LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cgtrfs.f
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00001 *> \brief \b CGTRFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CGTRFS + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgtrfs.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
00022 *                          IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
00023 *                          INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          TRANS
00027 *       INTEGER            INFO, LDB, LDX, N, NRHS
00028 *       ..
00029 *       .. Array Arguments ..
00030 *       INTEGER            IPIV( * )
00031 *       REAL               BERR( * ), FERR( * ), RWORK( * )
00032 *       COMPLEX            B( LDB, * ), D( * ), DF( * ), DL( * ),
00033 *      $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
00034 *      $                   WORK( * ), X( LDX, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> CGTRFS improves the computed solution to a system of linear
00044 *> equations when the coefficient matrix is tridiagonal, and provides
00045 *> error bounds and backward error estimates for the solution.
00046 *> \endverbatim
00047 *
00048 *  Arguments:
00049 *  ==========
00050 *
00051 *> \param[in] TRANS
00052 *> \verbatim
00053 *>          TRANS is CHARACTER*1
00054 *>          Specifies the form of the system of equations:
00055 *>          = 'N':  A * X = B     (No transpose)
00056 *>          = 'T':  A**T * X = B  (Transpose)
00057 *>          = 'C':  A**H * X = B  (Conjugate transpose)
00058 *> \endverbatim
00059 *>
00060 *> \param[in] N
00061 *> \verbatim
00062 *>          N is INTEGER
00063 *>          The order of the matrix A.  N >= 0.
00064 *> \endverbatim
00065 *>
00066 *> \param[in] NRHS
00067 *> \verbatim
00068 *>          NRHS is INTEGER
00069 *>          The number of right hand sides, i.e., the number of columns
00070 *>          of the matrix B.  NRHS >= 0.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] DL
00074 *> \verbatim
00075 *>          DL is COMPLEX array, dimension (N-1)
00076 *>          The (n-1) subdiagonal elements of A.
00077 *> \endverbatim
00078 *>
00079 *> \param[in] D
00080 *> \verbatim
00081 *>          D is COMPLEX array, dimension (N)
00082 *>          The diagonal elements of A.
00083 *> \endverbatim
00084 *>
00085 *> \param[in] DU
00086 *> \verbatim
00087 *>          DU is COMPLEX array, dimension (N-1)
00088 *>          The (n-1) superdiagonal elements of A.
00089 *> \endverbatim
00090 *>
00091 *> \param[in] DLF
00092 *> \verbatim
00093 *>          DLF is COMPLEX array, dimension (N-1)
00094 *>          The (n-1) multipliers that define the matrix L from the
00095 *>          LU factorization of A as computed by CGTTRF.
00096 *> \endverbatim
00097 *>
00098 *> \param[in] DF
00099 *> \verbatim
00100 *>          DF is COMPLEX array, dimension (N)
00101 *>          The n diagonal elements of the upper triangular matrix U from
00102 *>          the LU factorization of A.
00103 *> \endverbatim
00104 *>
00105 *> \param[in] DUF
00106 *> \verbatim
00107 *>          DUF is COMPLEX array, dimension (N-1)
00108 *>          The (n-1) elements of the first superdiagonal of U.
00109 *> \endverbatim
00110 *>
00111 *> \param[in] DU2
00112 *> \verbatim
00113 *>          DU2 is COMPLEX array, dimension (N-2)
00114 *>          The (n-2) elements of the second superdiagonal of U.
00115 *> \endverbatim
00116 *>
00117 *> \param[in] IPIV
00118 *> \verbatim
00119 *>          IPIV is INTEGER array, dimension (N)
00120 *>          The pivot indices; for 1 <= i <= n, row i of the matrix was
00121 *>          interchanged with row IPIV(i).  IPIV(i) will always be either
00122 *>          i or i+1; IPIV(i) = i indicates a row interchange was not
00123 *>          required.
00124 *> \endverbatim
00125 *>
00126 *> \param[in] B
00127 *> \verbatim
00128 *>          B is COMPLEX array, dimension (LDB,NRHS)
00129 *>          The right hand side matrix B.
00130 *> \endverbatim
00131 *>
00132 *> \param[in] LDB
00133 *> \verbatim
00134 *>          LDB is INTEGER
00135 *>          The leading dimension of the array B.  LDB >= max(1,N).
