1: *> \brief \b DSPRFS
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSPRFS + 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/dsprfs.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
22: * FERR, BERR, WORK, IWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, LDB, LDX, N, NRHS
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * ), 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: *> DSPRFS improves the computed solution to a system of linear
41: *> equations when the coefficient matrix is symmetric indefinite
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)*(2*n-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 factored form of the matrix A. AFP contains the block
83: *> diagonal matrix D and the multipliers used to obtain the
84: *> factor U or L from the factorization A = U*D*U**T or
85: *> A = L*D*L**T as computed by DSPTRF, stored as a packed
86: *> triangular matrix.
87: *> \endverbatim
88: *>
89: *> \param[in] IPIV
90: *> \verbatim
91: *> IPIV is INTEGER array, dimension (N)
92: *> Details of the interchanges and the block structure of D
93: *> as determined by DSPTRF.
94: *> \endverbatim
95: *>
96: *> \param[in] B
97: *> \verbatim
98: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
99: *> The right hand side matrix B.
100: *> \endverbatim
101: *>
102: *> \param[in] LDB
103: *> \verbatim
104: *> LDB is INTEGER
105: *> The leading dimension of the array B. LDB >= max(1,N).
106: *> \endverbatim
107: *>
108: *> \param[in,out] X
109: *> \verbatim
110: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
111: *> On entry, the solution matrix X, as computed by DSPTRS.
112: *> On exit, the improved solution matrix X.
113: *> \endverbatim
114: *>
115: *> \param[in] LDX
116: *> \verbatim
117: *> LDX is INTEGER
118: *> The leading dimension of the array X. LDX >= max(1,N).
119: *> \endverbatim
120: *>
121: *> \param[out] FERR
122: *> \verbatim
123: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
124: *> The estimated forward error bound for each solution vector
125: *> X(j) (the j-th column of the solution matrix X).
126: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
127: *> is an estimated upper bound for the magnitude of the largest
128: *> element in (X(j) - XTRUE) divided by the magnitude of the
129: *> largest element in X(j). The estimate is as reliable as
130: *> the estimate for RCOND, and is almost always a slight
131: *> overestimate of the true error.
132: *> \endverbatim
133: *>
134: *> \param[out] BERR
135: *> \verbatim
136: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
137: *> The componentwise relative backward error of each solution
138: *> vector X(j) (i.e., the smallest relative change in
139: *> any element of A or B that makes X(j) an exact solution).
140: *> \endverbatim
141: *>
142: *> \param[out] WORK
143: *> \verbatim
144: *> WORK is DOUBLE PRECISION array, dimension (3*N)
145: *> \endverbatim
146: *>
147: *> \param[out] IWORK
148: *> \verbatim
149: *> IWORK is INTEGER array, dimension (N)
150: *> \endverbatim
151: *>
152: *> \param[out] INFO
153: *> \verbatim
154: *> INFO is INTEGER
155: *> = 0: successful exit
156: *> < 0: if INFO = -i, the i-th argument had an illegal value
157: *> \endverbatim
158: *
159: *> \par Internal Parameters:
160: * =========================
161: *>
162: *> \verbatim
163: *> ITMAX is the maximum number of steps of iterative refinement.
164: *> \endverbatim
165: *
166: * Authors:
167: * ========
168: *
169: *> \author Univ. of Tennessee
170: *> \author Univ. of California Berkeley
171: *> \author Univ. of Colorado Denver
172: *> \author NAG Ltd.
173: *
174: *> \ingroup doubleOTHERcomputational
175: *
176: * =====================================================================
177: SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
178: $ FERR, BERR, WORK, IWORK, INFO )
179: *
180: * -- LAPACK computational routine --
181: * -- LAPACK is a software package provided by Univ. of Tennessee, --
182: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
183: *
184: * .. Scalar Arguments ..
185: CHARACTER UPLO
186: INTEGER INFO, LDB, LDX, N, NRHS
187: * ..
188: * .. Array Arguments ..
189: INTEGER IPIV( * ), IWORK( * )
190: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
191: $ FERR( * ), WORK( * ), X( LDX, * )
192: * ..
193: *
194: * =====================================================================
195: *
196: * .. Parameters ..
197: INTEGER ITMAX
198: PARAMETER ( ITMAX = 5 )
199: DOUBLE PRECISION ZERO
200: PARAMETER ( ZERO = 0.0D+0 )
201: DOUBLE PRECISION ONE
202: PARAMETER ( ONE = 1.0D+0 )
203: DOUBLE PRECISION TWO
204: PARAMETER ( TWO = 2.0D+0 )
205: DOUBLE PRECISION THREE
206: PARAMETER ( THREE = 3.0D+0 )
207: * ..
208: * .. Local Scalars ..
209: LOGICAL UPPER
210: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
211: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
212: * ..
213: * .. Local Arrays ..
214: INTEGER ISAVE( 3 )
215: * ..
216: * .. External Subroutines ..
217: EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
218: * ..
219: * .. Intrinsic Functions ..
220: INTRINSIC ABS, MAX
221: * ..
222: * .. External Functions ..
223: LOGICAL LSAME
224: DOUBLE PRECISION DLAMCH
225: EXTERNAL LSAME, DLAMCH
226: * ..
227: * .. Executable Statements ..
228: *
229: * Test the input parameters.
