Annotation of rpl/lapack/lapack/dpbrfs.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DPBRFS
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DPBRFS + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbrfs.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbrfs.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbrfs.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
! 22: * LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER UPLO
! 26: * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * INTEGER IWORK( * )
! 30: * DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 31: * $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
! 32: * ..
! 33: *
! 34: *
! 35: *> \par Purpose:
! 36: * =============
! 37: *>
! 38: *> \verbatim
! 39: *>
! 40: *> DPBRFS improves the computed solution to a system of linear
! 41: *> equations when the coefficient matrix is symmetric positive definite
! 42: *> and banded, 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] KD
! 63: *> \verbatim
! 64: *> KD is INTEGER
! 65: *> The number of superdiagonals of the matrix A if UPLO = 'U',
! 66: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in] NRHS
! 70: *> \verbatim
! 71: *> NRHS is INTEGER
! 72: *> The number of right hand sides, i.e., the number of columns
! 73: *> of the matrices B and X. NRHS >= 0.
! 74: *> \endverbatim
! 75: *>
! 76: *> \param[in] AB
! 77: *> \verbatim
! 78: *> AB is DOUBLE PRECISION array, dimension (LDAB,N)
! 79: *> The upper or lower triangle of the symmetric band matrix A,
! 80: *> stored in the first KD+1 rows of the array. The j-th column
! 81: *> of A is stored in the j-th column of the array AB as follows:
! 82: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
! 83: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
! 84: *> \endverbatim
! 85: *>
! 86: *> \param[in] LDAB
! 87: *> \verbatim
! 88: *> LDAB is INTEGER
! 89: *> The leading dimension of the array AB. LDAB >= KD+1.
! 90: *> \endverbatim
! 91: *>
! 92: *> \param[in] AFB
! 93: *> \verbatim
! 94: *> AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
! 95: *> The triangular factor U or L from the Cholesky factorization
! 96: *> A = U**T*U or A = L*L**T of the band matrix A as computed by
! 97: *> DPBTRF, in the same storage format as A (see AB).
! 98: *> \endverbatim
! 99: *>
! 100: *> \param[in] LDAFB
! 101: *> \verbatim
! 102: *> LDAFB is INTEGER
! 103: *> The leading dimension of the array AFB. LDAFB >= KD+1.
! 104: *> \endverbatim
! 105: *>
! 106: *> \param[in] B
! 107: *> \verbatim
! 108: *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
! 109: *> The right hand side matrix B.
! 110: *> \endverbatim
! 111: *>
! 112: *> \param[in] LDB
! 113: *> \verbatim
! 114: *> LDB is INTEGER
! 115: *> The leading dimension of the array B. LDB >= max(1,N).
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in,out] X
! 119: *> \verbatim
! 120: *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
! 121: *> On entry, the solution matrix X, as computed by DPBTRS.
! 122: *> On exit, the improved solution matrix X.
! 123: *> \endverbatim
! 124: *>
! 125: *> \param[in] LDX
! 126: *> \verbatim
! 127: *> LDX is INTEGER
! 128: *> The leading dimension of the array X. LDX >= max(1,N).
! 129: *> \endverbatim
! 130: *>
! 131: *> \param[out] FERR
! 132: *> \verbatim
! 133: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 134: *> The estimated forward error bound for each solution vector
! 135: *> X(j) (the j-th column of the solution matrix X).
! 136: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 137: *> is an estimated upper bound for the magnitude of the largest
! 138: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 139: *> largest element in X(j). The estimate is as reliable as
! 140: *> the estimate for RCOND, and is almost always a slight
! 141: *> overestimate of the true error.
! 142: *> \endverbatim
! 143: *>
! 144: *> \param[out] BERR
! 145: *> \verbatim
! 146: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 147: *> The componentwise relative backward error of each solution
! 148: *> vector X(j) (i.e., the smallest relative change in
! 149: *> any element of A or B that makes X(j) an exact solution).
! 150: *> \endverbatim
! 151: *>
! 152: *> \param[out] WORK
! 153: *> \verbatim
! 154: *> WORK is DOUBLE PRECISION array, dimension (3*N)
! 155: *> \endverbatim
! 156: *>
! 157: *> \param[out] IWORK
! 158: *> \verbatim
! 159: *> IWORK is INTEGER array, dimension (N)
! 160: *> \endverbatim
! 161: *>
! 162: *> \param[out] INFO
! 163: *> \verbatim
! 164: *> INFO is INTEGER
! 165: *> = 0: successful exit
! 166: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 167: *> \endverbatim
! 168: *
! 169: *> \par Internal Parameters:
! 170: * =========================
! 171: *>
! 172: *> \verbatim
! 173: *> ITMAX is the maximum number of steps of iterative refinement.
