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>
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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: *> \ingroup doubleOTHERcomputational
167: *
168: * =====================================================================
169: SUBROUTINE DPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
170: $ BERR, WORK, IWORK, INFO )
171: *
172: * -- LAPACK computational routine --
173: * -- LAPACK is a software package provided by Univ. of Tennessee, --
174: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175: *
176: * .. Scalar Arguments ..
177: CHARACTER UPLO
178: INTEGER INFO, LDB, LDX, N, NRHS
179: * ..
180: * .. Array Arguments ..
181: INTEGER IWORK( * )
182: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
183: $ FERR( * ), WORK( * ), X( LDX, * )
184: * ..
185: *
186: * =====================================================================
187: *
188: * .. Parameters ..
189: INTEGER ITMAX
190: PARAMETER ( ITMAX = 5 )
191: DOUBLE PRECISION ZERO
192: PARAMETER ( ZERO = 0.0D+0 )
193: DOUBLE PRECISION ONE
194: PARAMETER ( ONE = 1.0D+0 )
195: DOUBLE PRECISION TWO
196: PARAMETER ( TWO = 2.0D+0 )
197: DOUBLE PRECISION THREE
198: PARAMETER ( THREE = 3.0D+0 )
199: * ..
200: * .. Local Scalars ..
201: LOGICAL UPPER
202: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
203: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
204: * ..
205: * .. Local Arrays ..
206: INTEGER ISAVE( 3 )
207: * ..
208: * .. External Subroutines ..
209: EXTERNAL DAXPY, DCOPY, DLACN2, DPPTRS, DSPMV, XERBLA
210: * ..
211: * .. Intrinsic Functions ..
212: INTRINSIC ABS, MAX
213: * ..
214: * .. External Functions ..
215: LOGICAL LSAME
216: DOUBLE PRECISION DLAMCH
217: EXTERNAL LSAME, DLAMCH
218: * ..
219: * .. Executable Statements ..
220: *
221: * Test the input parameters.
222: *
223: INFO = 0
224: UPPER = LSAME( UPLO, 'U' )
225: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
226: INFO = -1
227: ELSE IF( N.LT.0 ) THEN
228: INFO = -2
229: ELSE IF( NRHS.LT.0 ) THEN
230: INFO = -3
231: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
232: INFO = -7
233: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
234: INFO = -9
235: END IF
236: IF( INFO.NE.0 ) THEN
237: CALL XERBLA( 'DPPRFS', -INFO )
238: RETURN
239: END IF
240: *
241: * Quick return if possible
242: *
243: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
244: DO 10 J = 1, NRHS
245: FERR( J ) = ZERO
246: BERR( J ) = ZERO
247: 10 CONTINUE
248: RETURN
249: END IF
250: *
251: * NZ = maximum number of nonzero elements in each row of A, plus 1
252: *
253: NZ = N + 1
254: EPS = DLAMCH( 'Epsilon' )
255: SAFMIN = DLAMCH( 'Safe minimum' )
256: SAFE1 = NZ*SAFMIN
257: SAFE2 = SAFE1 / EPS
258: *
259: * Do for each right hand side
260: *
261: DO 140 J = 1, NRHS
262: *
263: COUNT = 1
264: LSTRES = THREE
265: 20 CONTINUE
266: *
267: * Loop until stopping criterion is satisfied.
268: *
269: * Compute residual R = B - A * X
270: *
271: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
272: CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
273: $ 1 )
274: *
275: * Compute componentwise relative backward error from formula
276: *
277: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
278: *
279: * where abs(Z) is the componentwise absolute value of the matrix
280: * or vector Z. If the i-th component of the denominator is less
281: * than SAFE2, then SAFE1 is added to the i-th components of the
282: * numerator and denominator before dividing.
283: *
284: DO 30 I = 1, N
285: WORK( I ) = ABS( B( I, J ) )
286: 30 CONTINUE
287: *
288: * Compute abs(A)*abs(X) + abs(B).