230: *
231: INFO = 0
232: UPPER = LSAME( UPLO, 'U' )
233: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
234: INFO = -1
235: ELSE IF( N.LT.0 ) THEN
236: INFO = -2
237: ELSE IF( NRHS.LT.0 ) THEN
238: INFO = -3
239: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
240: INFO = -8
241: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
242: INFO = -10
243: END IF
244: IF( INFO.NE.0 ) THEN
245: CALL XERBLA( 'DSPRFS', -INFO )
246: RETURN
247: END IF
248: *
249: * Quick return if possible
250: *
251: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
252: DO 10 J = 1, NRHS
253: FERR( J ) = ZERO
254: BERR( J ) = ZERO
255: 10 CONTINUE
256: RETURN
257: END IF
258: *
259: * NZ = maximum number of nonzero elements in each row of A, plus 1
260: *
261: NZ = N + 1
262: EPS = DLAMCH( 'Epsilon' )
263: SAFMIN = DLAMCH( 'Safe minimum' )
264: SAFE1 = NZ*SAFMIN
265: SAFE2 = SAFE1 / EPS
266: *
267: * Do for each right hand side
268: *
269: DO 140 J = 1, NRHS
270: *
271: COUNT = 1
272: LSTRES = THREE
273: 20 CONTINUE
274: *
275: * Loop until stopping criterion is satisfied.
276: *
277: * Compute residual R = B - A * X
278: *
279: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
280: CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
281: $ 1 )
282: *
283: * Compute componentwise relative backward error from formula
284: *
285: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
286: *
287: * where abs(Z) is the componentwise absolute value of the matrix
288: * or vector Z. If the i-th component of the denominator is less
289: * than SAFE2, then SAFE1 is added to the i-th components of the
290: * numerator and denominator before dividing.
291: *
292: DO 30 I = 1, N
293: WORK( I ) = ABS( B( I, J ) )
294: 30 CONTINUE
295: *
296: * Compute abs(A)*abs(X) + abs(B).
297: *
298: KK = 1
299: IF( UPPER ) THEN
300: DO 50 K = 1, N
301: S = ZERO
302: XK = ABS( X( K, J ) )
303: IK = KK
304: DO 40 I = 1, K - 1
305: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
306: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
307: IK = IK + 1
308: 40 CONTINUE
309: WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
310: KK = KK + K
311: 50 CONTINUE
312: ELSE
313: DO 70 K = 1, N
314: S = ZERO
315: XK = ABS( X( K, J ) )
316: WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
317: IK = KK + 1
318: DO 60 I = K + 1, N
319: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
320: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
321: IK = IK + 1
322: 60 CONTINUE
323: WORK( K ) = WORK( K ) + S
324: KK = KK + ( N-K+1 )
325: 70 CONTINUE
326: END IF
327: S = ZERO
328: DO 80 I = 1, N
329: IF( WORK( I ).GT.SAFE2 ) THEN
330: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
331: ELSE
332: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
333: $ ( WORK( I )+SAFE1 ) )
334: END IF
335: 80 CONTINUE
336: BERR( J ) = S
337: *
338: * Test stopping criterion. Continue iterating if
339: * 1) The residual BERR(J) is larger than machine epsilon, and
340: * 2) BERR(J) decreased by at least a factor of 2 during the
341: * last iteration, and
342: * 3) At most ITMAX iterations tried.
343: *
344: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
345: $ COUNT.LE.ITMAX ) THEN
346: *
347: * Update solution and try again.
348: *
349: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
350: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
351: LSTRES = BERR( J )
352: COUNT = COUNT + 1
353: GO TO 20
354: END IF
355: *
356: * Bound error from formula
357: *
358: * norm(X - XTRUE) / norm(X) .le. FERR =
359: * norm( abs(inv(A))*
360: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
361: *
362: * where
363: * norm(Z) is the magnitude of the largest component of Z
364: * inv(A) is the inverse of A
365: * abs(Z) is the componentwise absolute value of the matrix or
366: * vector Z
367: * NZ is the maximum number of nonzeros in any row of A, plus 1
368: * EPS is machine epsilon
369: *
370: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
371: * is incremented by SAFE1 if the i-th component of
372: * abs(A)*abs(X) + abs(B) is less than SAFE2.
373: *
374: * Use DLACN2 to estimate the infinity-norm of the matrix
375: * inv(A) * diag(W),
376: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
377: *
378: DO 90 I = 1, N
379: IF( WORK( I ).GT.SAFE2 ) THEN
380: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
381: ELSE
382: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
383: END IF
384: 90 CONTINUE
385: *
386: KASE = 0
387: 100 CONTINUE
388: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
389: $ KASE, ISAVE )
390: IF( KASE.NE.0 ) THEN
391: IF( KASE.EQ.1 ) THEN
392: *
393: * Multiply by diag(W)*inv(A**T).
394: *
395: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
396: $ INFO )
397: DO 110 I = 1, N
398: WORK( N+I ) = WORK( I )*WORK( N+I )
399: 110 CONTINUE
400: ELSE IF( KASE.EQ.2 ) THEN
401: *
402: * Multiply by inv(A)*diag(W).
403: *
404: DO 120 I = 1, N
405: WORK( N+I ) = WORK( I )*WORK( N+I )
406: 120 CONTINUE
407: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
408: $ INFO )
409: END IF
410: GO TO 100
411: END IF
412: *
413: * Normalize error.
414: *
415: LSTRES = ZERO
416: DO 130 I = 1, N
417: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
418: 130 CONTINUE
419: IF( LSTRES.NE.ZERO )
420: $ FERR( J ) = FERR( J ) / LSTRES
421: *
422: 140 CONTINUE
423: *
424: RETURN
425: *
426: * End of DSPRFS
427: *
428: END
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