! 174: *> \endverbatim
! 175: *
! 176: * Authors:
! 177: * ========
! 178: *
! 179: *> \author Univ. of Tennessee
! 180: *> \author Univ. of California Berkeley
! 181: *> \author Univ. of Colorado Denver
! 182: *> \author NAG Ltd.
! 183: *
! 184: *> \date November 2011
! 185: *
! 186: *> \ingroup doubleOTHERcomputational
! 187: *
! 188: * =====================================================================
1.1 bertrand 189: SUBROUTINE DPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
190: $ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
191: *
1.9 ! bertrand 192: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 193: * -- LAPACK is a software package provided by Univ. of Tennessee, --
194: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 195: * November 2011
1.1 bertrand 196: *
197: * .. Scalar Arguments ..
198: CHARACTER UPLO
199: INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
200: * ..
201: * .. Array Arguments ..
202: INTEGER IWORK( * )
203: DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
204: $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
205: * ..
206: *
207: * =====================================================================
208: *
209: * .. Parameters ..
210: INTEGER ITMAX
211: PARAMETER ( ITMAX = 5 )
212: DOUBLE PRECISION ZERO
213: PARAMETER ( ZERO = 0.0D+0 )
214: DOUBLE PRECISION ONE
215: PARAMETER ( ONE = 1.0D+0 )
216: DOUBLE PRECISION TWO
217: PARAMETER ( TWO = 2.0D+0 )
218: DOUBLE PRECISION THREE
219: PARAMETER ( THREE = 3.0D+0 )
220: * ..
221: * .. Local Scalars ..
222: LOGICAL UPPER
223: INTEGER COUNT, I, J, K, KASE, L, NZ
224: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
225: * ..
226: * .. Local Arrays ..
227: INTEGER ISAVE( 3 )
228: * ..
229: * .. External Subroutines ..
230: EXTERNAL DAXPY, DCOPY, DLACN2, DPBTRS, DSBMV, XERBLA
231: * ..
232: * .. Intrinsic Functions ..
233: INTRINSIC ABS, MAX, MIN
234: * ..
235: * .. External Functions ..
236: LOGICAL LSAME
237: DOUBLE PRECISION DLAMCH
238: EXTERNAL LSAME, DLAMCH
239: * ..
240: * .. Executable Statements ..
241: *
242: * Test the input parameters.
243: *
244: INFO = 0
245: UPPER = LSAME( UPLO, 'U' )
246: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
247: INFO = -1
248: ELSE IF( N.LT.0 ) THEN
249: INFO = -2
250: ELSE IF( KD.LT.0 ) THEN
251: INFO = -3
252: ELSE IF( NRHS.LT.0 ) THEN
253: INFO = -4
254: ELSE IF( LDAB.LT.KD+1 ) THEN
255: INFO = -6
256: ELSE IF( LDAFB.LT.KD+1 ) THEN
257: INFO = -8
258: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
259: INFO = -10
260: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
261: INFO = -12
262: END IF
263: IF( INFO.NE.0 ) THEN
264: CALL XERBLA( 'DPBRFS', -INFO )
265: RETURN
266: END IF
267: *
268: * Quick return if possible
269: *
270: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
271: DO 10 J = 1, NRHS
272: FERR( J ) = ZERO
273: BERR( J ) = ZERO
274: 10 CONTINUE
275: RETURN
276: END IF
277: *
278: * NZ = maximum number of nonzero elements in each row of A, plus 1
279: *
280: NZ = MIN( N+1, 2*KD+2 )
281: EPS = DLAMCH( 'Epsilon' )
282: SAFMIN = DLAMCH( 'Safe minimum' )
283: SAFE1 = NZ*SAFMIN
284: SAFE2 = SAFE1 / EPS
285: *
286: * Do for each right hand side
287: *
288: DO 140 J = 1, NRHS
289: *
290: COUNT = 1
291: LSTRES = THREE
292: 20 CONTINUE
293: *
294: * Loop until stopping criterion is satisfied.