289: *
290: KK = 1
291: IF( UPPER ) THEN
292: DO 50 K = 1, N
293: S = ZERO
294: XK = ABS( X( K, J ) )
295: IK = KK
296: DO 40 I = 1, K - 1
297: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
298: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
299: IK = IK + 1
300: 40 CONTINUE
301: WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
302: KK = KK + K
303: 50 CONTINUE
304: ELSE
305: DO 70 K = 1, N
306: S = ZERO
307: XK = ABS( X( K, J ) )
308: WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
309: IK = KK + 1
310: DO 60 I = K + 1, N
311: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
312: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
313: IK = IK + 1
314: 60 CONTINUE
315: WORK( K ) = WORK( K ) + S
316: KK = KK + ( N-K+1 )
317: 70 CONTINUE
318: END IF
319: S = ZERO
320: DO 80 I = 1, N
321: IF( WORK( I ).GT.SAFE2 ) THEN
322: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
323: ELSE
324: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
325: $ ( WORK( I )+SAFE1 ) )
326: END IF
327: 80 CONTINUE
328: BERR( J ) = S
329: *
330: * Test stopping criterion. Continue iterating if
331: * 1) The residual BERR(J) is larger than machine epsilon, and
332: * 2) BERR(J) decreased by at least a factor of 2 during the
333: * last iteration, and
334: * 3) At most ITMAX iterations tried.
335: *
336: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
337: $ COUNT.LE.ITMAX ) THEN
338: *
339: * Update solution and try again.
340: *
341: CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
342: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
343: LSTRES = BERR( J )
344: COUNT = COUNT + 1
345: GO TO 20
346: END IF
347: *
348: * Bound error from formula
349: *
350: * norm(X - XTRUE) / norm(X) .le. FERR =
351: * norm( abs(inv(A))*
352: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
353: *
354: * where
355: * norm(Z) is the magnitude of the largest component of Z
356: * inv(A) is the inverse of A
357: * abs(Z) is the componentwise absolute value of the matrix or
358: * vector Z
359: * NZ is the maximum number of nonzeros in any row of A, plus 1
360: * EPS is machine epsilon
361: *
362: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
363: * is incremented by SAFE1 if the i-th component of
364: * abs(A)*abs(X) + abs(B) is less than SAFE2.
365: *
366: * Use DLACN2 to estimate the infinity-norm of the matrix
367: * inv(A) * diag(W),
368: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
369: *
370: DO 90 I = 1, N
371: IF( WORK( I ).GT.SAFE2 ) THEN
372: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
373: ELSE
374: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
375: END IF
376: 90 CONTINUE
377: *
378: KASE = 0
379: 100 CONTINUE
380: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
381: $ KASE, ISAVE )
382: IF( KASE.NE.0 ) THEN
383: IF( KASE.EQ.1 ) THEN
384: *
385: * Multiply by diag(W)*inv(A**T).
386: *
387: CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
388: DO 110 I = 1, N
389: WORK( N+I ) = WORK( I )*WORK( N+I )
390: 110 CONTINUE
391: ELSE IF( KASE.EQ.2 ) THEN
392: *
393: * Multiply by inv(A)*diag(W).
394: *
395: DO 120 I = 1, N
396: WORK( N+I ) = WORK( I )*WORK( N+I )
397: 120 CONTINUE
398: CALL DPPTRS( UPLO, N, 1, AFP, WORK( N+1 ), N, INFO )
399: END IF
400: GO TO 100
401: END IF
402: *
403: * Normalize error.
404: *
405: LSTRES = ZERO
406: DO 130 I = 1, N
407: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
408: 130 CONTINUE
409: IF( LSTRES.NE.ZERO )
410: $ FERR( J ) = FERR( J ) / LSTRES
411: *
412: 140 CONTINUE
413: *
414: RETURN
415: *
416: * End of DPPRFS
417: *
418: END
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