LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dgbrfs.f
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00001 *> \brief \b DGBRFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DGBRFS + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
00022 *                          IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
00023 *                          INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          TRANS
00027 *       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
00028 *       ..
00029 *       .. Array Arguments ..
00030 *       INTEGER            IPIV( * ), IWORK( * )
00031 *       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00032 *      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00033 *       ..
00034 *  
00035 *
00036 *> \par Purpose:
00037 *  =============
00038 *>
00039 *> \verbatim
00040 *>
00041 *> DGBRFS improves the computed solution to a system of linear
00042 *> equations when the coefficient matrix is banded, and provides
00043 *> error bounds and backward error estimates for the solution.
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] TRANS
00050 *> \verbatim
00051 *>          TRANS is CHARACTER*1
00052 *>          Specifies the form of the system of equations:
00053 *>          = 'N':  A * X = B     (No transpose)
00054 *>          = 'T':  A**T * X = B  (Transpose)
00055 *>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
00056 *> \endverbatim
00057 *>
00058 *> \param[in] N
00059 *> \verbatim
00060 *>          N is INTEGER
00061 *>          The order of the matrix A.  N >= 0.
00062 *> \endverbatim
00063 *>
00064 *> \param[in] KL
00065 *> \verbatim
00066 *>          KL is INTEGER
00067 *>          The number of subdiagonals within the band of A.  KL >= 0.
00068 *> \endverbatim
00069 *>
00070 *> \param[in] KU
00071 *> \verbatim
00072 *>          KU is INTEGER
00073 *>          The number of superdiagonals within the band of A.  KU >= 0.
00074 *> \endverbatim
00075 *>
00076 *> \param[in] NRHS
00077 *> \verbatim
00078 *>          NRHS is INTEGER
00079 *>          The number of right hand sides, i.e., the number of columns
00080 *>          of the matrices B and X.  NRHS >= 0.
00081 *> \endverbatim
00082 *>
00083 *> \param[in] AB
00084 *> \verbatim
00085 *>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
00086 *>          The original band matrix A, stored in rows 1 to KL+KU+1.
00087 *>          The j-th column of A is stored in the j-th column of the
00088 *>          array AB as follows:
00089 *>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
00090 *> \endverbatim
00091 *>
00092 *> \param[in] LDAB
00093 *> \verbatim
00094 *>          LDAB is INTEGER
00095 *>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
00096 *> \endverbatim
00097 *>
00098 *> \param[in] AFB
00099 *> \verbatim
00100 *>          AFB is DOUBLE PRECISION array, dimension (LDAFB,N)
00101 *>          Details of the LU factorization of the band matrix A, as
00102 *>          computed by DGBTRF.  U is stored as an upper triangular band
00103 *>          matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
00104 *>          the multipliers used during the factorization are stored in
00105 *>          rows KL+KU+2 to 2*KL+KU+1.
00106 *> \endverbatim
00107 *>
00108 *> \param[in] LDAFB
00109 *> \verbatim
00110 *>          LDAFB is INTEGER
00111 *>          The leading dimension of the array AFB.  LDAFB >= 2*KL*KU+1.
00112 *> \endverbatim
00113 *>
00114 *> \param[in] IPIV
00115 *> \verbatim
00116 *>          IPIV is INTEGER array, dimension (N)
00117 *>          The pivot indices from DGBTRF; for 1<=i<=N, row i of the
00118 *>          matrix was interchanged with row IPIV(i).
00119 *> \endverbatim
00120 *>
00121 *> \param[in] B
00122 *> \verbatim
00123 *>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
00124 *>          The right hand side matrix B.
00125 *> \endverbatim
00126 *>
00127 *> \param[in] LDB
00128 *> \verbatim
00129 *>          LDB is INTEGER
00130 *>          The leading dimension of the array B.  LDB >= max(1,N).
00131 *> \endverbatim
00132 *>
00133 *> \param[in,out] X
00134 *> \verbatim
00135 *>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
00136 *>          On entry, the solution matrix X, as computed by DGBTRS.
00137 *>          On exit, the improved solution matrix X.
00138 *> \endverbatim
00139 *>
00140 *> \param[in] LDX
00141 *> \verbatim
00142 *>          LDX is INTEGER
00143 *>          The leading dimension of the array X.  LDX >= max(1,N).
00144 *> \endverbatim
00145 *>
00146 *> \param[out] FERR
00147 *> \verbatim
00148 *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
00149 *>          The estimated forward error bound for each solution vector
00150 *>          X(j) (the j-th column of the solution matrix X).
00151 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00152 *>          is an estimated upper bound for the magnitude of the largest
00153 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00154 *>          largest element in X(j).  The estimate is as reliable as
00155 *>          the estimate for RCOND, and is almost always a slight
00156 *>          overestimate of the true error.
00157 *> \endverbatim
00158 *>
00159 *> \param[out] BERR
00160 *> \verbatim
00161 *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
00162 *>          The componentwise relative backward error of each solution
00163 *>          vector X(j) (i.e., the smallest relative change in
00164 *>          any element of A or B that makes X(j) an exact solution).