295: *
296: * Compute residual R = B - A * X
297: *
298: CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
299: CALL DSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
300: $ WORK( N+1 ), 1 )
301: *
302: * Compute componentwise relative backward error from formula
303: *
304: * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
305: *
306: * where abs(Z) is the componentwise absolute value of the matrix
307: * or vector Z. If the i-th component of the denominator is less
308: * than SAFE2, then SAFE1 is added to the i-th components of the
309: * numerator and denominator before dividing.
310: *
311: DO 30 I = 1, N
312: WORK( I ) = ABS( B( I, J ) )
313: 30 CONTINUE
314: *
315: * Compute abs(A)*abs(X) + abs(B).
316: *
317: IF( UPPER ) THEN
318: DO 50 K = 1, N
319: S = ZERO
320: XK = ABS( X( K, J ) )
321: L = KD + 1 - K
322: DO 40 I = MAX( 1, K-KD ), K - 1
323: WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
324: S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
325: 40 CONTINUE
326: WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
327: 50 CONTINUE
328: ELSE
329: DO 70 K = 1, N
330: S = ZERO
331: XK = ABS( X( K, J ) )
332: WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
333: L = 1 - K
334: DO 60 I = K + 1, MIN( N, K+KD )
335: WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
336: S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
337: 60 CONTINUE
338: WORK( K ) = WORK( K ) + S
339: 70 CONTINUE
340: END IF
341: S = ZERO
342: DO 80 I = 1, N
343: IF( WORK( I ).GT.SAFE2 ) THEN
344: S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
345: ELSE
346: S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
347: $ ( WORK( I )+SAFE1 ) )
348: END IF
349: 80 CONTINUE
350: BERR( J ) = S
351: *
352: * Test stopping criterion. Continue iterating if
353: * 1) The residual BERR(J) is larger than machine epsilon, and
354: * 2) BERR(J) decreased by at least a factor of 2 during the
355: * last iteration, and
356: * 3) At most ITMAX iterations tried.
357: *
358: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
359: $ COUNT.LE.ITMAX ) THEN
360: *
361: * Update solution and try again.
362: *
363: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
364: $ INFO )
365: CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
366: LSTRES = BERR( J )
367: COUNT = COUNT + 1
368: GO TO 20
369: END IF
370: *
371: * Bound error from formula
372: *
373: * norm(X - XTRUE) / norm(X) .le. FERR =
374: * norm( abs(inv(A))*
375: * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
376: *
377: * where
378: * norm(Z) is the magnitude of the largest component of Z
379: * inv(A) is the inverse of A
380: * abs(Z) is the componentwise absolute value of the matrix or
381: * vector Z
382: * NZ is the maximum number of nonzeros in any row of A, plus 1
383: * EPS is machine epsilon
384: *
385: * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
386: * is incremented by SAFE1 if the i-th component of
387: * abs(A)*abs(X) + abs(B) is less than SAFE2.
388: *
389: * Use DLACN2 to estimate the infinity-norm of the matrix
390: * inv(A) * diag(W),
391: * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
392: *
393: DO 90 I = 1, N
394: IF( WORK( I ).GT.SAFE2 ) THEN
395: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
396: ELSE
397: WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
398: END IF
399: 90 CONTINUE
400: *
401: KASE = 0
402: 100 CONTINUE
403: CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
404: $ KASE, ISAVE )
405: IF( KASE.NE.0 ) THEN
406: IF( KASE.EQ.1 ) THEN
407: *
1.8 bertrand 408: * Multiply by diag(W)*inv(A**T).
1.1 bertrand 409: *
410: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
411: $ INFO )
412: DO 110 I = 1, N
413: WORK( N+I ) = WORK( N+I )*WORK( I )
414: 110 CONTINUE
415: ELSE IF( KASE.EQ.2 ) THEN
416: *
417: * Multiply by inv(A)*diag(W).
418: *
419: DO 120 I = 1, N
420: WORK( N+I ) = WORK( N+I )*WORK( I )
421: 120 CONTINUE
422: CALL DPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
423: $ INFO )
424: END IF
425: GO TO 100
426: END IF
427: *
428: * Normalize error.
429: *
430: LSTRES = ZERO
431: DO 130 I = 1, N
432: LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
433: 130 CONTINUE
434: IF( LSTRES.NE.ZERO )
435: $ FERR( J ) = FERR( J ) / LSTRES
436: *
437: 140 CONTINUE
438: *
439: RETURN
440: *
441: * End of DPBRFS
442: *
443: END
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