LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cptrfs.f
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00001 *> \brief \b CPTRFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CPTRFS + 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 CPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
00022 *                          FERR, BERR, WORK, RWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          UPLO
00026 *       INTEGER            INFO, LDB, LDX, N, NRHS
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       REAL               BERR( * ), D( * ), DF( * ), FERR( * ),
00030 *      $                   RWORK( * )
00031 *       COMPLEX            B( LDB, * ), E( * ), EF( * ), WORK( * ),
00032 *      $                   X( LDX, * )
00033 *       ..
00034 *  
00035 *
00036 *> \par Purpose:
00037 *  =============
00038 *>
00039 *> \verbatim
00040 *>
00041 *> CPTRFS improves the computed solution to a system of linear
00042 *> equations when the coefficient matrix is Hermitian positive definite
00043 *> and tridiagonal, and provides error bounds and backward error
00044 *> estimates for the solution.
00045 *> \endverbatim
00046 *
00047 *  Arguments:
00048 *  ==========
00049 *
00050 *> \param[in] UPLO
00051 *> \verbatim
00052 *>          UPLO is CHARACTER*1
00053 *>          Specifies whether the superdiagonal or the subdiagonal of the
00054 *>          tridiagonal matrix A is stored and the form of the
00055 *>          factorization:
00056 *>          = 'U':  E is the superdiagonal of A, and A = U**H*D*U;
00057 *>          = 'L':  E is the subdiagonal of A, and A = L*D*L**H.
00058 *>          (The two forms are equivalent if A is real.)
00059 *> \endverbatim
00060 *>
00061 *> \param[in] N
00062 *> \verbatim
00063 *>          N is INTEGER
00064 *>          The order of the matrix A.  N >= 0.
00065 *> \endverbatim
00066 *>
00067 *> \param[in] NRHS
00068 *> \verbatim
00069 *>          NRHS is INTEGER
00070 *>          The number of right hand sides, i.e., the number of columns
00071 *>          of the matrix B.  NRHS >= 0.
00072 *> \endverbatim
00073 *>
00074 *> \param[in] D
00075 *> \verbatim
00076 *>          D is REAL array, dimension (N)
00077 *>          The n real diagonal elements of the tridiagonal matrix A.
00078 *> \endverbatim
00079 *>
00080 *> \param[in] E
00081 *> \verbatim
00082 *>          E is COMPLEX array, dimension (N-1)
00083 *>          The (n-1) off-diagonal elements of the tridiagonal matrix A
00084 *>          (see UPLO).
00085 *> \endverbatim
00086 *>
00087 *> \param[in] DF
00088 *> \verbatim
00089 *>          DF is REAL array, dimension (N)
00090 *>          The n diagonal elements of the diagonal matrix D from
00091 *>          the factorization computed by CPTTRF.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] EF
00095 *> \verbatim
00096 *>          EF is COMPLEX array, dimension (N-1)
00097 *>          The (n-1) off-diagonal elements of the unit bidiagonal
00098 *>          factor U or L from the factorization computed by CPTTRF
00099 *>          (see UPLO).
00100 *> \endverbatim
00101 *>
00102 *> \param[in] B
00103 *> \verbatim
00104 *>          B is COMPLEX array, dimension (LDB,NRHS)
00105 *>          The right hand side matrix B.
00106 *> \endverbatim
00107 *>
00108 *> \param[in] LDB
00109 *> \verbatim
00110 *>          LDB is INTEGER
00111 *>          The leading dimension of the array B.  LDB >= max(1,N).
00112 *> \endverbatim
00113 *>
00114 *> \param[in,out] X
00115 *> \verbatim
00116 *>          X is COMPLEX array, dimension (LDX,NRHS)
00117 *>          On entry, the solution matrix X, as computed by CPTTRS.
00118 *>          On exit, the improved solution matrix X.
00119 *> \endverbatim
00120 *>
00121 *> \param[in] LDX
00122 *> \verbatim
00123 *>          LDX is INTEGER
00124 *>          The leading dimension of the array X.  LDX >= max(1,N).
00125 *> \endverbatim
00126 *>
00127 *> \param[out] FERR
00128 *> \verbatim
00129 *>          FERR is REAL array, dimension (NRHS)
00130 *>          The forward error bound for each solution vector
00131 *>          X(j) (the j-th column of the solution matrix X).
00132 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00133 *>          is an estimated upper bound for the magnitude of the largest
00134 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00135 *>          largest element in X(j).
00136 *> \endverbatim
00137 *>
00138 *> \param[out] BERR
00139 *> \verbatim
00140 *>          BERR is REAL array, dimension (NRHS)
00141 *>          The componentwise relative backward error of each solution
00142 *>          vector X(j) (i.e., the smallest relative change in
00143 *>          any element of A or B that makes X(j) an exact solution).
