Annotation of rpl/lapack/lapack/zgbrfs.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b ZGBRFS
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZGBRFS + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbrfs.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbrfs.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbrfs.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
! 22: * IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
! 23: * INFO )
! 24: *
! 25: * .. Scalar Arguments ..
! 26: * CHARACTER TRANS
! 27: * INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * INTEGER IPIV( * )
! 31: * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
! 32: * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
! 33: * $ WORK( * ), X( LDX, * )
! 34: * ..
! 35: *
! 36: *
! 37: *> \par Purpose:
! 38: * =============
! 39: *>
! 40: *> \verbatim
! 41: *>
! 42: *> ZGBRFS improves the computed solution to a system of linear
! 43: *> equations when the coefficient matrix is banded, and provides
! 44: *> error bounds and backward error estimates for the solution.
! 45: *> \endverbatim
! 46: *
! 47: * Arguments:
! 48: * ==========
! 49: *
! 50: *> \param[in] TRANS
! 51: *> \verbatim
! 52: *> TRANS is CHARACTER*1
! 53: *> Specifies the form of the system of equations:
! 54: *> = 'N': A * X = B (No transpose)
! 55: *> = 'T': A**T * X = B (Transpose)
! 56: *> = 'C': A**H * X = B (Conjugate transpose)
! 57: *> \endverbatim
! 58: *>
! 59: *> \param[in] N
! 60: *> \verbatim
! 61: *> N is INTEGER
! 62: *> The order of the matrix A. N >= 0.
! 63: *> \endverbatim
! 64: *>
! 65: *> \param[in] KL
! 66: *> \verbatim
! 67: *> KL is INTEGER
! 68: *> The number of subdiagonals within the band of A. KL >= 0.
! 69: *> \endverbatim
! 70: *>
! 71: *> \param[in] KU
! 72: *> \verbatim
! 73: *> KU is INTEGER
! 74: *> The number of superdiagonals within the band of A. KU >= 0.
! 75: *> \endverbatim
! 76: *>
! 77: *> \param[in] NRHS
! 78: *> \verbatim
! 79: *> NRHS is INTEGER
! 80: *> The number of right hand sides, i.e., the number of columns
! 81: *> of the matrices B and X. NRHS >= 0.
! 82: *> \endverbatim
! 83: *>
! 84: *> \param[in] AB
! 85: *> \verbatim
! 86: *> AB is COMPLEX*16 array, dimension (LDAB,N)
! 87: *> The original band matrix A, stored in rows 1 to KL+KU+1.
! 88: *> The j-th column of A is stored in the j-th column of the
! 89: *> array AB as follows:
! 90: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[in] LDAB
! 94: *> \verbatim
! 95: *> LDAB is INTEGER
! 96: *> The leading dimension of the array AB. LDAB >= KL+KU+1.
! 97: *> \endverbatim
! 98: *>
! 99: *> \param[in] AFB
! 100: *> \verbatim
! 101: *> AFB is COMPLEX*16 array, dimension (LDAFB,N)
! 102: *> Details of the LU factorization of the band matrix A, as
! 103: *> computed by ZGBTRF. U is stored as an upper triangular band
! 104: *> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
! 105: *> the multipliers used during the factorization are stored in
! 106: *> rows KL+KU+2 to 2*KL+KU+1.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[in] LDAFB
! 110: *> \verbatim
! 111: *> LDAFB is INTEGER
! 112: *> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
! 113: *> \endverbatim
! 114: *>
! 115: *> \param[in] IPIV
! 116: *> \verbatim
! 117: *> IPIV is INTEGER array, dimension (N)
! 118: *> The pivot indices from ZGBTRF; for 1<=i<=N, row i of the
! 119: *> matrix was interchanged with row IPIV(i).
! 120: *> \endverbatim
! 121: *>
! 122: *> \param[in] B
! 123: *> \verbatim
! 124: *> B is COMPLEX*16 array, dimension (LDB,NRHS)
! 125: *> The right hand side matrix B.
