1: *> \brief <b> DPPSVX computes the solution to system of linear equations A * X = B for OTHER 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 DPPSVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppsvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppsvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppsvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
22: * X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER EQUED, FACT, UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * DOUBLE PRECISION RCOND
28: * ..
29: * .. Array Arguments ..
30: * INTEGER IWORK( * )
31: * DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
32: * $ FERR( * ), S( * ), WORK( * ), X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
42: *> compute the solution to a real system of linear equations
43: *> A * X = B,
44: *> where A is an N-by-N symmetric positive definite matrix stored in
45: *> packed format and X and B are N-by-NRHS matrices.
46: *>
47: *> Error bounds on the solution and a condition estimate are also
48: *> provided.
49: *> \endverbatim
50: *
51: *> \par Description:
52: * =================
53: *>
54: *> \verbatim
55: *>
56: *> The following steps are performed:
57: *>
58: *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
59: *> the system:
60: *> diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
61: *> Whether or not the system will be equilibrated depends on the
62: *> scaling of the matrix A, but if equilibration is used, A is
63: *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
64: *>
65: *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
66: *> factor the matrix A (after equilibration if FACT = 'E') as
67: *> A = U**T* U, if UPLO = 'U', or
68: *> A = L * L**T, if UPLO = 'L',
69: *> where U is an upper triangular matrix and L is a lower triangular
70: *> matrix.
71: *>
72: *> 3. If the leading i-by-i principal minor is not positive definite,
73: *> then the routine returns with INFO = i. Otherwise, the factored
74: *> form of A is used to estimate the condition number of the matrix
75: *> A. If the reciprocal of the condition number is less than machine
76: *> precision, INFO = N+1 is returned as a warning, but the routine
77: *> still goes on to solve for X and compute error bounds as
78: *> described below.
79: *>
80: *> 4. The system of equations is solved for X using the factored form
81: *> of A.
82: *>
83: *> 5. Iterative refinement is applied to improve the computed solution
84: *> matrix and calculate error bounds and backward error estimates
85: *> for it.
86: *>
87: *> 6. If equilibration was used, the matrix X is premultiplied by
88: *> diag(S) so that it solves the original system before
89: *> equilibration.
90: *> \endverbatim
91: *
92: * Arguments:
93: * ==========
94: *
95: *> \param[in] FACT
96: *> \verbatim
97: *> FACT is CHARACTER*1
98: *> Specifies whether or not the factored form of the matrix A is
99: *> supplied on entry, and if not, whether the matrix A should be
100: *> equilibrated before it is factored.
101: *> = 'F': On entry, AFP contains the factored form of A.
102: *> If EQUED = 'Y', the matrix A has been equilibrated
103: *> with scaling factors given by S. AP and AFP will not
104: *> be modified.
105: *> = 'N': The matrix A will be copied to AFP and factored.
106: *> = 'E': The matrix A will be equilibrated if necessary, then
107: *> copied to AFP and factored.
108: *> \endverbatim
109: *>
110: *> \param[in] UPLO
111: *> \verbatim
112: *> UPLO is CHARACTER*1
113: *> = 'U': Upper triangle of A is stored;
114: *> = 'L': Lower triangle of A is stored.
115: *> \endverbatim
116: *>
117: *> \param[in] N
118: *> \verbatim
119: *> N is INTEGER
120: *> The number of linear equations, i.e., the order of the
121: *> matrix A. N >= 0.
122: *> \endverbatim
123: *>
124: *> \param[in] NRHS
125: *> \verbatim
126: *> NRHS is INTEGER
127: *> The number of right hand sides, i.e., the number of columns
128: *> of the matrices B and X. NRHS >= 0.
129: *> \endverbatim
130: *>
131: *> \param[in,out] AP
132: *> \verbatim
133: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
134: *> On entry, the upper or lower triangle of the symmetric matrix
135: *> A, packed columnwise in a linear array, except if FACT = 'F'
136: *> and EQUED = 'Y', then A must contain the equilibrated matrix
137: *> diag(S)*A*diag(S). The j-th column of A is stored in the
138: *> array AP as follows:
139: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
140: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
141: *> See below for further details. A is not modified if
142: *> FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
143: *>
144: *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
145: *> diag(S)*A*diag(S).
146: *> \endverbatim
147: *>
148: *> \param[in,out] AFP
149: *> \verbatim
150: *> AFP is or output) DOUBLE PRECISION array, dimension
151: *> (N*(N+1)/2)
152: *> If FACT = 'F', then AFP is an input argument and on entry
153: *> contains the triangular factor U or L from the Cholesky
154: *> factorization A = U**T*U or A = L*L**T, in the same storage
155: *> format as A. If EQUED .ne. 'N', then AFP is the factored
156: *> form of the equilibrated matrix A.
