Annotation of rpl/lapack/lapack/zsprfs.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZSPRFS
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZSPRFS + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zsprfs.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zsprfs.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zsprfs.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
! 22: * FERR, BERR, WORK, RWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER UPLO
! 26: * INTEGER INFO, LDB, LDX, N, NRHS
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * INTEGER IPIV( * )
! 30: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 31: * COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
! 32: * $ X( LDX, * )
! 33: * ..
! 34: *
! 35: *
! 36: *> \par Purpose:
! 37: * =============
! 38: *>
! 39: *> \verbatim
! 40: *>
! 41: *> ZSPRFS improves the computed solution to a system of linear
! 42: *> equations when the coefficient matrix is symmetric indefinite
! 43: *> and packed, and provides error bounds and backward error estimates
! 44: *> for the solution.
! 45: *> \endverbatim
! 46: *
! 47: * Arguments:
! 48: * ==========
! 49: *
! 50: *> \param[in] UPLO
! 51: *> \verbatim
! 52: *> UPLO is CHARACTER*1
! 53: *> = 'U': Upper triangle of A is stored;
! 54: *> = 'L': Lower triangle of A is stored.
! 55: *> \endverbatim
! 56: *>
! 57: *> \param[in] N
! 58: *> \verbatim
! 59: *> N is INTEGER
! 60: *> The order of the matrix A. N >= 0.
! 61: *> \endverbatim
! 62: *>
! 63: *> \param[in] NRHS
! 64: *> \verbatim
! 65: *> NRHS is INTEGER
! 66: *> The number of right hand sides, i.e., the number of columns
! 67: *> of the matrices B and X. NRHS >= 0.
! 68: *> \endverbatim
! 69: *>
! 70: *> \param[in] AP
! 71: *> \verbatim
! 72: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
! 73: *> The upper or lower triangle of the symmetric matrix A, packed
! 74: *> columnwise in a linear array. The j-th column of A is stored
! 75: *> in the array AP as follows:
! 76: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 77: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 78: *> \endverbatim
! 79: *>
! 80: *> \param[in] AFP
! 81: *> \verbatim
! 82: *> AFP is COMPLEX*16 array, dimension (N*(N+1)/2)
! 83: *> The factored form of the matrix A. AFP contains the block
! 84: *> diagonal matrix D and the multipliers used to obtain the
! 85: *> factor U or L from the factorization A = U*D*U**T or
! 86: *> A = L*D*L**T as computed by ZSPTRF, stored as a packed
! 87: *> triangular matrix.
! 88: *> \endverbatim
! 89: *>
! 90: *> \param[in] IPIV
! 91: *> \verbatim
! 92: *> IPIV is INTEGER array, dimension (N)
! 93: *> Details of the interchanges and the block structure of D
! 94: *> as determined by ZSPTRF.
! 95: *> \endverbatim
! 96: *>
! 97: *> \param[in] B
! 98: *> \verbatim
! 99: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 100: *> The right hand side matrix B.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[in] LDB
! 104: *> \verbatim
! 105: *> LDB is INTEGER
! 106: *> The leading dimension of the array B. LDB >= max(1,N).
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[in,out] X
! 110: *> \verbatim
! 111: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 112: *> On entry, the solution matrix X, as computed by ZSPTRS.
! 113: *> On exit, the improved solution matrix X.
! 114: *> \endverbatim
! 115: *>
! 116: *> \param[in] LDX
! 117: *> \verbatim
! 118: *> LDX is INTEGER
! 119: *> The leading dimension of the array X. LDX >= max(1,N).
! 120: *> \endverbatim
! 121: *>
! 122: *> \param[out] FERR
! 123: *> \verbatim
! 124: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 125: *> The estimated forward error bound for each solution vector
! 126: *> X(j) (the j-th column of the solution matrix X).
! 127: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 128: *> is an estimated upper bound for the magnitude of the largest
! 129: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 130: *> largest element in X(j). The estimate is as reliable as
! 131: *> the estimate for RCOND, and is almost always a slight
! 132: *> overestimate of the true error.
! 133: *> \endverbatim
! 134: *>
! 135: *> \param[out] BERR
! 136: *> \verbatim
! 137: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 138: *> The componentwise relative backward error of each solution
! 139: *> vector X(j) (i.e., the smallest relative change in
! 140: *> any element of A or B that makes X(j) an exact solution).
