1: *> \brief \b DPPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DPPRFS + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpprfs.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpprfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
22: * BERR, WORK, IWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IWORK( * )
30: * DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
31: * $ FERR( * ), WORK( * ), X( LDX, * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> DPPRFS improves the computed solution to a system of linear
41: *> equations when the coefficient matrix is symmetric positive definite
42: *> and packed, and provides error bounds and backward error estimates
43: *> for the solution.
44: *> \endverbatim
45: *
46: * Arguments:
47: * ==========
48: *
49: *> \param[in] UPLO
50: *> \verbatim
51: *> UPLO is CHARACTER*1
52: *> = 'U': Upper triangle of A is stored;
53: *> = 'L': Lower triangle of A is stored.
54: *> \endverbatim
55: *>
56: *> \param[in] N
57: *> \verbatim
58: *> N is INTEGER
59: *> The order of the matrix A. N >= 0.
60: *> \endverbatim
61: *>
62: *> \param[in] NRHS
63: *> \verbatim
64: *> NRHS is INTEGER
65: *> The number of right hand sides, i.e., the number of columns
66: *> of the matrices B and X. NRHS >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in] AP
70: *> \verbatim
71: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
72: *> The upper or lower triangle of the symmetric matrix A, packed
73: *> columnwise in a linear array. The j-th column of A is stored
74: *> in the array AP as follows:
75: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
76: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
77: *> \endverbatim
78: *>
79: *> \param[in] AFP
80: *> \verbatim
81: *> AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
82: *> The triangular factor U or L from the Cholesky factorization
83: *> A = U**T*U or A = L*L**T, as computed by DPPTRF/ZPPTRF,
84: *> packed columnwise in a linear array in the same format as A
85: *> (see AP).
86: *> \endverbatim
87: *>
88: *> \param[in] B
89: *> \verbatim
90: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
91: *> The right hand side matrix B.
92: *> \endverbatim
93: *>
94: *> \param[in] LDB
95: *> \verbatim
96: *> LDB is INTEGER
97: *> The leading dimension of the array B. LDB >= max(1,N).
98: *> \endverbatim
99: *>
100: *> \param[in,out] X
101: *> \verbatim
102: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
103: *> On entry, the solution matrix X, as computed by DPPTRS.
104: *> On exit, the improved solution matrix X.
105: *> \endverbatim
106: *>
107: *> \param[in] LDX
108: *> \verbatim
109: *> LDX is INTEGER
110: *> The leading dimension of the array X. LDX >= max(1,N).
111: *> \endverbatim
112: *>
113: *> \param[out] FERR
114: *> \verbatim
115: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
116: *> The estimated forward error bound for each solution vector
117: *> X(j) (the j-th column of the solution matrix X).
118: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
119: *> is an estimated upper bound for the magnitude of the largest
120: *> element in (X(j) - XTRUE) divided by the magnitude of the
121: *> largest element in X(j). The estimate is as reliable as
122: *> the estimate for RCOND, and is almost always a slight
123: *> overestimate of the true error.
124: *> \endverbatim
125: *>
126: *> \param[out] BERR
127: *> \verbatim
128: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
129: *> The componentwise relative backward error of each solution
130: *> vector X(j) (i.e., the smallest relative change in
131: *> any element of A or B that makes X(j) an exact solution).
132: *> \endverbatim
133: *>
134: *> \param[out] WORK
135: *> \verbatim
136: *> WORK is DOUBLE PRECISION array, dimension (3*N)
137: *> \endverbatim
138: *>
139: *> \param[out] IWORK
140: *> \verbatim
141: *> IWORK is INTEGER array, dimension (N)
142: *> \endverbatim
143: *>
144: *> \param[out] INFO
145: *> \verbatim
146: *> INFO is INTEGER
147: *> = 0: successful exit
148: *> < 0: if INFO = -i, the i-th argument had an illegal value
149: *> \endverbatim
150: *
151: *> \par Internal Parameters:
152: * =========================
153: *>
154: *> \verbatim
155: *> ITMAX is the maximum number of steps of iterative refinement.