157: *>
158: *> If FACT = 'N', then AFP is an output argument and on exit
159: *> returns the triangular factor U or L from the Cholesky
160: *> factorization A = U**T * U or A = L * L**T of the original
161: *> matrix A.
162: *>
163: *> If FACT = 'E', then AFP is an output argument and on exit
164: *> returns the triangular factor U or L from the Cholesky
165: *> factorization A = U**T * U or A = L * L**T of the equilibrated
166: *> matrix A (see the description of AP for the form of the
167: *> equilibrated matrix).
168: *> \endverbatim
169: *>
170: *> \param[in,out] EQUED
171: *> \verbatim
172: *> EQUED is or output) CHARACTER*1
173: *> Specifies the form of equilibration that was done.
174: *> = 'N': No equilibration (always true if FACT = 'N').
175: *> = 'Y': Equilibration was done, i.e., A has been replaced by
176: *> diag(S) * A * diag(S).
177: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
178: *> output argument.
179: *> \endverbatim
180: *>
181: *> \param[in,out] S
182: *> \verbatim
183: *> S is or output) DOUBLE PRECISION array, dimension (N)
184: *> The scale factors for A; not accessed if EQUED = 'N'. S is
185: *> an input argument if FACT = 'F'; otherwise, S is an output
186: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
187: *> must be positive.
188: *> \endverbatim
189: *>
190: *> \param[in,out] B
191: *> \verbatim
192: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
193: *> On entry, the N-by-NRHS right hand side matrix B.
194: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
195: *> B is overwritten by diag(S) * B.
196: *> \endverbatim
197: *>
198: *> \param[in] LDB
199: *> \verbatim
200: *> LDB is INTEGER
201: *> The leading dimension of the array B. LDB >= max(1,N).
202: *> \endverbatim
203: *>
204: *> \param[out] X
205: *> \verbatim
206: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
207: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
208: *> the original system of equations. Note that if EQUED = 'Y',
209: *> A and B are modified on exit, and the solution to the
210: *> equilibrated system is inv(diag(S))*X.
211: *> \endverbatim
212: *>
213: *> \param[in] LDX
214: *> \verbatim
215: *> LDX is INTEGER
216: *> The leading dimension of the array X. LDX >= max(1,N).
217: *> \endverbatim
218: *>
219: *> \param[out] RCOND
220: *> \verbatim
221: *> RCOND is DOUBLE PRECISION
222: *> The estimate of the reciprocal condition number of the matrix
223: *> A after equilibration (if done). If RCOND is less than the
224: *> machine precision (in particular, if RCOND = 0), the matrix
225: *> is singular to working precision. This condition is
226: *> indicated by a return code of INFO > 0.
227: *> \endverbatim
228: *>
229: *> \param[out] FERR
230: *> \verbatim
231: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
232: *> The estimated forward error bound for each solution vector
233: *> X(j) (the j-th column of the solution matrix X).
234: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
235: *> is an estimated upper bound for the magnitude of the largest
236: *> element in (X(j) - XTRUE) divided by the magnitude of the
237: *> largest element in X(j). The estimate is as reliable as
238: *> the estimate for RCOND, and is almost always a slight
239: *> overestimate of the true error.
240: *> \endverbatim
241: *>
242: *> \param[out] BERR
243: *> \verbatim
244: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
245: *> The componentwise relative backward error of each solution
246: *> vector X(j) (i.e., the smallest relative change in
247: *> any element of A or B that makes X(j) an exact solution).
248: *> \endverbatim
249: *>
250: *> \param[out] WORK
251: *> \verbatim
252: *> WORK is DOUBLE PRECISION array, dimension (3*N)
253: *> \endverbatim
254: *>
255: *> \param[out] IWORK
256: *> \verbatim
257: *> IWORK is INTEGER array, dimension (N)
258: *> \endverbatim
259: *>
260: *> \param[out] INFO
261: *> \verbatim
262: *> INFO is INTEGER
263: *> = 0: successful exit
264: *> < 0: if INFO = -i, the i-th argument had an illegal value
265: *> > 0: if INFO = i, and i is
266: *> <= N: the leading minor of order i of A is
267: *> not positive definite, so the factorization
268: *> could not be completed, and the solution has not
269: *> been computed. RCOND = 0 is returned.
