Annotation of rpl/lapack/lapack/zherfs.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZHERFS
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZHERFS + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zherfs.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zherfs.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zherfs.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
! 22: * X, LDX, FERR, BERR, WORK, RWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER UPLO
! 26: * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * INTEGER IPIV( * )
! 30: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 31: * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
! 32: * $ WORK( * ), X( LDX, * )
! 33: * ..
! 34: *
! 35: *
! 36: *> \par Purpose:
! 37: * =============
! 38: *>
! 39: *> \verbatim
! 40: *>
! 41: *> ZHERFS improves the computed solution to a system of linear
! 42: *> equations when the coefficient matrix is Hermitian indefinite, and
! 43: *> provides error bounds and backward error estimates 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] A
! 70: *> \verbatim
! 71: *> A is COMPLEX*16 array, dimension (LDA,N)
! 72: *> The Hermitian matrix A. If UPLO = 'U', the leading N-by-N
! 73: *> upper triangular part of A contains the upper triangular part
! 74: *> of the matrix A, and the strictly lower triangular part of A
! 75: *> is not referenced. If UPLO = 'L', the leading N-by-N lower
! 76: *> triangular part of A contains the lower triangular part of
! 77: *> the matrix A, and the strictly upper triangular part of A is
! 78: *> not referenced.
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[in] LDA
! 82: *> \verbatim
! 83: *> LDA is INTEGER
! 84: *> The leading dimension of the array A. LDA >= max(1,N).
! 85: *> \endverbatim
! 86: *>
! 87: *> \param[in] AF
! 88: *> \verbatim
! 89: *> AF is COMPLEX*16 array, dimension (LDAF,N)
! 90: *> The factored form of the matrix A. AF contains the block
! 91: *> diagonal matrix D and the multipliers used to obtain the
! 92: *> factor U or L from the factorization A = U*D*U**H or
! 93: *> A = L*D*L**H as computed by ZHETRF.
! 94: *> \endverbatim
! 95: *>
! 96: *> \param[in] LDAF
! 97: *> \verbatim
! 98: *> LDAF is INTEGER
! 99: *> The leading dimension of the array AF. LDAF >= max(1,N).
! 100: *> \endverbatim
! 101: *>
! 102: *> \param[in] IPIV
! 103: *> \verbatim
! 104: *> IPIV is INTEGER array, dimension (N)
! 105: *> Details of the interchanges and the block structure of D
! 106: *> as determined by ZHETRF.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[in] B
! 110: *> \verbatim
! 111: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 112: *> The right hand side matrix B.
! 113: *> \endverbatim
! 114: *>
! 115: *> \param[in] LDB
! 116: *> \verbatim
! 117: *> LDB is INTEGER
! 118: *> The leading dimension of the array B. LDB >= max(1,N).
! 119: *> \endverbatim
! 120: *>
! 121: *> \param[in,out] X
! 122: *> \verbatim
! 123: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 124: *> On entry, the solution matrix X, as computed by ZHETRS.
! 125: *> On exit, the improved solution matrix X.
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[in] LDX
! 129: *> \verbatim
! 130: *> LDX is INTEGER
! 131: *> The leading dimension of the array X. LDX >= max(1,N).
! 132: *> \endverbatim
! 133: *>
! 134: *> \param[out] FERR
! 135: *> \verbatim
! 136: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 137: *> The estimated forward error bound for each solution vector
! 138: *> X(j) (the j-th column of the solution matrix X).
! 139: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 140: *> is an estimated upper bound for the magnitude of the largest
! 141: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 142: *> largest element in X(j). The estimate is as reliable as
! 143: *> the estimate for RCOND, and is almost always a slight
! 144: *> overestimate of the true error.
! 145: *> \endverbatim
! 146: *>
! 147: *> \param[out] BERR
! 148: *> \verbatim
! 149: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 150: *> The componentwise relative backward error of each solution
! 151: *> vector X(j) (i.e., the smallest relative change in
! 152: *> any element of A or B that makes X(j) an exact solution).