00136 *> \endverbatim
00137 *>
00138 *> \param[in,out] X
00139 *> \verbatim
00140 *>          X is COMPLEX array, dimension (LDX,NRHS)
00141 *>          On entry, the solution matrix X, as computed by CGTTRS.
00142 *>          On exit, the improved solution matrix X.
00143 *> \endverbatim
00144 *>
00145 *> \param[in] LDX
00146 *> \verbatim
00147 *>          LDX is INTEGER
00148 *>          The leading dimension of the array X.  LDX >= max(1,N).
00149 *> \endverbatim
00150 *>
00151 *> \param[out] FERR
00152 *> \verbatim
00153 *>          FERR is REAL array, dimension (NRHS)
00154 *>          The estimated forward error bound for each solution vector
00155 *>          X(j) (the j-th column of the solution matrix X).
00156 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00157 *>          is an estimated upper bound for the magnitude of the largest
00158 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00159 *>          largest element in X(j).  The estimate is as reliable as
00160 *>          the estimate for RCOND, and is almost always a slight
00161 *>          overestimate of the true error.
00162 *> \endverbatim
00163 *>
00164 *> \param[out] BERR
00165 *> \verbatim
00166 *>          BERR is REAL array, dimension (NRHS)
00167 *>          The componentwise relative backward error of each solution
00168 *>          vector X(j) (i.e., the smallest relative change in
00169 *>          any element of A or B that makes X(j) an exact solution).
00170 *> \endverbatim
00171 *>
00172 *> \param[out] WORK
00173 *> \verbatim
00174 *>          WORK is COMPLEX array, dimension (2*N)
00175 *> \endverbatim
00176 *>
00177 *> \param[out] RWORK
00178 *> \verbatim
00179 *>          RWORK is REAL array, dimension (N)
00180 *> \endverbatim
00181 *>
00182 *> \param[out] INFO
00183 *> \verbatim
00184 *>          INFO is INTEGER
00185 *>          = 0:  successful exit
00186 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00187 *> \endverbatim
00188 *
00189 *> \par Internal Parameters:
00190 *  =========================
00191 *>
00192 *> \verbatim
00193 *>  ITMAX is the maximum number of steps of iterative refinement.
00194 *> \endverbatim
00195 *
00196 *  Authors:
00197 *  ========
00198 *
00199 *> \author Univ. of Tennessee 
00200 *> \author Univ. of California Berkeley 
00201 *> \author Univ. of Colorado Denver 
00202 *> \author NAG Ltd. 
00203 *
00204 *> \date November 2011
00205 *
00206 *> \ingroup complexOTHERcomputational
00207 *
00208 *  =====================================================================
00209       SUBROUTINE CGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
00210      $                   IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
00211      $                   INFO )
00212 *
00213 *  -- LAPACK computational routine (version 3.4.0) --
00214 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00215 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00216 *     November 2011
00217 *
00218 *     .. Scalar Arguments ..
00219       CHARACTER          TRANS
00220       INTEGER            INFO, LDB, LDX, N, NRHS
00221 *     ..
00222 *     .. Array Arguments ..
00223       INTEGER            IPIV( * )
00224       REAL               BERR( * ), FERR( * ), RWORK( * )
00225       COMPLEX            B( LDB, * ), D( * ), DF( * ), DL( * ),
00226      $                   DLF( * ), DU( * ), DU2( * ), DUF( * ),
00227      $                   WORK( * ), X( LDX, * )
00228 *     ..
00229 *
00230 *  =====================================================================
00231 *
00232 *     .. Parameters ..
00233       INTEGER            ITMAX
00234       PARAMETER          ( ITMAX = 5 )
00235       REAL               ZERO, ONE
00236       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00237       REAL               TWO
00238       PARAMETER          ( TWO = 2.0E+0 )
00239       REAL               THREE
00240       PARAMETER          ( THREE = 3.0E+0 )
00241 *     ..
00242 *     .. Local Scalars ..
00243       LOGICAL            NOTRAN
00244       CHARACTER          TRANSN, TRANST
00245       INTEGER            COUNT, I, J, KASE, NZ
00246       REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
00247       COMPLEX            ZDUM
00248 *     ..
00249 *     .. Local Arrays ..
00250       INTEGER            ISAVE( 3 )
00251 *     ..
00252 *     .. External Subroutines ..
00253       EXTERNAL           CAXPY, CCOPY, CGTTRS, CLACN2, CLAGTM, XERBLA
00254 *     ..
00255 *     .. Intrinsic Functions ..