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[in] LDB
! 129: *> \verbatim
! 130: *> LDB is INTEGER
! 131: *> The leading dimension of the array B. LDB >= max(1,N).
! 132: *> \endverbatim
! 133: *>
! 134: *> \param[in,out] X
! 135: *> \verbatim
! 136: *> X is COMPLEX*16 array, dimension (LDX,NRHS)
! 137: *> On entry, the solution matrix X, as computed by ZGBTRS.
! 138: *> On exit, the improved solution matrix X.
! 139: *> \endverbatim
! 140: *>
! 141: *> \param[in] LDX
! 142: *> \verbatim
! 143: *> LDX is INTEGER
! 144: *> The leading dimension of the array X. LDX >= max(1,N).
! 145: *> \endverbatim
! 146: *>
! 147: *> \param[out] FERR
! 148: *> \verbatim
! 149: *> FERR is DOUBLE PRECISION array, dimension (NRHS)
! 150: *> The estimated forward error bound for each solution vector
! 151: *> X(j) (the j-th column of the solution matrix X).
! 152: *> If XTRUE is the true solution corresponding to X(j), FERR(j)
! 153: *> is an estimated upper bound for the magnitude of the largest
! 154: *> element in (X(j) - XTRUE) divided by the magnitude of the
! 155: *> largest element in X(j). The estimate is as reliable as
! 156: *> the estimate for RCOND, and is almost always a slight
! 157: *> overestimate of the true error.
! 158: *> \endverbatim
! 159: *>
! 160: *> \param[out] BERR
! 161: *> \verbatim
! 162: *> BERR is DOUBLE PRECISION array, dimension (NRHS)
! 163: *> The componentwise relative backward error of each solution
! 164: *> vector X(j) (i.e., the smallest relative change in
! 165: *> any element of A or B that makes X(j) an exact solution).
! 166: *> \endverbatim
! 167: *>
! 168: *> \param[out] WORK
! 169: *> \verbatim
! 170: *> WORK is COMPLEX*16 array, dimension (2*N)
! 171: *> \endverbatim
! 172: *>
! 173: *> \param[out] RWORK
! 174: *> \verbatim
! 175: *> RWORK is DOUBLE PRECISION array, dimension (N)
! 176: *> \endverbatim
! 177: *>
! 178: *> \param[out] INFO
! 179: *> \verbatim
! 180: *> INFO is INTEGER
! 181: *> = 0: successful exit
! 182: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 183: *> \endverbatim
! 184: *
! 185: *> \par Internal Parameters:
! 186: * =========================
! 187: *>
! 188: *> \verbatim
! 189: *> ITMAX is the maximum number of steps of iterative refinement.
! 190: *> \endverbatim
! 191: *
! 192: * Authors:
! 193: * ========
! 194: *
! 195: *> \author Univ. of Tennessee
! 196: *> \author Univ. of California Berkeley
! 197: *> \author Univ. of Colorado Denver
! 198: *> \author NAG Ltd.
! 199: *
! 200: *> \date November 2011
! 201: *
! 202: *> \ingroup complex16GBcomputational
! 203: *
! 204: * =====================================================================
1.1 bertrand 205: SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
206: $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
207: $ INFO )
208: *
1.8 ! bertrand 209: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 210: * -- LAPACK is a software package provided by Univ. of Tennessee, --
211: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 212: * November 2011
1.1 bertrand 213: *
214: * .. Scalar Arguments ..
215: CHARACTER TRANS
216: INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
217: * ..
218: * .. Array Arguments ..
219: INTEGER IPIV( * )
220: DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
221: COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
222: $ WORK( * ), X( LDX, * )
223: * ..
224: *
225: * =====================================================================
226: *
227: * .. Parameters ..
