1: *> \brief <b> ZPPSVX 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 ZPPSVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zppsvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zppsvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zppsvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
22: * X, LDX, RCOND, FERR, BERR, WORK, RWORK, 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: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
31: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
32: * $ X( LDX, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> ZPPSVX uses the Cholesky factorization A = U**H * U or A = L * L**H to
42: *> compute the solution to a complex system of linear equations
43: *> A * X = B,
44: *> where A is an N-by-N Hermitian 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**H * U , if UPLO = 'U', or
68: *> A = L * L**H, if UPLO = 'L',
69: *> where U is an upper triangular matrix, L is a lower triangular
70: *> matrix, and **H indicates conjugate transpose.
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 COMPLEX*16 array, dimension (N*(N+1)/2)
134: *> On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 array, dimension (N*(N+1)/2)
151: *> If FACT = 'F', then AFP is an input argument and on entry
152: *> contains the triangular factor U or L from the Cholesky
153: *> factorization A = U**H*U or A = L*L**H, in the same storage
154: *> format as A. If EQUED .ne. 'N', then AFP is the factored
155: *> form of the equilibrated matrix A.
156: *>
157: *> If FACT = 'N', then AFP is an output argument and on exit
158: *> returns the triangular factor U or L from the Cholesky
159: *> factorization A = U**H * U or A = L * L**H of the original
160: *> matrix A.
161: *>
162: *> If FACT = 'E', then AFP is an output argument and on exit
163: *> returns the triangular factor U or L from the Cholesky
164: *> factorization A = U**H * U or A = L * L**H of the equilibrated
165: *> matrix A (see the description of AP for the form of the
166: *> equilibrated matrix).
167: *> \endverbatim
168: *>
169: *> \param[in,out] EQUED
170: *> \verbatim
171: *> EQUED is CHARACTER*1
172: *> Specifies the form of equilibration that was done.
173: *> = 'N': No equilibration (always true if FACT = 'N').
174: *> = 'Y': Equilibration was done, i.e., A has been replaced by
175: *> diag(S) * A * diag(S).
176: *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
177: *> output argument.
178: *> \endverbatim
179: *>
180: *> \param[in,out] S
181: *> \verbatim
182: *> S is DOUBLE PRECISION array, dimension (N)
183: *> The scale factors for A; not accessed if EQUED = 'N'. S is
184: *> an input argument if FACT = 'F'; otherwise, S is an output
185: *> argument. If FACT = 'F' and EQUED = 'Y', each element of S
186: *> must be positive.
187: *> \endverbatim
188: *>
189: *> \param[in,out] B
190: *> \verbatim
191: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
192: *> On entry, the N-by-NRHS right hand side matrix B.
193: *> On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
194: *> B is overwritten by diag(S) * B.
195: *> \endverbatim
196: *>
197: *> \param[in] LDB
198: *> \verbatim
199: *> LDB is INTEGER
200: *> The leading dimension of the array B. LDB >= max(1,N).
201: *> \endverbatim
202: *>
203: *> \param[out] X
204: *> \verbatim
205: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
206: *> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
207: *> the original system of equations. Note that if EQUED = 'Y',
208: *> A and B are modified on exit, and the solution to the
209: *> equilibrated system is inv(diag(S))*X.
210: *> \endverbatim
211: *>
212: *> \param[in] LDX
213: *> \verbatim
214: *> LDX is INTEGER
215: *> The leading dimension of the array X. LDX >= max(1,N).
216: *> \endverbatim
217: *>
218: *> \param[out] RCOND
219: *> \verbatim
220: *> RCOND is DOUBLE PRECISION
221: *> The estimate of the reciprocal condition number of the matrix
222: *> A after equilibration (if done). If RCOND is less than the
223: *> machine precision (in particular, if RCOND = 0), the matrix
224: *> is singular to working precision. This condition is
225: *> indicated by a return code of INFO > 0.
226: *> \endverbatim
227: *>
228: *> \param[out] FERR
229: *> \verbatim
230: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
231: *> The estimated forward error bound for each solution vector
232: *> X(j) (the j-th column of the solution matrix X).
233: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
234: *> is an estimated upper bound for the magnitude of the largest
235: *> element in (X(j) - XTRUE) divided by the magnitude of the
236: *> largest element in X(j). The estimate is as reliable as
237: *> the estimate for RCOND, and is almost always a slight
238: *> overestimate of the true error.
239: *> \endverbatim
240: *>
241: *> \param[out] BERR
242: *> \verbatim
243: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
244: *> The componentwise relative backward error of each solution
245: *> vector X(j) (i.e., the smallest relative change in
246: *> any element of A or B that makes X(j) an exact solution).
247: *> \endverbatim
248: *>
249: *> \param[out] WORK
250: *> \verbatim
251: *> WORK is COMPLEX*16 array, dimension (2*N)
252: *> \endverbatim
253: *>
254: *> \param[out] RWORK
255: *> \verbatim
256: *> RWORK is DOUBLE PRECISION array, dimension (N)
257: *> \endverbatim
258: *>
259: *> \param[out] INFO
260: *> \verbatim
261: *> INFO is INTEGER
262: *> = 0: successful exit
263: *> < 0: if INFO = -i, the i-th argument had an illegal value
264: *> > 0: if INFO = i, and i is
265: *> <= N: the leading minor of order i of A is
266: *> not positive definite, so the factorization
267: *> could not be completed, and the solution has not
268: *> been computed. RCOND = 0 is returned.
269: *> = N+1: U is nonsingular, but RCOND is less than machine
270: *> precision, meaning that the matrix is singular
271: *> to working precision. Nevertheless, the
272: *> solution and error bounds are computed because
273: *> there are a number of situations where the
274: *> computed solution can be more accurate than the
275: *> value of RCOND would suggest.