00165 *> \endverbatim
00166 *>
00167 *> \param[out] WORK
00168 *> \verbatim
00169 *>          WORK is DOUBLE PRECISION array, dimension (3*N)
00170 *> \endverbatim
00171 *>
00172 *> \param[out] IWORK
00173 *> \verbatim
00174 *>          IWORK is INTEGER array, dimension (N)
00175 *> \endverbatim
00176 *>
00177 *> \param[out] INFO
00178 *> \verbatim
00179 *>          INFO is INTEGER
00180 *>          = 0:  successful exit
00181 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00182 *> \endverbatim
00183 *
00184 *> \par Internal Parameters:
00185 *  =========================
00186 *>
00187 *> \verbatim
00188 *>  ITMAX is the maximum number of steps of iterative refinement.
00189 *> \endverbatim
00190 *
00191 *  Authors:
00192 *  ========
00193 *
00194 *> \author Univ. of Tennessee 
00195 *> \author Univ. of California Berkeley 
00196 *> \author Univ. of Colorado Denver 
00197 *> \author NAG Ltd. 
00198 *
00199 *> \date November 2011
00200 *
00201 *> \ingroup doubleGBcomputational
00202 *
00203 *  =====================================================================
00204       SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
00205      $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
00206      $                   INFO )
00207 *
00208 *  -- LAPACK computational routine (version 3.4.0) --
00209 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00210 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00211 *     November 2011
00212 *
00213 *     .. Scalar Arguments ..
00214       CHARACTER          TRANS
00215       INTEGER            INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
00216 *     ..
00217 *     .. Array Arguments ..
00218       INTEGER            IPIV( * ), IWORK( * )
00219       DOUBLE PRECISION   AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00220      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00221 *     ..
00222 *
00223 *  =====================================================================
00224 *
00225 *     .. Parameters ..
00226       INTEGER            ITMAX
00227       PARAMETER          ( ITMAX = 5 )
00228       DOUBLE PRECISION   ZERO
00229       PARAMETER          ( ZERO = 0.0D+0 )
00230       DOUBLE PRECISION   ONE
00231       PARAMETER          ( ONE = 1.0D+0 )
00232       DOUBLE PRECISION   TWO
00233       PARAMETER          ( TWO = 2.0D+0 )
00234       DOUBLE PRECISION   THREE
00235       PARAMETER          ( THREE = 3.0D+0 )
00236 *     ..
00237 *     .. Local Scalars ..
00238       LOGICAL            NOTRAN
00239       CHARACTER          TRANST
00240       INTEGER            COUNT, I, J, K, KASE, KK, NZ
00241       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00242 *     ..
00243 *     .. Local Arrays ..
00244       INTEGER            ISAVE( 3 )
00245 *     ..
00246 *     .. External Subroutines ..
00247       EXTERNAL           DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2, XERBLA
00248 *     ..
00249 *     .. Intrinsic Functions ..
00250       INTRINSIC          ABS, MAX, MIN
00251 *     ..
00252 *     .. External Functions ..
00253       LOGICAL            LSAME
00254       DOUBLE PRECISION   DLAMCH
00255       EXTERNAL           LSAME, DLAMCH
00256 *     ..
00257 *     .. Executable Statements ..
00258 *
00259 *     Test the input parameters.
00260 *
00261       INFO = 0
00262       NOTRAN = LSAME( TRANS, 'N' )
00263       IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
00264      $    LSAME( TRANS, 'C' ) ) THEN
00265          INFO = -1
00266       ELSE IF( N.LT.0 ) THEN
00267          INFO = -2
00268       ELSE IF( KL.LT.0 ) THEN
00269          INFO = -3
00270       ELSE IF( KU.LT.0 ) THEN
00271          INFO = -4
00272       ELSE IF( NRHS.LT.0 ) THEN
00273          INFO = -5
00274       ELSE IF( LDAB.LT.KL+KU+1 ) THEN
00275          INFO = -7
00276       ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
00277          INFO = -9
00278       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00279          INFO = -12
00280       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00281          INFO = -14
00282       END IF
00283       IF( INFO.NE.0 ) THEN
00284          CALL XERBLA( 'DGBRFS', -INFO )
00285          RETURN
00286       END IF
00287 *
00288 *     Quick return if possible
00289 *
00290       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00291          DO 10 J = 1, NRHS
00292             FERR( J ) = ZERO
00293             BERR( J ) = ZERO
00294    10    CONTINUE
00295          RETURN
00296       END IF
00297 *
00298       IF( NOTRAN ) THEN
00299          TRANST = 'T'
00300       ELSE
00301          TRANST = 'N'
00302       END IF
00303 *
00304 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00305 *
00306       NZ = MIN( KL+KU+2, N+1 )
00307       EPS = DLAMCH( 'Epsilon' )
00308       SAFMIN = DLAMCH( 'Safe minimum' )
00309       SAFE1 = NZ*SAFMIN
00310       SAFE2 = SAFE1 / EPS
00311 *
00312 *     Do for each right hand side
00313 *
00314       DO 140 J = 1, NRHS
00315 *
00316          COUNT = 1
00317          LSTRES = THREE
00318    20    CONTINUE
00319 *
00320 *        Loop until stopping criterion is satisfied.