! 141: *> \endverbatim
! 142: *>
! 143: *> \param[out] WORK
! 144: *> \verbatim
! 145: *> WORK is COMPLEX*16 array, dimension (2*N)
! 146: *> \endverbatim
! 147: *>
! 148: *> \param[out] RWORK
! 149: *> \verbatim
! 150: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 151: *> \endverbatim
! 152: *>
! 153: *> \param[out] INFO
! 154: *> \verbatim
! 155: *> INFO is INTEGER
! 156: *> = 0: successful exit
! 157: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 158: *> \endverbatim
! 159: *
! 160: *> \par Internal Parameters:
! 161: * =========================
! 162: *>
! 163: *> \verbatim
! 164: *> ITMAX is the maximum number of steps of iterative refinement.
! 165: *> \endverbatim
! 166: *
! 167: * Authors:
! 168: * ========
! 169: *
! 170: *> \author Univ. of Tennessee
! 171: *> \author Univ. of California Berkeley
! 172: *> \author Univ. of Colorado Denver
! 173: *> \author NAG Ltd.
! 174: *
! 175: *> \date November 2011
! 176: *
! 177: *> \ingroup complex16OTHERcomputational
! 178: *
! 179: * =====================================================================
1.1 bertrand 180: SUBROUTINE ZSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
181: $ FERR, BERR, WORK, RWORK, INFO )
182: *
1.9 ! bertrand 183: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 184: * -- LAPACK is a software package provided by Univ. of Tennessee, --
185: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 186: * November 2011
1.1 bertrand 187: *
188: * .. Scalar Arguments ..
189: CHARACTER UPLO
190: INTEGER INFO, LDB, LDX, N, NRHS
191: * ..
192: * .. Array Arguments ..
193: INTEGER IPIV( * )
194: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
195: COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
196: $ X( LDX, * )
197: * ..
198: *
199: * =====================================================================
200: *
201: * .. Parameters ..
202: INTEGER ITMAX
203: PARAMETER ( ITMAX = 5 )
204: DOUBLE PRECISION ZERO
205: PARAMETER ( ZERO = 0.0D+0 )
206: COMPLEX*16 ONE
207: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
208: DOUBLE PRECISION TWO
209: PARAMETER ( TWO = 2.0D+0 )
210: DOUBLE PRECISION THREE
211: PARAMETER ( THREE = 3.0D+0 )
212: * ..
213: * .. Local Scalars ..
214: LOGICAL UPPER
215: INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
216: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
217: COMPLEX*16 ZDUM
218: * ..
219: * .. Local Arrays ..
220: INTEGER ISAVE( 3 )
221: * ..
222: * .. External Subroutines ..
223: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSPMV, ZSPTRS
224: * ..
225: * .. Intrinsic Functions ..
226: INTRINSIC ABS, DBLE, DIMAG, MAX
227: * ..
228: * .. External Functions ..
229: LOGICAL LSAME
230: DOUBLE PRECISION DLAMCH
231: EXTERNAL LSAME, DLAMCH
232: * ..
233: * .. Statement Functions ..
234: DOUBLE PRECISION CABS1
235: * ..
236: * .. Statement Function definitions ..
237: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
238: * ..
239: * .. Executable Statements ..
240: *
241: * Test the input parameters.
242: *
243: INFO = 0
244: UPPER = LSAME( UPLO, 'U' )
245: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
246: INFO = -1
247: ELSE IF( N.LT.0 ) THEN
248: INFO = -2
249: ELSE IF( NRHS.LT.0 ) THEN
250: INFO = -3
251: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
252: INFO = -8
253: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
254: INFO = -10
255: END IF
256: IF( INFO.NE.0 ) THEN
257: CALL XERBLA( 'ZSPRFS', -INFO )
258: RETURN
259: END IF
260: *
261: * Quick return if possible
262: *
263: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
264: DO 10 J = 1, NRHS
265: FERR( J ) = ZERO
266: BERR( J ) = ZERO
267: 10 CONTINUE
268: RETURN
269: END IF
270: *
271: * NZ = maximum number of nonzero elements in each row of A, plus 1
272: *
273: NZ = N + 1
274: EPS = DLAMCH( 'Epsilon' )
275: SAFMIN = DLAMCH( 'Safe minimum' )
276: SAFE1 = NZ*SAFMIN
277: SAFE2 = SAFE1 / EPS
278: *
279: * Do for each right hand side
280: *
281: DO 140 J = 1, NRHS
282: *
283: COUNT = 1
284: LSTRES = THREE
285: 20 CONTINUE
286: *
287: * Loop until stopping criterion is satisfied.