00144 *> \endverbatim
00145 *>
00146 *> \param[out] WORK
00147 *> \verbatim
00148 *>          WORK is COMPLEX array, dimension (N)
00149 *> \endverbatim
00150 *>
00151 *> \param[out] RWORK
00152 *> \verbatim
00153 *>          RWORK is REAL array, dimension (N)
00154 *> \endverbatim
00155 *>
00156 *> \param[out] INFO
00157 *> \verbatim
00158 *>          INFO is INTEGER
00159 *>          = 0:  successful exit
00160 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00161 *> \endverbatim
00162 *
00163 *> \par Internal Parameters:
00164 *  =========================
00165 *>
00166 *> \verbatim
00167 *>  ITMAX is the maximum number of steps of iterative refinement.
00168 *> \endverbatim
00169 *
00170 *  Authors:
00171 *  ========
00172 *
00173 *> \author Univ. of Tennessee 
00174 *> \author Univ. of California Berkeley 
00175 *> \author Univ. of Colorado Denver 
00176 *> \author NAG Ltd. 
00177 *
00178 *> \date November 2011
00179 *
00180 *> \ingroup complexOTHERcomputational
00181 *
00182 *  =====================================================================
00183       SUBROUTINE CPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
00184      $                   FERR, BERR, WORK, RWORK, INFO )
00185 *
00186 *  -- LAPACK computational routine (version 3.4.0) --
00187 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00188 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00189 *     November 2011
00190 *
00191 *     .. Scalar Arguments ..
00192       CHARACTER          UPLO
00193       INTEGER            INFO, LDB, LDX, N, NRHS
00194 *     ..
00195 *     .. Array Arguments ..
00196       REAL               BERR( * ), D( * ), DF( * ), FERR( * ),
00197      $                   RWORK( * )
00198       COMPLEX            B( LDB, * ), E( * ), EF( * ), WORK( * ),
00199      $                   X( LDX, * )
00200 *     ..
00201 *
00202 *  =====================================================================
00203 *
00204 *     .. Parameters ..
00205       INTEGER            ITMAX
00206       PARAMETER          ( ITMAX = 5 )
00207       REAL               ZERO
00208       PARAMETER          ( ZERO = 0.0E+0 )
00209       REAL               ONE
00210       PARAMETER          ( ONE = 1.0E+0 )
00211       REAL               TWO
00212       PARAMETER          ( TWO = 2.0E+0 )
00213       REAL               THREE
00214       PARAMETER          ( THREE = 3.0E+0 )
00215 *     ..
00216 *     .. Local Scalars ..
00217       LOGICAL            UPPER
00218       INTEGER            COUNT, I, IX, J, NZ
00219       REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
00220       COMPLEX            BI, CX, DX, EX, ZDUM
00221 *     ..
00222 *     .. External Functions ..
00223       LOGICAL            LSAME
00224       INTEGER            ISAMAX
00225       REAL               SLAMCH
00226       EXTERNAL           LSAME, ISAMAX, SLAMCH
00227 *     ..
00228 *     .. External Subroutines ..
00229       EXTERNAL           CAXPY, CPTTRS, XERBLA
00230 *     ..
00231 *     .. Intrinsic Functions ..
00232       INTRINSIC          ABS, AIMAG, CMPLX, CONJG, MAX, REAL
00233 *     ..
00234 *     .. Statement Functions ..
00235       REAL               CABS1
00236 *     ..
00237 *     .. Statement Function definitions ..
00238       CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
00239 *     ..
00240 *     .. Executable Statements ..
00241 *
00242 *     Test the input parameters.
00243 *
00244       INFO = 0
00245       UPPER = LSAME( UPLO, 'U' )
00246       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00247          INFO = -1
00248       ELSE IF( N.LT.0 ) THEN
00249          INFO = -2
00250       ELSE IF( NRHS.LT.0 ) THEN
00251          INFO = -3
00252       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00253          INFO = -9
00254       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00255          INFO = -11
00256       END IF
00257       IF( INFO.NE.0 ) THEN
00258          CALL XERBLA( 'CPTRFS', -INFO )
00259          RETURN
00260       END IF
00261 *
00262 *     Quick return if possible
00263 *
00264       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00265          DO 10 J = 1, NRHS
00266             FERR( J ) = ZERO
00267             BERR( J ) = ZERO
00268    10    CONTINUE
00269          RETURN
00270       END IF
00271 *
00272 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00273 *
00274       NZ = 4
00275       EPS = SLAMCH( 'Epsilon' )
00276       SAFMIN = SLAMCH( 'Safe minimum' )
00277       SAFE1 = NZ*SAFMIN
00278       SAFE2 = SAFE1 / EPS
00279 *
00280 *     Do for each right hand side
00281 *
00282       DO 100 J = 1, NRHS
00283 *
00284          COUNT = 1
00285          LSTRES = THREE
00286    20    CONTINUE
00287 *
00288 *        Loop until stopping criterion is satisfied.