270: *> = N+1: U is nonsingular, but RCOND is less than machine
271: *> precision, meaning that the matrix is singular
272: *> to working precision. Nevertheless, the
273: *> solution and error bounds are computed because
274: *> there are a number of situations where the
275: *> computed solution can be more accurate than the
276: *> value of RCOND would suggest.
277: *> \endverbatim
278: *
279: * Authors:
280: * ========
281: *
282: *> \author Univ. of Tennessee
283: *> \author Univ. of California Berkeley
284: *> \author Univ. of Colorado Denver
285: *> \author NAG Ltd.
286: *
287: *> \date November 2011
288: *
289: *> \ingroup doubleOTHERsolve
290: *
291: *> \par Further Details:
292: * =====================
293: *>
294: *> \verbatim
295: *>
296: *> The packed storage scheme is illustrated by the following example
297: *> when N = 4, UPLO = 'U':
298: *>
299: *> Two-dimensional storage of the symmetric matrix A:
300: *>
301: *> a11 a12 a13 a14
302: *> a22 a23 a24
303: *> a33 a34 (aij = conjg(aji))
304: *> a44
305: *>
306: *> Packed storage of the upper triangle of A:
307: *>
308: *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
309: *> \endverbatim
310: *>
311: * =====================================================================
312: SUBROUTINE DPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
313: $ X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
314: *
315: * -- LAPACK driver routine (version 3.4.0) --
316: * -- LAPACK is a software package provided by Univ. of Tennessee, --
317: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
318: * November 2011
319: *
320: * .. Scalar Arguments ..
321: CHARACTER EQUED, FACT, UPLO
322: INTEGER INFO, LDB, LDX, N, NRHS
323: DOUBLE PRECISION RCOND
324: * ..
325: * .. Array Arguments ..
326: INTEGER IWORK( * )
327: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
328: $ FERR( * ), S( * ), WORK( * ), X( LDX, * )
329: * ..
330: *
331: * =====================================================================
332: *
333: * .. Parameters ..
334: DOUBLE PRECISION ZERO, ONE
335: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
336: * ..
337: * .. Local Scalars ..
338: LOGICAL EQUIL, NOFACT, RCEQU
339: INTEGER I, INFEQU, J
340: DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
341: * ..
342: * .. External Functions ..
343: LOGICAL LSAME
344: DOUBLE PRECISION DLAMCH, DLANSP
345: EXTERNAL LSAME, DLAMCH, DLANSP
346: * ..
347: * .. External Subroutines ..
348: EXTERNAL DCOPY, DLACPY, DLAQSP, DPPCON, DPPEQU, DPPRFS,
349: $ DPPTRF, DPPTRS, XERBLA
350: * ..
351: * .. Intrinsic Functions ..
352: INTRINSIC MAX, MIN
353: * ..
354: * .. Executable Statements ..
355: *
356: INFO = 0
357: NOFACT = LSAME( FACT, 'N' )
358: EQUIL = LSAME( FACT, 'E' )
359: IF( NOFACT .OR. EQUIL ) THEN
360: EQUED = 'N'
361: RCEQU = .FALSE.
362: ELSE
363: RCEQU = LSAME( EQUED, 'Y' )
364: SMLNUM = DLAMCH( 'Safe minimum' )
365: BIGNUM = ONE / SMLNUM
366: END IF
367: *
368: * Test the input parameters.
369: *
370: IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
371: $ THEN
372: INFO = -1
373: ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
374: $ THEN
375: INFO = -2
376: ELSE IF( N.LT.0 ) THEN
377: INFO = -3
378: ELSE IF( NRHS.LT.0 ) THEN
379: INFO = -4
380: ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
381: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
382: INFO = -7
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 = -8
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 = -10
402: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
403: INFO = -12
404: END IF
405: END IF
406: END IF
407: *
408: IF( INFO.NE.0 ) THEN
409: CALL XERBLA( 'DPPSVX', -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 DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
418: IF( INFEQU.EQ.0 ) THEN
419: *
420: * Equilibrate the matrix.
421: *
422: CALL DLAQSP( UPLO, N, AP, 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 DCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
442: CALL DPPTRF( UPLO, N, AFP, 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 = DLANSP( 'I', UPLO, N, AP, WORK )
455: *
456: * Compute the reciprocal of the condition number of A.
457: *
458: CALL DPPCON( UPLO, N, AFP, 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 DPPTRS( UPLO, N, NRHS, AFP, 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 DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR,
469: $ 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 DPPSVX
493: *
494: END
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