276: *> \endverbatim
277: *
278: * Authors:
279: * ========
280: *
281: *> \author Univ. of Tennessee
282: *> \author Univ. of California Berkeley
283: *> \author Univ. of Colorado Denver
284: *> \author NAG Ltd.
285: *
286: *> \ingroup complex16OTHERsolve
287: *
288: *> \par Further Details:
289: * =====================
290: *>
291: *> \verbatim
292: *>
293: *> The packed storage scheme is illustrated by the following example
294: *> when N = 4, UPLO = 'U':
295: *>
296: *> Two-dimensional storage of the Hermitian matrix A:
297: *>
298: *> a11 a12 a13 a14
299: *> a22 a23 a24
300: *> a33 a34 (aij = conjg(aji))
301: *> a44
302: *>
303: *> Packed storage of the upper triangle of A:
304: *>
305: *> AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
306: *> \endverbatim
307: *>
308: * =====================================================================
309: SUBROUTINE ZPPSVX( FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB,
310: $ X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO )
311: *
312: * -- LAPACK driver routine --
313: * -- LAPACK is a software package provided by Univ. of Tennessee, --
314: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
315: *
316: * .. Scalar Arguments ..
317: CHARACTER EQUED, FACT, UPLO
318: INTEGER INFO, LDB, LDX, N, NRHS
319: DOUBLE PRECISION RCOND
320: * ..
321: * .. Array Arguments ..
322: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * ), S( * )
323: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), 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, ZLANHP
341: EXTERNAL LSAME, DLAMCH, ZLANHP
342: * ..
343: * .. External Subroutines ..
344: EXTERNAL XERBLA, ZCOPY, ZLACPY, ZLAQHP, ZPPCON, ZPPEQU,
345: $ ZPPRFS, ZPPTRF, ZPPTRS
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( LSAME( FACT, 'F' ) .AND. .NOT.
377: $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
378: INFO = -7
379: ELSE
380: IF( RCEQU ) THEN
381: SMIN = BIGNUM
382: SMAX = ZERO
383: DO 10 J = 1, N
384: SMIN = MIN( SMIN, S( J ) )
385: SMAX = MAX( SMAX, S( J ) )
386: 10 CONTINUE
387: IF( SMIN.LE.ZERO ) THEN
388: INFO = -8
389: ELSE IF( N.GT.0 ) THEN
390: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
391: ELSE
392: SCOND = ONE
393: END IF
394: END IF
395: IF( INFO.EQ.0 ) THEN
396: IF( LDB.LT.MAX( 1, N ) ) THEN
397: INFO = -10
398: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
399: INFO = -12
400: END IF
401: END IF
402: END IF
403: *
404: IF( INFO.NE.0 ) THEN
405: CALL XERBLA( 'ZPPSVX', -INFO )
406: RETURN
407: END IF
408: *
409: IF( EQUIL ) THEN
410: *
411: * Compute row and column scalings to equilibrate the matrix A.
412: *
413: CALL ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFEQU )
414: IF( INFEQU.EQ.0 ) THEN
415: *
416: * Equilibrate the matrix.
417: *
418: CALL ZLAQHP( UPLO, N, AP, S, SCOND, AMAX, EQUED )
419: RCEQU = LSAME( EQUED, 'Y' )
420: END IF
421: END IF
422: *
423: * Scale the right-hand side.
424: *
425: IF( RCEQU ) THEN
426: DO 30 J = 1, NRHS
427: DO 20 I = 1, N
428: B( I, J ) = S( I )*B( I, J )
429: 20 CONTINUE
430: 30 CONTINUE
431: END IF
432: *
433: IF( NOFACT .OR. EQUIL ) THEN
434: *
435: * Compute the Cholesky factorization A = U**H * U or A = L * L**H.
436: *
437: CALL ZCOPY( N*( N+1 ) / 2, AP, 1, AFP, 1 )
438: CALL ZPPTRF( UPLO, N, AFP, INFO )
439: *
440: * Return if INFO is non-zero.
441: *
442: IF( INFO.GT.0 )THEN
443: RCOND = ZERO
444: RETURN
445: END IF
446: END IF
447: *
448: * Compute the norm of the matrix A.
449: *
450: ANORM = ZLANHP( 'I', UPLO, N, AP, RWORK )
451: *
452: * Compute the reciprocal of the condition number of A.
453: *
454: CALL ZPPCON( UPLO, N, AFP, ANORM, RCOND, WORK, RWORK, INFO )
455: *
456: * Compute the solution matrix X.
457: *
458: CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
459: CALL ZPPTRS( UPLO, N, NRHS, AFP, X, LDX, INFO )
460: *
461: * Use iterative refinement to improve the computed solution and
462: * compute error bounds and backward error estimates for it.
463: *
464: CALL ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR,
465: $ WORK, RWORK, INFO )
466: *
467: * Transform the solution matrix X to a solution of the original
468: * system.
469: *
470: IF( RCEQU ) THEN
471: DO 50 J = 1, NRHS
472: DO 40 I = 1, N
473: X( I, J ) = S( I )*X( I, J )
474: 40 CONTINUE
475: 50 CONTINUE
476: DO 60 J = 1, NRHS
477: FERR( J ) = FERR( J ) / SCOND
478: 60 CONTINUE
479: END IF
480: *
481: * Set INFO = N+1 if the matrix is singular to working precision.
482: *
483: IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
484: $ INFO = N + 1
485: *
486: RETURN
487: *
488: * End of ZPPSVX
489: *
490: END
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