verbatim
113: *> UPLO is CHARACTER*1
114: *> = 'U': Upper triangle of A is stored;
115: *> = 'L': Lower triangle of A is stored.
116: *> \endverbatim
117: *>
118: *> \param[in] N
119: *> \verbatim
120: *> N is INTEGER
121: *> The number of linear equations, i.e., the order of the
122: *> matrix A. N >= 0.
123: *> \endverbatim
124: *>
125: *> \param[in] NRHS
126: *> \verbatim
127: *> NRHS is INTEGER
128: *> The number of right hand sides, i.e., the number of columns
129: *> of the matrices B and X. NRHS >= 0.
130: *> \endverbatim
131: *>
132: *> \param[in,out] A
133: *> \verbatim
134: *> A is COMPLEX*16 array, dimension (LDA,N)
135: *> On entry, the Hermitian matrix A, except if FACT = 'F' and
136: *> EQUED = 'Y', then A must contain the equilibrated matrix
137: *> diag(S)*A*diag(S). If UPLO = 'U', the leading
138: *> N-by-N upper triangular part of A contains the upper
139: *> triangular part of the matrix A, and the strictly lower
140: *> triangular part of A is not referenced. If UPLO = 'L', the
141: *> leading N-by-N lower triangular part of A contains the lower
142: *> triangular part of the matrix A, and the strictly upper
143: *> triangular part of A is not referenced. A is not modified if
144: *> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
145: *>
146: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
147: *> diag(S)*A*diag(S).
148: *> \endverbatim
149: *>
150: *> \param[in] LDA
151: *> \verbatim
152: *> LDA is INTEGER
153: *> The leading dimension of the array A. LDA >= max(1,N).
154: *> \endverbatim
155: *>
156: *> \param[in,out] AF
157: *> \verbatim
158: *> AF is COMPLEX*16 array, dimension (LDAF,N)
159: *> If FACT = 'F', then AF is an input argument and on entry
160: *> contains the triangular factor U or L from the Cholesky
161: *> factorization A = U**H *U or A = L*L**H, in the same storage
162: *> format as A. If EQUED .ne. 'N', then AF is the factored form
163: *> of the equilibrated matrix diag(S)*A*diag(S).
164: *>
165: *> If FACT = 'N', then AF is an output argument and on exit
166: *> returns the triangular factor U or L from the Cholesky
167: *> factorization A = U**H *U or A = L*L**H of the original
168: *> matrix A.
169: *>
170: *> If FACT = 'E', then AF is an output argument and on exit
171: *> returns the triangular factor U or L from the Cholesky
172: *> factorization A = U**H *U or A = L*L**H of the equilibrated
173: *> matrix A (see the description of A for the form of the
174: *> equilibrated matrix).
175: *> \endverbatim
176: *>
177: *> \param[in] LDAF
178: *> \verbatim
179: *> LDAF is INTEGER
180: *> The leading dimension of the array AF. LDAF >= max(1,N).
181: *> \endverbatim
182: *>
183: *> \param[in,out] EQUED
184: *> \verbatim
185: *> EQUED is CHARACTER*1
186: *> Specifies the form of equilibration that was done.
187: *> = 'N': No equilibration (always true if FACT = 'N').
188: *> = 'Y': Equilibration was done, i.e., A has been replaced by
189: *> diag(S) * A * diag(S).
190: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
191: *> output argument.
192: *> \endverbatim
193: *>
194: *> \param[in,out] S
195: *> \verbatim
196: *> S is DOUBLE PRECISION array, dimension (N)
197: *> The scale factors for A; not accessed if EQUED = 'N'. S is
198: *> an input argument if FACT = 'F'; otherwise, S is an output
199: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
200: *> must be positive.
201: *> \endverbatim
202: *>
203: *> \param[in,out] B
204: *> \verbatim
205: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
206: *> On entry, the N-by-NRHS righthand side matrix B.
207: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
208: *> B is overwritten by diag(S) * B.
209: *> \endverbatim
210: *>
211: *> \param[in] LDB
212: *> \verbatim
213: *> LDB is INTEGER
214: *> The leading dimension of the array B. LDB >= max(1,N).
215: *> \endverbatim
216: *>
217: *> \param[out] X
218: *> \verbatim
219: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
220: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
221: *> the original system of equations. Note that if EQUED = 'Y',
222: *> A and B are modified on exit, and the solution to the
223: *> equilibrated system is inv(diag(S))*X.
