1: *> \brief <b> DPOSVX 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 DPOSVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dposvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dposvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dposvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
22: * S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
23: * IWORK, 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: * INTEGER IWORK( * )
32: * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
33: * $ BERR( * ), FERR( * ), S( * ), WORK( * ),
34: * $ X( LDX, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> DPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
44: *> compute the solution to a real system of linear equations
45: *> A * X = B,
46: *> where A is an N-by-N symmetric positive definite matrix and X and B
47: *> are N-by-NRHS matrices.
48: *>
49: *> Error bounds on the solution and a condition estimate are also
50: *> provided.
51: *> \endverbatim
52: *
53: *> \par Description:
54: * =================
55: *>
56: *> \verbatim
57: *>
58: *> The following steps are performed:
59: *>
60: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
61: *> the system:
62: *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
63: *> Whether or not the system will be equilibrated depends on the
64: *> scaling of the matrix A, but if equilibration is used, A is
65: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
66: *>
67: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
68: *> factor the matrix A (after equilibration if FACT = 'E') as
69: *> A = U**T* U, if UPLO = 'U', or
70: *> A = L * L**T, if UPLO = 'L',
71: *> where U is an upper triangular matrix and L is a lower triangular
72: *> matrix.
73: *>
74: *> 3. If the leading i-by-i principal minor is not positive definite,
75: *> then the routine returns with INFO = i. Otherwise, the factored
76: *> form of A is used to estimate the condition number of the matrix
77: *> A. If the reciprocal of the condition number is less than machine
78: *> precision, INFO = N+1 is returned as a warning, but the routine
79: *> still goes on to solve for X and compute error bounds as
80: *> described below.
81: *>
82: *> 4. The system of equations is solved for X using the factored form
83: *> of A.
84: *>
85: *> 5. Iterative refinement is applied to improve the computed solution
86: *> matrix and calculate error bounds and backward error estimates
87: *> for it.
88: *>
89: *> 6. If equilibration was used, the matrix X is premultiplied by
90: *> diag(S) so that it solves the original system before
91: *> equilibration.
92: *> \endverbatim
93: *
94: * Arguments:
95: * ==========
96: *
97: *> \param[in] FACT
98: *> \verbatim
99: *> FACT is CHARACTER*1
100: *> Specifies whether or not the factored form of the matrix A is
101: *> supplied on entry, and if not, whether the matrix A should be
102: *> equilibrated before it is factored.
103: *> = 'F': On entry, AF contains the factored form of A.
104: *> If EQUED = 'Y', the matrix A has been equilibrated
105: *> with scaling factors given by S. A and AF will not
106: *> be modified.
107: *> = 'N': The matrix A will be copied to AF and factored.
108: *> = 'E': The matrix A will be equilibrated if necessary, then
109: *> copied to AF and factored.
110: *> \endverbatim
111: *>
112: *> \param[in] UPLO
113: *> \verbatim
114: *> UPLO is CHARACTER*1
115: *> = 'U': Upper triangle of A is stored;
116: *> = 'L': Lower triangle of A is stored.
117: *> \endverbatim
118: *>
119: *> \param[in] N
120: *> \verbatim
121: *> N is INTEGER
122: *> The number of linear equations, i.e., the order of the
123: *> matrix A. N >= 0.
124: *> \endverbatim
125: *>
126: *> \param[in] NRHS
127: *> \verbatim
128: *> NRHS is INTEGER
129: *> The number of right hand sides, i.e., the number of columns
130: *> of the matrices B and X. NRHS >= 0.
131: *> \endverbatim
132: *>
133: *> \param[in,out] A
134: *> \verbatim
135: *> A is DOUBLE PRECISION array, dimension (LDA,N)
136: *> On entry, the symmetric matrix A, except if FACT = 'F' and
137: *> EQUED = 'Y', then A must contain the equilibrated matrix
138: *> diag(S)*A*diag(S). If UPLO = 'U', the leading
139: *> N-by-N upper triangular part of A contains the upper
140: *> triangular part of the matrix A, and the strictly lower
141: *> triangular part of A is not referenced. If UPLO = 'L', the
142: *> leading N-by-N lower triangular part of A contains the lower
143: *> triangular part of the matrix A, and the strictly upper
144: *> triangular part of A is not referenced. A is not modified if
145: *> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
146: *>
147: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
148: *> diag(S)*A*diag(S).