228: INTEGER ITMAX
229: PARAMETER ( ITMAX = 5 )
230: DOUBLE PRECISION ZERO
231: PARAMETER ( ZERO = 0.0D+0 )
232: COMPLEX*16 CONE
233: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
234: DOUBLE PRECISION TWO
235: PARAMETER ( TWO = 2.0D+0 )
236: DOUBLE PRECISION THREE
237: PARAMETER ( THREE = 3.0D+0 )
238: * ..
239: * .. Local Scalars ..
240: LOGICAL NOTRAN
241: CHARACTER TRANSN, TRANST
242: INTEGER COUNT, I, J, K, KASE, KK, NZ
243: DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
244: COMPLEX*16 ZDUM
245: * ..
246: * .. Local Arrays ..
247: INTEGER ISAVE( 3 )
248: * ..
249: * .. External Subroutines ..
250: EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGBMV, ZGBTRS, ZLACN2
251: * ..
252: * .. Intrinsic Functions ..
253: INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
254: * ..
255: * .. External Functions ..
256: LOGICAL LSAME
257: DOUBLE PRECISION DLAMCH
258: EXTERNAL LSAME, DLAMCH
259: * ..
260: * .. Statement Functions ..
261: DOUBLE PRECISION CABS1
262: * ..
263: * .. Statement Function definitions ..
264: CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
265: * ..
266: * .. Executable Statements ..
267: *
268: * Test the input parameters.
269: *
270: INFO = 0
271: NOTRAN = LSAME( TRANS, 'N' )
272: IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
273: $ LSAME( TRANS, 'C' ) ) THEN
274: INFO = -1
275: ELSE IF( N.LT.0 ) THEN
276: INFO = -2
277: ELSE IF( KL.LT.0 ) THEN
278: INFO = -3
279: ELSE IF( KU.LT.0 ) THEN
280: INFO = -4
281: ELSE IF( NRHS.LT.0 ) THEN
282: INFO = -5
283: ELSE IF( LDAB.LT.KL+KU+1 ) THEN
284: INFO = -7
285: ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
286: INFO = -9
287: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
288: INFO = -12
289: ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
290: INFO = -14
291: END IF
292: IF( INFO.NE.0 ) THEN
293: CALL XERBLA( 'ZGBRFS', -INFO )
294: RETURN
295: END IF
296: *
297: * Quick return if possible
298: *
299: IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
300: DO 10 J = 1, NRHS
301: FERR( J ) = ZERO
302: BERR( J ) = ZERO
303: 10 CONTINUE
304: RETURN
305: END IF
306: *
307: IF( NOTRAN ) THEN
308: TRANSN = 'N'
309: TRANST = 'C'
310: ELSE
311: TRANSN = 'C'
312: TRANST = 'N'
313: END IF
314: *
315: * NZ = maximum number of nonzero elements in each row of A, plus 1
316: *
317: NZ = MIN( KL+KU+2, N+1 )
318: EPS = DLAMCH( 'Epsilon' )
319: SAFMIN = DLAMCH( 'Safe minimum' )
320: SAFE1 = NZ*SAFMIN
321: SAFE2 = SAFE1 / EPS
322: *
323: * Do for each right hand side
324: *
325: DO 140 J = 1, NRHS
326: *
327: COUNT = 1
328: LSTRES = THREE
329: 20 CONTINUE
330: *
331: * Loop until stopping criterion is satisfied.
332: *
333: * Compute residual R = B - op(A) * X,
334: * where op(A) = A, A**T, or A**H, depending on TRANS.
335: *
336: CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
337: CALL ZGBMV( TRANS, N, N, KL, KU, -CONE, AB, LDAB, X( 1, J ), 1,
338: $ CONE, WORK, 1 )
339: *
340: * Compute componentwise relative backward error from formula
341: *
342: * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
343: *
344: * where abs(Z) is the componentwise absolute value of the matrix
345: * or vector Z. If the i-th component of the denominator is less
346: * than SAFE2, then SAFE1 is added to the i-th components of the
347: * numerator and denominator before dividing.
348: *
349: DO 30 I = 1, N
350: RWORK( I ) = CABS1( B( I, J ) )
351: 30 CONTINUE
352: *
353: * Compute abs(op(A))*abs(X) + abs(B).