156: *> \endverbatim
157: *
158: * Authors:
159: * ========
160: *
161: *> \author Univ. of Tennessee
162: *> \author Univ. of California Berkeley
163: *> \author Univ. of Colorado Denver
164: *> \author NAG Ltd.
165: *
166: *> \date November 2011
167: *
168: *> \ingroup doubleOTHERcomputational
169: *
170: * =====================================================================
171: SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
172: $ BERR, WORK, IWORK, INFO )
173: *
174: * -- LAPACK computational routine (version 3.4.0) --
175: * -- LAPACK is a software package provided by Univ. of Tennessee, --
176: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177: * November 2011
178: *
179: * .. Scalar Arguments ..
180: CHARACTER UPLO
181: INTEGER INFO, LDB, LDX, N, NRHS
182: * ..
183: * .. Array Arguments ..
184: INTEGER IWORK( * )
185: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
186: $ FERR( * ), WORK( * ), X( LDX, * )
187: * ..
188: *
189: * =====================================================================
190: *
191: * .. Parameters ..
192: INTEGER ITMAX
193: PARAMETER ( ITMAX = 5 )
194: DOUBLE PRECISION ZERO
195: PARAMETER ( ZERO = 0.0D+0 )
196: DOUBLE PRECISION ONE
197: PARAMETER ( ONE = 1.0D+0 )
198: DOUBLE PRECISION TWO
199: PARAMETER ( TWO = 2.0D+0 )
200: DOUBLE PRECISION THREE
201: PARAMETER ( THREE = 3.0D+0 )
202: * ..
203: * .. Local Scalars ..
204: LOGICAL UPPER
205: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
206: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
207: * ..
208: * .. Local Arrays ..
209: INTEGER ISAVE( 3 )
210: * ..
211: * .. External Subroutines ..
212: EXTERNAL DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV, XERBLA
213: * ..
214: * .. Intrinsic Functions ..
215: INTRINSIC ABS, MAX
216: * ..
217: * .. External Functions ..
218: LOGICAL LSAME
219: DOUBLE PRECISION DLAMCH
220: EXTERNAL LSAME, DLAMCH
221: * ..
222: * .. Executable Statements ..
223: *
224: * Test the input parameters.
225: *
226: INFO = 0
227: UPPER = LSAME( UPLO, 'U' )
228: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
229: INFO = -1
230: ELSE IF( N.LT.0 ) THEN
231: INFO = -2
232: ELSE IF( NRHS.LT.0 ) THEN
233: INFO = -3
234: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
235: INFO = -7
236: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
237: INFO = -9
238: END IF
239: IF( INFO.NE.0 ) THEN
240: CALL XERBLA( 'DPPRFS', -INFO )
241: RETURN
242: END IF
243: *
244: * Quick return if possible
245: *
246: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
247: DO 10 J = 1, NRHS
248: FERR( J ) = ZERO
249: BERR( J ) = ZERO
250: 10 CONTINUE
251: RETURN
252: END IF
253: *
254: * NZ = maximum number of nonzero elements in each row of A, plus 1
255: *
256: NZ = N + 1
257: EPS = DLAMCH( 'Epsilon' )
258: SAFMIN = DLAMCH( 'Safe minimum' )
259: SAFE1 = NZ*SAFMIN
260: SAFE2 = SAFE1 / EPS
261: *
262: * Do for each right hand side
263: *
264: DO 140 J = 1, NRHS
265: *
266: COUNT = 1
267: LSTRES = THREE
268: 20 CONTINUE
269: *
270: * Loop until stopping criterion is satisfied.
271: *
272: * Compute residual R = B - A * X
273: *
274: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
275: CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
276: $ 1 )
277: *
278: * Compute componentwise relative backward error from formula
279: *
280: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
281: *
282: * where abs(Z) is the componentwise absolute value of the matrix
283: * or vector Z. If the i-th component of the denominator is less
284: * than SAFE2, then SAFE1 is added to the i-th components of the
285: * numerator and denominator before dividing.