00321 *
00322 *        Compute residual R = B - op(A) * X,
00323 *        where op(A) = A, A**T, or A**H, depending on TRANS.
00324 *
00325          CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
00326          CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1,
00327      $               ONE, WORK( N+1 ), 1 )
00328 *
00329 *        Compute componentwise relative backward error from formula
00330 *
00331 *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
00332 *
00333 *        where abs(Z) is the componentwise absolute value of the matrix
00334 *        or vector Z.  If the i-th component of the denominator is less
00335 *        than SAFE2, then SAFE1 is added to the i-th components of the
00336 *        numerator and denominator before dividing.
00337 *
00338          DO 30 I = 1, N
00339             WORK( I ) = ABS( B( I, J ) )
00340    30    CONTINUE
00341 *
00342 *        Compute abs(op(A))*abs(X) + abs(B).
00343 *
00344          IF( NOTRAN ) THEN
00345             DO 50 K = 1, N
00346                KK = KU + 1 - K
00347                XK = ABS( X( K, J ) )
00348                DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
00349                   WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
00350    40          CONTINUE
00351    50       CONTINUE
00352          ELSE
00353             DO 70 K = 1, N
00354                S = ZERO
00355                KK = KU + 1 - K
00356                DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
00357                   S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
00358    60          CONTINUE
00359                WORK( K ) = WORK( K ) + S
00360    70       CONTINUE
00361          END IF
00362          S = ZERO
00363          DO 80 I = 1, N
00364             IF( WORK( I ).GT.SAFE2 ) THEN
00365                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
00366             ELSE
00367                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
00368      $             ( WORK( I )+SAFE1 ) )
00369             END IF
00370    80    CONTINUE
00371          BERR( J ) = S
00372 *
00373 *        Test stopping criterion. Continue iterating if
00374 *           1) The residual BERR(J) is larger than machine epsilon, and
00375 *           2) BERR(J) decreased by at least a factor of 2 during the
00376 *              last iteration, and
00377 *           3) At most ITMAX iterations tried.
00378 *
00379          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00380      $       COUNT.LE.ITMAX ) THEN
00381 *
00382 *           Update solution and try again.
00383 *
00384             CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
00385      $                   WORK( N+1 ), N, INFO )
00386             CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
00387             LSTRES = BERR( J )
00388             COUNT = COUNT + 1
00389             GO TO 20
00390          END IF
00391 *
00392 *        Bound error from formula
00393 *
00394 *        norm(X - XTRUE) / norm(X) .le. FERR =
00395 *        norm( abs(inv(op(A)))*
00396 *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
00397 *
00398 *        where
00399 *          norm(Z) is the magnitude of the largest component of Z
00400 *          inv(op(A)) is the inverse of op(A)
00401 *          abs(Z) is the componentwise absolute value of the matrix or
00402 *             vector Z
00403 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00404 *          EPS is machine epsilon
00405 *
00406 *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
00407 *        is incremented by SAFE1 if the i-th component of
00408 *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
00409 *
00410 *        Use DLACN2 to estimate the infinity-norm of the matrix
00411 *           inv(op(A)) * diag(W),
00412 *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
00413 *
00414          DO 90 I = 1, N
00415             IF( WORK( I ).GT.SAFE2 ) THEN
00416                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
00417             ELSE
00418                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
00419             END IF
00420    90    CONTINUE
00421 *
00422          KASE = 0
00423   100    CONTINUE
00424          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
00425      $                KASE, ISAVE )
00426          IF( KASE.NE.0 ) THEN
00427             IF( KASE.EQ.1 ) THEN
00428 *
00429 *              Multiply by diag(W)*inv(op(A)**T).
00430 *
00431                CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
00432      $                      WORK( N+1 ), N, INFO )
00433                DO 110 I = 1, N
00434                   WORK( N+I ) = WORK( N+I )*WORK( I )
00435   110          CONTINUE
00436             ELSE
00437 *
00438 *              Multiply by inv(op(A))*diag(W).
00439 *
00440                DO 120 I = 1, N
00441                   WORK( N+I ) = WORK( N+I )*WORK( I )
00442   120          CONTINUE
00443                CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
00444      $                      WORK( N+1 ), N, INFO )
00445             END IF
00446             GO TO 100
00447          END IF
00448 *
00449 *        Normalize error.
00450 *
00451          LSTRES = ZERO
00452          DO 130 I = 1, N
00453             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
00454   130    CONTINUE
00455          IF( LSTRES.NE.ZERO )
00456      $      FERR( J ) = FERR( J ) / LSTRES
00457 *
00458   140 CONTINUE
00459 *
00460       RETURN
00461 *
00462 *     End of DGBRFS
00463 *
00464       END
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