354: *
355: IF( NOTRAN ) THEN
356: DO 50 K = 1, N
357: KK = KU + 1 - K
358: XK = CABS1( X( K, J ) )
359: DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
360: RWORK( I ) = RWORK( I ) + CABS1( AB( KK+I, K ) )*XK
361: 40 CONTINUE
362: 50 CONTINUE
363: ELSE
364: DO 70 K = 1, N
365: S = ZERO
366: KK = KU + 1 - K
367: DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
368: S = S + CABS1( AB( KK+I, K ) )*CABS1( X( I, J ) )
369: 60 CONTINUE
370: RWORK( K ) = RWORK( K ) + S
371: 70 CONTINUE
372: END IF
373: S = ZERO
374: DO 80 I = 1, N
375: IF( RWORK( I ).GT.SAFE2 ) THEN
376: S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
377: ELSE
378: S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
379: $ ( RWORK( I )+SAFE1 ) )
380: END IF
381: 80 CONTINUE
382: BERR( J ) = S
383: *
384: * Test stopping criterion. Continue iterating if
385: * 1) The residual BERR(J) is larger than machine epsilon, and
386: * 2) BERR(J) decreased by at least a factor of 2 during the
387: * last iteration, and
388: * 3) At most ITMAX iterations tried.
389: *
390: IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
391: $ COUNT.LE.ITMAX ) THEN
392: *
393: * Update solution and try again.
394: *
395: CALL ZGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV, WORK, N,
396: $ INFO )
397: CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
398: LSTRES = BERR( J )
399: COUNT = COUNT + 1
400: GO TO 20
401: END IF
402: *
403: * Bound error from formula
404: *
405: * norm(X - XTRUE) / norm(X) .le. FERR =
406: * norm( abs(inv(op(A)))*
407: * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
408: *
409: * where
410: * norm(Z) is the magnitude of the largest component of Z
411: * inv(op(A)) is the inverse of op(A)
412: * abs(Z) is the componentwise absolute value of the matrix or
413: * vector Z
414: * NZ is the maximum number of nonzeros in any row of A, plus 1
415: * EPS is machine epsilon
416: *
417: * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
418: * is incremented by SAFE1 if the i-th component of
419: * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
420: *
421: * Use ZLACN2 to estimate the infinity-norm of the matrix
422: * inv(op(A)) * diag(W),
423: * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
424: *
425: DO 90 I = 1, N
426: IF( RWORK( I ).GT.SAFE2 ) THEN
427: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
428: ELSE
429: RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
430: $ SAFE1
431: END IF
432: 90 CONTINUE
433: *
434: KASE = 0
435: 100 CONTINUE
436: CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
437: IF( KASE.NE.0 ) THEN
438: IF( KASE.EQ.1 ) THEN
439: *
440: * Multiply by diag(W)*inv(op(A)**H).
441: *
442: CALL ZGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
443: $ WORK, N, INFO )
444: DO 110 I = 1, N
445: WORK( I ) = RWORK( I )*WORK( I )
446: 110 CONTINUE
447: ELSE
448: *
449: * Multiply by inv(op(A))*diag(W).
450: *
451: DO 120 I = 1, N
452: WORK( I ) = RWORK( I )*WORK( I )
453: 120 CONTINUE
454: CALL ZGBTRS( TRANSN, N, KL, KU, 1, AFB, LDAFB, IPIV,
455: $ WORK, N, INFO )
456: END IF
457: GO TO 100
458: END IF
459: *
460: * Normalize error.
461: *
462: LSTRES = ZERO
463: DO 130 I = 1, N
464: LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
465: 130 CONTINUE
466: IF( LSTRES.NE.ZERO )
467: $ FERR( J ) = FERR( J ) / LSTRES
468: *
469: 140 CONTINUE
470: *
471: RETURN
472: *
473: * End of ZGBRFS
474: *
475: END
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