286: *
287: DO 30 I = 1, N
288: WORK( I ) = ABS( B( I, J ) )
289: 30 CONTINUE
290: *
291: * Compute abs(A)*abs(X) + abs(B).
292: *
293: KK = 1
294: IF( UPPER ) THEN
295: DO 50 K = 1, N
296: S = ZERO
297: XK = ABS( X( K, J ) )
298: IK = KK
299: DO 40 I = 1, K - 1
300: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
301: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
302: IK = IK + 1
303: 40 CONTINUE
304: WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
305: KK = KK + K
306: 50 CONTINUE
307: ELSE
308: DO 70 K = 1, N
309: S = ZERO
310: XK = ABS( X( K, J ) )
311: WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
312: IK = KK + 1
313: DO 60 I = K + 1, N
314: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
315: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
316: IK = IK + 1
317: 60 CONTINUE
318: WORK( K ) = WORK( K ) + S
319: KK = KK + ( N-K+1 )
320: 70 CONTINUE
321: END IF
322: S = ZERO
323: DO 80 I = 1, N
324: IF( WORK( I ).GT.SAFE2 ) THEN
325: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
326: ELSE
327: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
328: $ ( WORK( I )+SAFE1 ) )
329: END IF
330: 80 CONTINUE
331: BERR( J ) = S
332: *
333: * Test stopping criterion. Continue iterating if
334: * 1) The residual BERR(J) is larger than machine epsilon, and
335: * 2) BERR(J) decreased by at least a factor of 2 during the
336: * last iteration, and
337: * 3) At most ITMAX iterations tried.
338: *
339: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
340: $ COUNT.LE.ITMAX ) THEN
341: *
342: * Update solution and try again.
343: *
344: CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
345: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
346: LSTRES = BERR( J )
347: COUNT = COUNT + 1
348: GO TO 20
349: END IF
350: *
351: * Bound error from formula
352: *
353: * norm(X - XTRUE) / norm(X) .le. FERR =
354: * norm( abs(inv(A))*
355: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
356: *
357: * where
358: * norm(Z) is the magnitude of the largest component of Z
359: * inv(A) is the inverse of A
360: * abs(Z) is the componentwise absolute value of the matrix or
361: * vector Z
362: * NZ is the maximum number of nonzeros in any row of A, plus 1
363: * EPS is machine epsilon
364: *
365: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
366: * is incremented by SAFE1 if the i-th component of
367: * abs(A)*abs(X) + abs(B) is less than SAFE2.
368: *
369: * Use DLACN2 to estimate the infinity-norm of the matrix
370: * inv(A) * diag(W),
371: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
372: *
373: DO 90 I = 1, N
374: IF( WORK( I ).GT.SAFE2 ) THEN
375: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
376: ELSE
377: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
378: END IF
379: 90 CONTINUE
380: *
381: KASE = 0
382: 100 CONTINUE
383: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
384: $ KASE, ISAVE )
385: IF( KASE.NE.0 ) THEN
386: IF( KASE.EQ.1 ) THEN
387: *
388: * Multiply by diag(W)*inv(A**T).
389: *
390: CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
391: DO 110 I = 1, N
392: WORK( N+I ) = WORK( I )*WORK( N+I )
393: 110 CONTINUE
394: ELSE IF( KASE.EQ.2 ) THEN
395: *
396: * Multiply by inv(A)*diag(W).
397: *
398: DO 120 I = 1, N
399: WORK( N+I ) = WORK( I )*WORK( N+I )
400: 120 CONTINUE
401: CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
402: END IF
403: GO TO 100
404: END IF
405: *
406: * Normalize error.
407: *
408: LSTRES = ZERO
409: DO 130 I = 1, N
410: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
411: 130 CONTINUE
412: IF( LSTRES.NE.ZERO )
413: $ FERR( J ) = FERR( J ) / LSTRES
414: *
415: 140 CONTINUE
416: *
417: RETURN
418: *
419: * End of DPPRFS
420: *
421: END
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