149: *> \endverbatim
150: *>
151: *> \param[in] LDA
152: *> \verbatim
153: *> LDA is INTEGER
154: *> The leading dimension of the array A. LDA >= max(1,N).
155: *> \endverbatim
156: *>
157: *> \param[in,out] AF
158: *> \verbatim
159: *> AF is or output) DOUBLE PRECISION array, dimension (LDAF,N)
160: *> If FACT = 'F', then AF is an input argument and on entry
161: *> contains the triangular factor U or L from the Cholesky
162: *> factorization A = U**T*U or A = L*L**T, in the same storage
163: *> format as A. If EQUED .ne. 'N', then AF is the factored form
164: *> of the equilibrated matrix diag(S)*A*diag(S).
165: *>
166: *> If FACT = 'N', then AF is an output argument and on exit
167: *> returns the triangular factor U or L from the Cholesky
168: *> factorization A = U**T*U or A = L*L**T of the original
169: *> matrix A.
170: *>
171: *> If FACT = 'E', then AF is an output argument and on exit
172: *> returns the triangular factor U or L from the Cholesky
173: *> factorization A = U**T*U or A = L*L**T of the equilibrated
174: *> matrix A (see the description of A for the form of the
175: *> equilibrated matrix).
176: *> \endverbatim
177: *>
178: *> \param[in] LDAF
179: *> \verbatim
180: *> LDAF is INTEGER
181: *> The leading dimension of the array AF. LDAF >= max(1,N).
182: *> \endverbatim
183: *>
184: *> \param[in,out] EQUED
185: *> \verbatim
186: *> EQUED is or output) CHARACTER*1
187: *> Specifies the form of equilibration that was done.
188: *> = 'N': No equilibration (always true if FACT = 'N').
189: *> = 'Y': Equilibration was done, i.e., A has been replaced by
190: *> diag(S) * A * diag(S).
191: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
192: *> output argument.
193: *> \endverbatim
194: *>
195: *> \param[in,out] S
196: *> \verbatim
197: *> S is or output) DOUBLE PRECISION array, dimension (N)
198: *> The scale factors for A; not accessed if EQUED = 'N'. S is
199: *> an input argument if FACT = 'F'; otherwise, S is an output
200: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
201: *> must be positive.
202: *> \endverbatim
203: *>
204: *> \param[in,out] B
205: *> \verbatim
206: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
207: *> On entry, the N-by-NRHS right hand side matrix B.
208: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
209: *> B is overwritten by diag(S) * B.
210: *> \endverbatim
211: *>
212: *> \param[in] LDB
213: *> \verbatim
214: *> LDB is INTEGER
215: *> The leading dimension of the array B. LDB >= max(1,N).
216: *> \endverbatim
217: *>
218: *> \param[out] X
219: *> \verbatim
220: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
221: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
222: *> the original system of equations. Note that if EQUED = 'Y',
223: *> A and B are modified on exit, and the solution to the
224: *> equilibrated system is inv(diag(S))*X.
225: *> \endverbatim
226: *>
227: *> \param[in] LDX
228: *> \verbatim
229: *> LDX is INTEGER
230: *> The leading dimension of the array X. LDX >= max(1,N).
231: *> \endverbatim
232: *>
233: *> \param[out] RCOND
234: *> \verbatim
235: *> RCOND is DOUBLE PRECISION
236: *> The estimate of the reciprocal condition number of the matrix
237: *> A after equilibration (if done). If RCOND is less than the
238: *> machine precision (in particular, if RCOND = 0), the matrix
239: *> is singular to working precision. This condition is
240: *> indicated by a return code of INFO > 0.
241: *> \endverbatim
242: *>
243: *> \param[out] FERR
244: *> \verbatim
245: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
246: *> The estimated forward error bound for each solution vector
247: *> X(j) (the j-th column of the solution matrix X).