! 153: *> \endverbatim
! 154: *>
! 155: *> \param[out] WORK
! 156: *> \verbatim
! 157: *> WORK is COMPLEX*16 array, dimension (2*N)
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[out] RWORK
! 161: *> \verbatim
! 162: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 163: *> \endverbatim
! 164: *>
! 165: *> \param[out] INFO
! 166: *> \verbatim
! 167: *> INFO is INTEGER
! 168: *> = 0: successful exit
! 169: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 170: *> \endverbatim
! 171: *
! 172: *> \par Internal Parameters:
! 173: * =========================
! 174: *>
! 175: *> \verbatim
! 176: *> ITMAX is the maximum number of steps of iterative refinement.
! 177: *> \endverbatim
! 178: *
! 179: * Authors:
! 180: * ========
! 181: *
! 182: *> \author Univ. of Tennessee
! 183: *> \author Univ. of California Berkeley
! 184: *> \author Univ. of Colorado Denver
! 185: *> \author NAG Ltd.
! 186: *
! 187: *> \date November 2011
! 188: *
! 189: *> \ingroup complex16HEcomputational
! 190: *
! 191: * =====================================================================
1.1 bertrand 192: SUBROUTINE ZHERFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
193: $ X, LDX, FERR, BERR, WORK, RWORK, INFO )
194: *
1.9 ! bertrand 195: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 196: * -- LAPACK is a software package provided by Univ. of Tennessee, --
197: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 198: * November 2011
1.1 bertrand 199: *
200: * .. Scalar Arguments ..
201: CHARACTER UPLO
202: INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
203: * ..
204: * .. Array Arguments ..
205: INTEGER IPIV( * )
206: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
207: COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
208: $ WORK( * ), X( LDX, * )
209: * ..
210: *
211: * =====================================================================
212: *
213: * .. Parameters ..
214: INTEGER ITMAX
215: PARAMETER ( ITMAX = 5 )
216: DOUBLE PRECISION ZERO
217: PARAMETER ( ZERO = 0.0D+0 )
218: COMPLEX*16 ONE
219: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
220: DOUBLE PRECISION TWO
221: PARAMETER ( TWO = 2.0D+0 )
222: DOUBLE PRECISION THREE
223: PARAMETER ( THREE = 3.0D+0 )
224: * ..
225: * .. Local Scalars ..
226: LOGICAL UPPER
227: INTEGER COUNT, I, J, K, KASE, NZ
228: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
229: COMPLEX*16 ZDUM
230: * ..
231: * .. Local Arrays ..
232: INTEGER ISAVE( 3 )
233: * ..
234: * .. External Subroutines ..
235: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZHETRS, ZLACN2
236: * ..
237: * .. Intrinsic Functions ..
238: INTRINSIC ABS, DBLE, DIMAG, MAX
239: * ..
240: * .. External Functions ..
241: LOGICAL LSAME
242: DOUBLE PRECISION DLAMCH
243: EXTERNAL LSAME, DLAMCH
244: * ..
245: * .. Statement Functions ..
246: DOUBLE PRECISION CABS1
247: * ..
248: * .. Statement Function definitions ..
249: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
250: * ..
251: * .. Executable Statements ..
252: *
253: * Test the input parameters.
254: *
255: INFO = 0
256: UPPER = LSAME( UPLO, 'U' )
257: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
258: INFO = -1
259: ELSE IF( N.LT.0 ) THEN
260: INFO = -2
261: ELSE IF( NRHS.LT.0 ) THEN
262: INFO = -3
263: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
264: INFO = -5
265: ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
266: INFO = -7
267: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
268: INFO = -10
269: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
270: INFO = -12
271: END IF
272: IF( INFO.NE.0 ) THEN
273: CALL XERBLA( 'ZHERFS', -INFO )
274: RETURN
275: END IF
276: *
277: * Quick return if possible
278: *
279: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
280: DO 10 J = 1, NRHS
281: FERR( J ) = ZERO
282: BERR( J ) = ZERO
283: 10 CONTINUE
284: RETURN
285: END IF
286: *
287: * NZ = maximum number of nonzero elements in each row of A, plus 1
288: *
289: NZ = N + 1
290: EPS = DLAMCH( 'Epsilon' )
291: SAFMIN = DLAMCH( 'Safe minimum' )
292: SAFE1 = NZ*SAFMIN
293: SAFE2 = SAFE1 / EPS
294: *
295: * Do for each right hand side
296: *
297: DO 140 J = 1, NRHS
298: *
299: COUNT = 1
300: LSTRES = THREE
301: 20 CONTINUE
302: *
303: * Loop until stopping criterion is satisfied.