288: *
289: * Compute residual R = B - A * X
290: *
291: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
292: CALL ZSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
293: *
294: * Compute componentwise relative backward error from formula
295: *
296: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
297: *
298: * where abs(Z) is the componentwise absolute value of the matrix
299: * or vector Z. If the i-th component of the denominator is less
300: * than SAFE2, then SAFE1 is added to the i-th components of the
301: * numerator and denominator before dividing.
302: *
303: DO 30 I = 1, N
304: RWORK( I ) = CABS1( B( I, J ) )
305: 30 CONTINUE
306: *
307: * Compute abs(A)*abs(X) + abs(B).
308: *
309: KK = 1
310: IF( UPPER ) THEN
311: DO 50 K = 1, N
312: S = ZERO
313: XK = CABS1( X( K, J ) )
314: IK = KK
315: DO 40 I = 1, K - 1
316: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
317: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
318: IK = IK + 1
319: 40 CONTINUE
320: RWORK( K ) = RWORK( K ) + CABS1( AP( KK+K-1 ) )*XK + S
321: KK = KK + K
322: 50 CONTINUE
323: ELSE
324: DO 70 K = 1, N
325: S = ZERO
326: XK = CABS1( X( K, J ) )
327: RWORK( K ) = RWORK( K ) + CABS1( AP( KK ) )*XK
328: IK = KK + 1
329: DO 60 I = K + 1, N
330: RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
331: S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
332: IK = IK + 1
333: 60 CONTINUE
334: RWORK( K ) = RWORK( K ) + S
335: KK = KK + ( N-K+1 )
336: 70 CONTINUE
337: END IF
338: S = ZERO
339: DO 80 I = 1, N
340: IF( RWORK( I ).GT.SAFE2 ) THEN
341: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
342: ELSE
343: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
344: $ ( RWORK( I )+SAFE1 ) )
345: END IF
346: 80 CONTINUE
347: BERR( J ) = S
348: *
349: * Test stopping criterion. Continue iterating if
350: * 1) The residual BERR(J) is larger than machine epsilon, and
351: * 2) BERR(J) decreased by at least a factor of 2 during the
352: * last iteration, and
353: * 3) At most ITMAX iterations tried.
354: *
355: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
356: $ COUNT.LE.ITMAX ) THEN
357: *
358: * Update solution and try again.
359: *
360: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
361: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
362: LSTRES = BERR( J )
363: COUNT = COUNT + 1
364: GO TO 20
365: END IF
366: *
367: * Bound error from formula
368: *
369: * norm(X - XTRUE) / norm(X) .le. FERR =
370: * norm( abs(inv(A))*
371: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
372: *
373: * where
374: * norm(Z) is the magnitude of the largest component of Z
375: * inv(A) is the inverse of A
376: * abs(Z) is the componentwise absolute value of the matrix or
377: * vector Z
378: * NZ is the maximum number of nonzeros in any row of A, plus 1
379: * EPS is machine epsilon
380: *
381: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
382: * is incremented by SAFE1 if the i-th component of
383: * abs(A)*abs(X) + abs(B) is less than SAFE2.
384: *
385: * Use ZLACN2 to estimate the infinity-norm of the matrix
386: * inv(A) * diag(W),
387: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
388: *
389: DO 90 I = 1, N
390: IF( RWORK( I ).GT.SAFE2 ) THEN
391: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
392: ELSE
393: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
394: $ SAFE1
395: END IF
396: 90 CONTINUE
397: *
398: KASE = 0
399: 100 CONTINUE
400: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
401: IF( KASE.NE.0 ) THEN
402: IF( KASE.EQ.1 ) THEN
403: *
1.8 bertrand 404: * Multiply by diag(W)*inv(A**T).
1.1 bertrand 405: *
406: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
407: DO 110 I = 1, N
408: WORK( I ) = RWORK( I )*WORK( I )
409: 110 CONTINUE
410: ELSE IF( KASE.EQ.2 ) THEN
411: *
412: * Multiply by inv(A)*diag(W).
413: *
414: DO 120 I = 1, N
415: WORK( I ) = RWORK( I )*WORK( I )
416: 120 CONTINUE
417: CALL ZSPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
418: END IF
419: GO TO 100
420: END IF
421: *
422: * Normalize error.
423: *
424: LSTRES = ZERO
425: DO 130 I = 1, N
426: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
427: 130 CONTINUE
428: IF( LSTRES.NE.ZERO )
429: $ FERR( J ) = FERR( J ) / LSTRES
430: *
431: 140 CONTINUE
432: *
433: RETURN
434: *
435: * End of ZSPRFS
436: *
437: END
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