224: *> \endverbatim
225: *>
226: *> \param[in] LDX
227: *> \verbatim
228: *> LDX is INTEGER
229: *> The leading dimension of the array X. LDX >= max(1,N).
230: *> \endverbatim
231: *>
232: *> \param[out] RCOND
233: *> \verbatim
234: *> RCOND is DOUBLE PRECISION
235: *> The estimate of the reciprocal condition number of the matrix
236: *> A after equilibration (if done). If RCOND is less than the
237: *> machine precision (in particular, if RCOND = 0), the matrix
238: *> is singular to working precision. This condition is
239: *> indicated by a return code of INFO > 0.
240: *> \endverbatim
241: *>
242: *> \param[out] FERR
243: *> \verbatim
244: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
245: *> The estimated forward error bound for each solution vector
246: *> X(j) (the j-th column of the solution matrix X).
247: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
248: *> is an estimated upper bound for the magnitude of the largest
249: *> element in (X(j) - XTRUE) divided by the magnitude of the
250: *> largest element in X(j). The estimate is as reliable as
251: *> the estimate for RCOND, and is almost always a slight
252: *> overestimate of the true error.
253: *> \endverbatim
254: *>
255: *> \param[out] BERR
256: *> \verbatim
257: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
258: *> The componentwise relative backward error of each solution
259: *> vector X(j) (i.e., the smallest relative change in
260: *> any element of A or B that makes X(j) an exact solution).
261: *> \endverbatim
262: *>
263: *> \param[out] WORK
264: *> \verbatim
265: *> WORK is COMPLEX*16 array, dimension (2*N)
266: *> \endverbatim
267: *>
268: *> \param[out] RWORK
269: *> \verbatim
270: *> RWORK is DOUBLE PRECISION array, dimension (N)
271: *> \endverbatim
272: *>
273: *> \param[out] INFO
274: *> \verbatim
275: *> INFO is INTEGER
276: *> = 0: successful exit
277: *> < 0: if INFO = -i, the i-th argument had an illegal value
278: *> > 0: if INFO = i, and i is
279: *> <= N: the leading minor of order i of A is
280: *> not positive definite, so the factorization
281: *> could not be completed, and the solution has not
282: *> been computed. RCOND = 0 is returned.
283: *> = N+1: U is nonsingular, but RCOND is less than machine
284: *> precision, meaning that the matrix is singular
285: *> to working precision. Nevertheless, the
286: *> solution and error bounds are computed because
287: *> there are a number of situations where the
288: *> computed solution can be more accurate than the
289: *> value of RCOND would suggest.
290: *> \endverbatim
291: *
292: * Authors:
293: * ========
294: *
295: *> \author Univ. of Tennessee
296: *> \author Univ. of California Berkeley
297: *> \author Univ. of Colorado Denver
298: *> \author NAG Ltd.
299: *
300: *> \ingroup complex16POsolve
301: *
302: * =====================================================================
303: SUBROUTINE ZPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
304: $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
305: $ RWORK, INFO )
306: *
307: * -- LAPACK driver routine --
308: * -- LAPACK is a software package provided by Univ. of Tennessee, --
309: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
310: *
311: * .. Scalar Arguments ..
312: CHARACTER EQUED, FACT, UPLO
313: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
314: DOUBLE PRECISION RCOND
315: * ..
316: * .. Array Arguments ..
317: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
318: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
319: $ WORK( * ), X( LDX, * )
320: * ..
321: *
322: * =====================================================================
323: *
324: * .. Parameters ..
325: DOUBLE PRECISION ZERO, ONE
326: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
327: * ..
328: * .. Local Scalars ..
329: LOGICAL EQUIL, NOFACT, RCEQU
330: INTEGER I, INFEQU, J
331: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
332: * ..
333: * .. External Functions ..
334: LOGICAL LSAME
335: DOUBLE PRECISION DLAMCH, ZLANHE
336: EXTERNAL LSAME, DLAMCH, ZLANHE
337: * ..
338: * .. External Subroutines ..
339: EXTERNAL XERBLA, ZLACPY, ZLAQHE, ZPOCON, ZPOEQU, ZPORFS,
340: $ ZPOTRF, ZPOTRS
341: * ..
342: * .. Intrinsic Functions ..