304: *
305: * Compute residual R = B - A * X
306: *
307: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
308: CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
309: *
310: * Compute componentwise relative backward error from formula
311: *
312: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
313: *
314: * where abs(Z) is the componentwise absolute value of the matrix
315: * or vector Z. If the i-th component of the denominator is less
316: * than SAFE2, then SAFE1 is added to the i-th components of the
317: * numerator and denominator before dividing.
318: *
319: DO 30 I = 1, N
320: RWORK( I ) = CABS1( B( I, J ) )
321: 30 CONTINUE
322: *
323: * Compute abs(A)*abs(X) + abs(B).
324: *
325: IF( UPPER ) THEN
326: DO 50 K = 1, N
327: S = ZERO
328: XK = CABS1( X( K, J ) )
329: DO 40 I = 1, K - 1
330: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
331: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
332: 40 CONTINUE
333: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
334: 50 CONTINUE
335: ELSE
336: DO 70 K = 1, N
337: S = ZERO
338: XK = CABS1( X( K, J ) )
339: RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
340: DO 60 I = K + 1, N
341: RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
342: S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
343: 60 CONTINUE
344: RWORK( K ) = RWORK( K ) + S
345: 70 CONTINUE
346: END IF
347: S = ZERO
348: DO 80 I = 1, N
349: IF( RWORK( I ).GT.SAFE2 ) THEN
350: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
351: ELSE
352: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
353: $ ( RWORK( I )+SAFE1 ) )
354: END IF
355: 80 CONTINUE
356: BERR( J ) = S
357: *
358: * Test stopping criterion. Continue iterating if
359: * 1) The residual BERR(J) is larger than machine epsilon, and
360: * 2) BERR(J) decreased by at least a factor of 2 during the
361: * last iteration, and
362: * 3) At most ITMAX iterations tried.
363: *
364: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
365: $ COUNT.LE.ITMAX ) THEN
366: *
367: * Update solution and try again.
368: *
369: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
370: CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
371: LSTRES = BERR( J )
372: COUNT = COUNT + 1
373: GO TO 20
374: END IF
375: *
376: * Bound error from formula
377: *
378: * norm(X - XTRUE) / norm(X) .le. FERR =
379: * norm( abs(inv(A))*
380: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
381: *
382: * where
383: * norm(Z) is the magnitude of the largest component of Z
384: * inv(A) is the inverse of A
385: * abs(Z) is the componentwise absolute value of the matrix or
386: * vector Z
387: * NZ is the maximum number of nonzeros in any row of A, plus 1
388: * EPS is machine epsilon
389: *
390: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
391: * is incremented by SAFE1 if the i-th component of
392: * abs(A)*abs(X) + abs(B) is less than SAFE2.
393: *
394: * Use ZLACN2 to estimate the infinity-norm of the matrix
395: * inv(A) * diag(W),
396: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
397: *
398: DO 90 I = 1, N
399: IF( RWORK( I ).GT.SAFE2 ) THEN
400: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
401: ELSE
402: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
403: $ SAFE1
404: END IF
405: 90 CONTINUE
406: *
407: KASE = 0
408: 100 CONTINUE
409: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
410: IF( KASE.NE.0 ) THEN
411: IF( KASE.EQ.1 ) THEN
412: *
1.8 bertrand 413: * Multiply by diag(W)*inv(A**H).
1.1 bertrand 414: *
415: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
416: DO 110 I = 1, N
417: WORK( I ) = RWORK( I )*WORK( I )
418: 110 CONTINUE
419: ELSE IF( KASE.EQ.2 ) THEN
420: *
421: * Multiply by inv(A)*diag(W).
422: *
423: DO 120 I = 1, N
424: WORK( I ) = RWORK( I )*WORK( I )
425: 120 CONTINUE
426: CALL ZHETRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
427: END IF
428: GO TO 100
429: END IF
430: *
431: * Normalize error.
432: *
433: LSTRES = ZERO
434: DO 130 I = 1, N
435: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
436: 130 CONTINUE
437: IF( LSTRES.NE.ZERO )
438: $ FERR( J ) = FERR( J ) / LSTRES
439: *
440: 140 CONTINUE
441: *
442: RETURN
443: *
444: * End of ZHERFS
445: *
446: END
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