Annotation of rpl/lapack/lapack/dsprfs.f, revision 1.9
1.9 ! bertrand 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
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsprfs.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsprfs.f">
! 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: *> \date November 2011
! 175: *
! 176: *> \ingroup doubleOTHERcomputational
! 177: *
! 178: * =====================================================================
1.1 bertrand 179: SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
180: $ FERR, BERR, WORK, IWORK, INFO )
181: *
1.9 ! bertrand 182: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 183: * -- LAPACK is a software package provided by Univ. of Tennessee, --
184: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 185: * November 2011
1.1 bertrand 186: *
187: * .. Scalar Arguments ..
188: CHARACTER UPLO
189: INTEGER INFO, LDB, LDX, N, NRHS
190: * ..
191: * .. Array Arguments ..
192: INTEGER IPIV( * ), IWORK( * )
193: DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
194: $ FERR( * ), WORK( * ), X( LDX, * )
195: * ..
196: *
197: * =====================================================================
198: *
199: * .. Parameters ..
200: INTEGER ITMAX
201: PARAMETER ( ITMAX = 5 )
202: DOUBLE PRECISION ZERO
203: PARAMETER ( ZERO = 0.0D+0 )
204: DOUBLE PRECISION ONE
205: PARAMETER ( ONE = 1.0D+0 )
206: DOUBLE PRECISION TWO
207: PARAMETER ( TWO = 2.0D+0 )
208: DOUBLE PRECISION THREE
209: PARAMETER ( THREE = 3.0D+0 )
210: * ..
211: * .. Local Scalars ..
212: LOGICAL UPPER
213: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
214: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
215: * ..
216: * .. Local Arrays ..
217: INTEGER ISAVE( 3 )
218: * ..
219: * .. External Subroutines ..
220: EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
221: * ..
222: * .. Intrinsic Functions ..
223: INTRINSIC ABS, MAX
224: * ..
225: * .. External Functions ..
226: LOGICAL LSAME
227: DOUBLE PRECISION DLAMCH
228: EXTERNAL LSAME, DLAMCH
229: * ..
230: * .. Executable Statements ..
231: *
232: * Test the input parameters.
233: *
234: INFO = 0
235: UPPER = LSAME( UPLO, 'U' )
236: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
237: INFO = -1
238: ELSE IF( N.LT.0 ) THEN
239: INFO = -2
240: ELSE IF( NRHS.LT.0 ) THEN
241: INFO = -3
242: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
243: INFO = -8
244: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
245: INFO = -10
246: END IF
247: IF( INFO.NE.0 ) THEN
248: CALL XERBLA( 'DSPRFS', -INFO )
249: RETURN
250: END IF
251: *
252: * Quick return if possible
253: *
254: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
255: DO 10 J = 1, NRHS
256: FERR( J ) = ZERO
257: BERR( J ) = ZERO
258: 10 CONTINUE
259: RETURN
260: END IF
261: *
262: * NZ = maximum number of nonzero elements in each row of A, plus 1
263: *
264: NZ = N + 1
265: EPS = DLAMCH( 'Epsilon' )
266: SAFMIN = DLAMCH( 'Safe minimum' )
267: SAFE1 = NZ*SAFMIN
268: SAFE2 = SAFE1 / EPS
269: *
270: * Do for each right hand side
271: *
272: DO 140 J = 1, NRHS
273: *
274: COUNT = 1
275: LSTRES = THREE
276: 20 CONTINUE
277: *
278: * Loop until stopping criterion is satisfied.
279: *
280: * Compute residual R = B - A * X
281: *
282: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
283: CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
284: $ 1 )
285: *
286: * Compute componentwise relative backward error from formula
287: *
288: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
289: *
290: * where abs(Z) is the componentwise absolute value of the matrix
291: * or vector Z. If the i-th component of the denominator is less
292: * than SAFE2, then SAFE1 is added to the i-th components of the
293: * numerator and denominator before dividing.