00256       INTRINSIC          ABS, AIMAG, CMPLX, MAX, REAL
00257 *     ..
00258 *     .. External Functions ..
00259       LOGICAL            LSAME
00260       REAL               SLAMCH
00261       EXTERNAL           LSAME, SLAMCH
00262 *     ..
00263 *     .. Statement Functions ..
00264       REAL               CABS1
00265 *     ..
00266 *     .. Statement Function definitions ..
00267       CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
00268 *     ..
00269 *     .. Executable Statements ..
00270 *
00271 *     Test the input parameters.
00272 *
00273       INFO = 0
00274       NOTRAN = LSAME( TRANS, 'N' )
00275       IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
00276      $    LSAME( TRANS, 'C' ) ) THEN
00277          INFO = -1
00278       ELSE IF( N.LT.0 ) THEN
00279          INFO = -2
00280       ELSE IF( NRHS.LT.0 ) THEN
00281          INFO = -3
00282       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00283          INFO = -13
00284       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00285          INFO = -15
00286       END IF
00287       IF( INFO.NE.0 ) THEN
00288          CALL XERBLA( 'CGTRFS', -INFO )
00289          RETURN
00290       END IF
00291 *
00292 *     Quick return if possible
00293 *
00294       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00295          DO 10 J = 1, NRHS
00296             FERR( J ) = ZERO
00297             BERR( J ) = ZERO
00298    10    CONTINUE
00299          RETURN
00300       END IF
00301 *
00302       IF( NOTRAN ) THEN
00303          TRANSN = 'N'
00304          TRANST = 'C'
00305       ELSE
00306          TRANSN = 'C'
00307          TRANST = 'N'
00308       END IF
00309 *
00310 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00311 *
00312       NZ = 4
00313       EPS = SLAMCH( 'Epsilon' )
00314       SAFMIN = SLAMCH( 'Safe minimum' )
00315       SAFE1 = NZ*SAFMIN
00316       SAFE2 = SAFE1 / EPS
00317 *
00318 *     Do for each right hand side
00319 *
00320       DO 110 J = 1, NRHS
00321 *
00322          COUNT = 1
00323          LSTRES = THREE
00324    20    CONTINUE
00325 *
00326 *        Loop until stopping criterion is satisfied.
00327 *
00328 *        Compute residual R = B - op(A) * X,
00329 *        where op(A) = A, A**T, or A**H, depending on TRANS.
00330 *
00331          CALL CCOPY( N, B( 1, J ), 1, WORK, 1 )
00332          CALL CLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
00333      $                WORK, N )
00334 *
00335 *        Compute abs(op(A))*abs(x) + abs(b) for use in the backward
00336 *        error bound.
00337 *
00338          IF( NOTRAN ) THEN
00339             IF( N.EQ.1 ) THEN
00340                RWORK( 1 ) = CABS1( B( 1, J ) ) +
00341      $                      CABS1( D( 1 ) )*CABS1( X( 1, J ) )
00342             ELSE
00343                RWORK( 1 ) = CABS1( B( 1, J ) ) +
00344      $                      CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
00345      $                      CABS1( DU( 1 ) )*CABS1( X( 2, J ) )
00346                DO 30 I = 2, N - 1
00347                   RWORK( I ) = CABS1( B( I, J ) ) +
00348      $                         CABS1( DL( I-1 ) )*CABS1( X( I-1, J ) ) +
00349      $                         CABS1( D( I ) )*CABS1( X( I, J ) ) +
00350      $                         CABS1( DU( I ) )*CABS1( X( I+1, J ) )
00351    30          CONTINUE
00352                RWORK( N ) = CABS1( B( N, J ) ) +
00353      $                      CABS1( DL( N-1 ) )*CABS1( X( N-1, J ) ) +
00354      $                      CABS1( D( N ) )*CABS1( X( N, J ) )
00355             END IF
00356          ELSE
00357             IF( N.EQ.1 ) THEN
00358                RWORK( 1 ) = CABS1( B( 1, J ) ) +
00359      $                      CABS1( D( 1 ) )*CABS1( X( 1, J ) )
00360             ELSE
00361                RWORK( 1 ) = CABS1( B( 1, J ) ) +
00362      $                      CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
00363      $                      CABS1( DL( 1 ) )*CABS1( X( 2, J ) )
00364                DO 40 I = 2, N - 1
00365                   RWORK( I ) = CABS1( B( I, J ) ) +
00366      $                         CABS1( DU( I-1 ) )*CABS1( X( I-1, J ) ) +
00367      $                         CABS1( D( I ) )*CABS1( X( I, J ) ) +
00368      $                         CABS1( DL( I ) )*CABS1( X( I+1, J ) )
00369    40          CONTINUE
00370                RWORK( N ) = CABS1( B( N, J ) ) +
00371      $                      CABS1( DU( N-1 ) )*CABS1( X( N-1, J ) ) +
00372      $                      CABS1( D( N ) )*CABS1( X( N, J ) )
00373             END IF
00374          END IF
00375 *
00376 *        Compute componentwise relative backward error from formula
00377 *
00378 *        max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
00379 *
00380 *        where abs(Z) is the componentwise absolute value of the matrix
00381 *        or vector Z.  If the i-th component of the denominator is less
00382 *        than SAFE2, then SAFE1 is added to the i-th components of the
00383 *        numerator and denominator before dividing.