343: INTRINSIC MAX, MIN
344: * ..
345: * .. Executable Statements ..
346: *
347: INFO = 0
348: NOFACT = LSAME( FACT, 'N' )
349: EQUIL = LSAME( FACT, 'E' )
350: IF( NOFACT .OR. EQUIL ) THEN
351: EQUED = 'N'
352: RCEQU = .FALSE.
353: ELSE
354: RCEQU = LSAME( EQUED, 'Y' )
355: SMLNUM = DLAMCH( 'Safe minimum' )
356: BIGNUM = ONE / SMLNUM
357: END IF
358: *
359: * Test the input parameters.
360: *
361: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
362: $ THEN
363: INFO = -1
364: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
365: $ THEN
366: INFO = -2
367: ELSE IF( N.LT.0 ) THEN
368: INFO = -3
369: ELSE IF( NRHS.LT.0 ) THEN
370: INFO = -4
371: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
372: INFO = -6
373: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
374: INFO = -8
375: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
376: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
377: INFO = -9
378: ELSE
379: IF( RCEQU ) THEN
380: SMIN = BIGNUM
381: SMAX = ZERO
382: DO 10 J = 1, N
383: SMIN = MIN( SMIN, S( J ) )
384: SMAX = MAX( SMAX, S( J ) )
385: 10 CONTINUE
386: IF( SMIN.LE.ZERO ) THEN
387: INFO = -10
388: ELSE IF( N.GT.0 ) THEN
389: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
390: ELSE
391: SCOND = ONE
392: END IF
393: END IF
394: IF( INFO.EQ.0 ) THEN
395: IF( LDB.LT.MAX( 1, N ) ) THEN
396: INFO = -12
397: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
398: INFO = -14
399: END IF
400: END IF
401: END IF
402: *
403: IF( INFO.NE.0 ) THEN
404: CALL XERBLA( 'ZPOSVX', -INFO )
405: RETURN
406: END IF
407: *
408: IF( EQUIL ) THEN
409: *
410: * Compute row and column scalings to equilibrate the matrix A.
411: *
412: CALL ZPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
413: IF( INFEQU.EQ.0 ) THEN
414: *
415: * Equilibrate the matrix.
416: *
417: CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
418: RCEQU = LSAME( EQUED, 'Y' )
419: END IF
420: END IF
421: *
422: * Scale the right hand side.
423: *
424: IF( RCEQU ) THEN
425: DO 30 J = 1, NRHS
426: DO 20 I = 1, N
427: B( I, J ) = S( I )*B( I, J )
428: 20 CONTINUE
429: 30 CONTINUE
430: END IF
431: *
432: IF( NOFACT .OR. EQUIL ) THEN
433: *
434: * Compute the Cholesky factorization A = U**H *U or A = L*L**H.
435: *
436: CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
437: CALL ZPOTRF( UPLO, N, AF, LDAF, INFO )
438: *
439: * Return if INFO is non-zero.
440: *
441: IF( INFO.GT.0 )THEN
442: RCOND = ZERO
443: RETURN
444: END IF
445: END IF
446: *
447: * Compute the norm of the matrix A.
448: *
449: ANORM = ZLANHE( '1', UPLO, N, A, LDA, RWORK )
450: *
451: * Compute the reciprocal of the condition number of A.
452: *
453: CALL ZPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, RWORK, INFO )
454: *
455: * Compute the solution matrix X.
456: *
457: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
458: CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
459: *
460: * Use iterative refinement to improve the computed solution and
461: * compute error bounds and backward error estimates for it.
462: *
463: CALL ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
464: $ FERR, BERR, WORK, RWORK, INFO )
465: *
466: * Transform the solution matrix X to a solution of the original
467: * system.
468: *
469: IF( RCEQU ) THEN
470: DO 50 J = 1, NRHS
471: DO 40 I = 1, N
472: X( I, J ) = S( I )*X( I, J )
473: 40 CONTINUE
474: 50 CONTINUE
475: DO 60 J = 1, NRHS
476: FERR( J ) = FERR( J ) / SCOND
477: 60 CONTINUE
478: END IF
479: *
480: * Set INFO = N+1 if the matrix is singular to working precision.
481: *
482: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
483: $ INFO = N + 1
484: *
485: RETURN
486: *
487: * End of ZPOSVX
488: *
489: END
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