294: *
295: DO 30 I = 1, N
296: WORK( I ) = ABS( B( I, J ) )
297: 30 CONTINUE
298: *
299: * Compute abs(A)*abs(X) + abs(B).
300: *
301: KK = 1
302: IF( UPPER ) THEN
303: DO 50 K = 1, N
304: S = ZERO
305: XK = ABS( X( K, J ) )
306: IK = KK
307: DO 40 I = 1, K - 1
308: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
309: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
310: IK = IK + 1
311: 40 CONTINUE
312: WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
313: KK = KK + K
314: 50 CONTINUE
315: ELSE
316: DO 70 K = 1, N
317: S = ZERO
318: XK = ABS( X( K, J ) )
319: WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
320: IK = KK + 1
321: DO 60 I = K + 1, N
322: WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
323: S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
324: IK = IK + 1
325: 60 CONTINUE
326: WORK( K ) = WORK( K ) + S
327: KK = KK + ( N-K+1 )
328: 70 CONTINUE
329: END IF
330: S = ZERO
331: DO 80 I = 1, N
332: IF( WORK( I ).GT.SAFE2 ) THEN
333: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
334: ELSE
335: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
336: $ ( WORK( I )+SAFE1 ) )
337: END IF
338: 80 CONTINUE
339: BERR( J ) = S
340: *
341: * Test stopping criterion. Continue iterating if
342: * 1) The residual BERR(J) is larger than machine epsilon, and
343: * 2) BERR(J) decreased by at least a factor of 2 during the
344: * last iteration, and
345: * 3) At most ITMAX iterations tried.
346: *
347: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
348: $ COUNT.LE.ITMAX ) THEN
349: *
350: * Update solution and try again.
351: *
352: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
353: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
354: LSTRES = BERR( J )
355: COUNT = COUNT + 1
356: GO TO 20
357: END IF
358: *
359: * Bound error from formula
360: *
361: * norm(X - XTRUE) / norm(X) .le. FERR =
362: * norm( abs(inv(A))*
363: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
364: *
365: * where
366: * norm(Z) is the magnitude of the largest component of Z
367: * inv(A) is the inverse of A
368: * abs(Z) is the componentwise absolute value of the matrix or
369: * vector Z
370: * NZ is the maximum number of nonzeros in any row of A, plus 1
371: * EPS is machine epsilon
372: *
373: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
374: * is incremented by SAFE1 if the i-th component of
375: * abs(A)*abs(X) + abs(B) is less than SAFE2.
376: *
377: * Use DLACN2 to estimate the infinity-norm of the matrix
378: * inv(A) * diag(W),
379: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
380: *
381: DO 90 I = 1, N
382: IF( WORK( I ).GT.SAFE2 ) THEN
383: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
384: ELSE
385: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
386: END IF
387: 90 CONTINUE
388: *
389: KASE = 0
390: 100 CONTINUE
391: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
392: $ KASE, ISAVE )
393: IF( KASE.NE.0 ) THEN
394: IF( KASE.EQ.1 ) THEN
395: *
1.8 bertrand 396: * Multiply by diag(W)*inv(A**T).
1.1 bertrand 397: *
398: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
399: $ INFO )
400: DO 110 I = 1, N
401: WORK( N+I ) = WORK( I )*WORK( N+I )
402: 110 CONTINUE
403: ELSE IF( KASE.EQ.2 ) THEN
404: *
405: * Multiply by inv(A)*diag(W).
406: *
407: DO 120 I = 1, N
408: WORK( N+I ) = WORK( I )*WORK( N+I )
409: 120 CONTINUE
410: CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
411: $ INFO )
412: END IF
413: GO TO 100
414: END IF
415: *
416: * Normalize error.
417: *
418: LSTRES = ZERO
419: DO 130 I = 1, N
420: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
421: 130 CONTINUE
422: IF( LSTRES.NE.ZERO )
423: $ FERR( J ) = FERR( J ) / LSTRES
424: *
425: 140 CONTINUE
426: *
427: RETURN
428: *
429: * End of DSPRFS
430: *
431: END
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