00384 *
00385          S = ZERO
00386          DO 50 I = 1, N
00387             IF( RWORK( I ).GT.SAFE2 ) THEN
00388                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
00389             ELSE
00390                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
00391      $             ( RWORK( I )+SAFE1 ) )
00392             END IF
00393    50    CONTINUE
00394          BERR( J ) = S
00395 *
00396 *        Test stopping criterion. Continue iterating if
00397 *           1) The residual BERR(J) is larger than machine epsilon, and
00398 *           2) BERR(J) decreased by at least a factor of 2 during the
00399 *              last iteration, and
00400 *           3) At most ITMAX iterations tried.
00401 *
00402          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00403      $       COUNT.LE.ITMAX ) THEN
00404 *
00405 *           Update solution and try again.
00406 *
00407             CALL CGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, WORK, N,
00408      $                   INFO )
00409             CALL CAXPY( N, CMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
00410             LSTRES = BERR( J )
00411             COUNT = COUNT + 1
00412             GO TO 20
00413          END IF
00414 *
00415 *        Bound error from formula
00416 *
00417 *        norm(X - XTRUE) / norm(X) .le. FERR =
00418 *        norm( abs(inv(op(A)))*
00419 *           ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
00420 *
00421 *        where
00422 *          norm(Z) is the magnitude of the largest component of Z
00423 *          inv(op(A)) is the inverse of op(A)
00424 *          abs(Z) is the componentwise absolute value of the matrix or
00425 *             vector Z
00426 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00427 *          EPS is machine epsilon
00428 *
00429 *        The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
00430 *        is incremented by SAFE1 if the i-th component of
00431 *        abs(op(A))*abs(X) + abs(B) is less than SAFE2.
00432 *
00433 *        Use CLACN2 to estimate the infinity-norm of the matrix
00434 *           inv(op(A)) * diag(W),
00435 *        where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
00436 *
00437          DO 60 I = 1, N
00438             IF( RWORK( I ).GT.SAFE2 ) THEN
00439                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
00440             ELSE
00441                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
00442      $                      SAFE1
00443             END IF
00444    60    CONTINUE
00445 *
00446          KASE = 0
00447    70    CONTINUE
00448          CALL CLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
00449          IF( KASE.NE.0 ) THEN
00450             IF( KASE.EQ.1 ) THEN
00451 *
00452 *              Multiply by diag(W)*inv(op(A)**H).
00453 *
00454                CALL CGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
00455      $                      N, INFO )
00456                DO 80 I = 1, N
00457                   WORK( I ) = RWORK( I )*WORK( I )
00458    80          CONTINUE
00459             ELSE
00460 *
00461 *              Multiply by inv(op(A))*diag(W).
00462 *
00463                DO 90 I = 1, N
00464                   WORK( I ) = RWORK( I )*WORK( I )
00465    90          CONTINUE
00466                CALL CGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
00467      $                      N, INFO )
00468             END IF
00469             GO TO 70
00470          END IF
00471 *
00472 *        Normalize error.
00473 *
00474          LSTRES = ZERO
00475          DO 100 I = 1, N
00476             LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
00477   100    CONTINUE
00478          IF( LSTRES.NE.ZERO )
00479      $      FERR( J ) = FERR( J ) / LSTRES
00480 *
00481   110 CONTINUE
00482 *
00483       RETURN
00484 *
00485 *     End of CGTRFS
00486 *
00487       END
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