248: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
249: *> is an estimated upper bound for the magnitude of the largest
250: *> element in (X(j) - XTRUE) divided by the magnitude of the
251: *> largest element in X(j). The estimate is as reliable as
252: *> the estimate for RCOND, and is almost always a slight
253: *> overestimate of the true error.
254: *> \endverbatim
255: *>
256: *> \param[out] BERR
257: *> \verbatim
258: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
259: *> The componentwise relative backward error of each solution
260: *> vector X(j) (i.e., the smallest relative change in
261: *> any element of A or B that makes X(j) an exact solution).
262: *> \endverbatim
263: *>
264: *> \param[out] WORK
265: *> \verbatim
266: *> WORK is DOUBLE PRECISION array, dimension (3*N)
267: *> \endverbatim
268: *>
269: *> \param[out] IWORK
270: *> \verbatim
271: *> IWORK is INTEGER array, dimension (N)
272: *> \endverbatim
273: *>
274: *> \param[out] INFO
275: *> \verbatim
276: *> INFO is INTEGER
277: *> = 0: successful exit
278: *> < 0: if INFO = -i, the i-th argument had an illegal value
279: *> > 0: if INFO = i, and i is
280: *> <= N: the leading minor of order i of A is
281: *> not positive definite, so the factorization
282: *> could not be completed, and the solution has not
283: *> been computed. RCOND = 0 is returned.
284: *> = N+1: U is nonsingular, but RCOND is less than machine
285: *> precision, meaning that the matrix is singular
286: *> to working precision. Nevertheless, the
287: *> solution and error bounds are computed because
288: *> there are a number of situations where the
289: *> computed solution can be more accurate than the
290: *> value of RCOND would suggest.
291: *> \endverbatim
292: *
293: * Authors:
294: * ========
295: *
296: *> \author Univ. of Tennessee
297: *> \author Univ. of California Berkeley
298: *> \author Univ. of Colorado Denver
299: *> \author NAG Ltd.
300: *
301: *> \date November 2011
302: *
303: *> \ingroup doublePOsolve
304: *
305: * =====================================================================
306: SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
307: $ S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK,
308: $ IWORK, INFO )
309: *
310: * -- LAPACK driver routine (version 3.4.0) --
311: * -- LAPACK is a software package provided by Univ. of Tennessee, --
312: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
313: * November 2011
314: *
315: * .. Scalar Arguments ..
316: CHARACTER EQUED, FACT, UPLO
317: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
318: DOUBLE PRECISION RCOND
319: * ..
320: * .. Array Arguments ..
321: INTEGER IWORK( * )
322: DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
323: $ BERR( * ), FERR( * ), S( * ), WORK( * ),
324: $ X( LDX, * )
325: * ..
326: *
327: * =====================================================================
328: *
329: * .. Parameters ..
330: DOUBLE PRECISION ZERO, ONE
331: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
332: * ..
333: * .. Local Scalars ..
334: LOGICAL EQUIL, NOFACT, RCEQU
335: INTEGER I, INFEQU, J
336: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
337: * ..
338: * .. External Functions ..
339: LOGICAL LSAME
340: DOUBLE PRECISION DLAMCH, DLANSY
341: EXTERNAL LSAME, DLAMCH, DLANSY
342: * ..
343: * .. External Subroutines ..
344: EXTERNAL DLACPY, DLAQSY, DPOCON, DPOEQU, DPORFS, DPOTRF,
345: $ DPOTRS, XERBLA
346: * ..
347: * .. Intrinsic Functions ..
348: INTRINSIC MAX, MIN
349: * ..
350: * .. Executable Statements ..
351: *
352: INFO = 0
353: NOFACT = LSAME( FACT, 'N' )
354: EQUIL = LSAME( FACT, 'E' )
355: IF( NOFACT .OR. EQUIL ) THEN
356: EQUED = 'N'
357: RCEQU = .FALSE.
358: ELSE
359: RCEQU = LSAME( EQUED, 'Y' )
360: SMLNUM = DLAMCH( 'Safe minimum' )
361: BIGNUM = ONE / SMLNUM
362: END IF
363: *
364: * Test the input parameters.
365: *
366: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
367: $ THEN
368: INFO = -1
369: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
370: $ THEN
371: INFO = -2
372: ELSE IF( N.LT.0 ) THEN
373: INFO = -3
374: ELSE IF( NRHS.LT.0 ) THEN
375: INFO = -4
376: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
377: INFO = -6
378: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
379: INFO = -8
380: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
381: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
382: INFO = -9
383: ELSE
384: IF( RCEQU ) THEN
385: SMIN = BIGNUM
386: SMAX = ZERO
387: DO 10 J = 1, N
388: SMIN = MIN( SMIN, S( J ) )
389: SMAX = MAX( SMAX, S( J ) )
390: 10 CONTINUE
391: IF( SMIN.LE.ZERO ) THEN
392: INFO = -10
393: ELSE IF( N.GT.0 ) THEN
394: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
395: ELSE
396: SCOND = ONE
397: END IF
398: END IF
399: IF( INFO.EQ.0 ) THEN
400: IF( LDB.LT.MAX( 1, N ) ) THEN
401: INFO = -12
402: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
403: INFO = -14
404: END IF
405: END IF
406: END IF
407: *
408: IF( INFO.NE.0 ) THEN
409: CALL XERBLA( 'DPOSVX', -INFO )
410: RETURN
411: END IF
412: *
413: IF( EQUIL ) THEN
414: *
415: * Compute row and column scalings to equilibrate the matrix A.
416: *
417: CALL DPOEQU( N, A, LDA, S, SCOND, AMAX, INFEQU )
418: IF( INFEQU.EQ.0 ) THEN
419: *
420: * Equilibrate the matrix.
421: *
422: CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
423: RCEQU = LSAME( EQUED, 'Y' )
424: END IF
425: END IF
426: *
427: * Scale the right hand side.
428: *
429: IF( RCEQU ) THEN
430: DO 30 J = 1, NRHS
431: DO 20 I = 1, N
432: B( I, J ) = S( I )*B( I, J )
433: 20 CONTINUE
434: 30 CONTINUE
435: END IF
436: *
437: IF( NOFACT .OR. EQUIL ) THEN
438: *
439: * Compute the Cholesky factorization A = U**T *U or A = L*L**T.
440: *
441: CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF )
442: CALL DPOTRF( UPLO, N, AF, LDAF, INFO )
443: *
444: * Return if INFO is non-zero.
445: *
446: IF( INFO.GT.0 )THEN
447: RCOND = ZERO
448: RETURN
449: END IF
450: END IF
451: *
452: * Compute the norm of the matrix A.
453: *
454: ANORM = DLANSY( '1', UPLO, N, A, LDA, WORK )
455: *
456: * Compute the reciprocal of the condition number of A.
457: *
458: CALL DPOCON( UPLO, N, AF, LDAF, ANORM, RCOND, WORK, IWORK, INFO )
459: *
460: * Compute the solution matrix X.
461: *
462: CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
463: CALL DPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO )
464: *
465: * Use iterative refinement to improve the computed solution and
466: * compute error bounds and backward error estimates for it.
467: *
468: CALL DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX,
469: $ FERR, BERR, WORK, IWORK, INFO )
470: *
471: * Transform the solution matrix X to a solution of the original
472: * system.
473: *
474: IF( RCEQU ) THEN
475: DO 50 J = 1, NRHS
476: DO 40 I = 1, N
477: X( I, J ) = S( I )*X( I, J )
478: 40 CONTINUE
479: 50 CONTINUE
480: DO 60 J = 1, NRHS
481: FERR( J ) = FERR( J ) / SCOND
482: 60 CONTINUE
483: END IF
484: *
485: * Set INFO = N+1 if the matrix is singular to working precision.
486: *
487: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
488: $ INFO = N + 1
489: *
490: RETURN
491: *
492: * End of DPOSVX
493: *
494: END
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