00289 *
00290 *        Compute residual R = B - A * X.  Also compute
00291 *        abs(A)*abs(x) + abs(b) for use in the backward error bound.
00292 *
00293          IF( UPPER ) THEN
00294             IF( N.EQ.1 ) THEN
00295                BI = B( 1, J )
00296                DX = D( 1 )*X( 1, J )
00297                WORK( 1 ) = BI - DX
00298                RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
00299             ELSE
00300                BI = B( 1, J )
00301                DX = D( 1 )*X( 1, J )
00302                EX = E( 1 )*X( 2, J )
00303                WORK( 1 ) = BI - DX - EX
00304                RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
00305      $                      CABS1( E( 1 ) )*CABS1( X( 2, J ) )
00306                DO 30 I = 2, N - 1
00307                   BI = B( I, J )
00308                   CX = CONJG( E( I-1 ) )*X( I-1, J )
00309                   DX = D( I )*X( I, J )
00310                   EX = E( I )*X( I+1, J )
00311                   WORK( I ) = BI - CX - DX - EX
00312                   RWORK( I ) = CABS1( BI ) +
00313      $                         CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
00314      $                         CABS1( DX ) + CABS1( E( I ) )*
00315      $                         CABS1( X( I+1, J ) )
00316    30          CONTINUE
00317                BI = B( N, J )
00318                CX = CONJG( E( N-1 ) )*X( N-1, J )
00319                DX = D( N )*X( N, J )
00320                WORK( N ) = BI - CX - DX
00321                RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
00322      $                      CABS1( X( N-1, J ) ) + CABS1( DX )
00323             END IF
00324          ELSE
00325             IF( N.EQ.1 ) THEN
00326                BI = B( 1, J )
00327                DX = D( 1 )*X( 1, J )
00328                WORK( 1 ) = BI - DX
00329                RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
00330             ELSE
00331                BI = B( 1, J )
00332                DX = D( 1 )*X( 1, J )
00333                EX = CONJG( E( 1 ) )*X( 2, J )
00334                WORK( 1 ) = BI - DX - EX
00335                RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
00336      $                      CABS1( E( 1 ) )*CABS1( X( 2, J ) )
00337                DO 40 I = 2, N - 1
00338                   BI = B( I, J )
00339                   CX = E( I-1 )*X( I-1, J )
00340                   DX = D( I )*X( I, J )
00341                   EX = CONJG( E( I ) )*X( I+1, J )
00342                   WORK( I ) = BI - CX - DX - EX
00343                   RWORK( I ) = CABS1( BI ) +
00344      $                         CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
00345      $                         CABS1( DX ) + CABS1( E( I ) )*
00346      $                         CABS1( X( I+1, J ) )
00347    40          CONTINUE
00348                BI = B( N, J )
00349                CX = E( N-1 )*X( N-1, J )
00350                DX = D( N )*X( N, J )
00351                WORK( N ) = BI - CX - DX
00352                RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
00353      $                      CABS1( X( N-1, J ) ) + CABS1( DX )
00354             END IF
00355          END IF
00356 *
00357 *        Compute componentwise relative backward error from formula
00358 *
00359 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00360 *
00361 *        where abs(Z) is the componentwise absolute value of the matrix
00362 *        or vector Z.  If the i-th component of the denominator is less
00363 *        than SAFE2, then SAFE1 is added to the i-th components of the
00364 *        numerator and denominator before dividing.
00365 *
00366          S = ZERO
00367          DO 50 I = 1, N
00368             IF( RWORK( I ).GT.SAFE2 ) THEN
00369                S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
00370             ELSE
00371                S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
00372      $             ( RWORK( I )+SAFE1 ) )
00373             END IF
00374    50    CONTINUE
00375          BERR( J ) = S
00376 *
00377 *        Test stopping criterion. Continue iterating if
00378 *           1) The residual BERR(J) is larger than machine epsilon, and
00379 *           2) BERR(J) decreased by at least a factor of 2 during the
00380 *              last iteration, and
00381 *           3) At most ITMAX iterations tried.
00382 *
00383          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00384      $       COUNT.LE.ITMAX ) THEN
00385 *
00386 *           Update solution and try again.
00387 *
00388             CALL CPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO )
00389             CALL CAXPY( N, CMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
00390             LSTRES = BERR( J )
00391             COUNT = COUNT + 1
00392             GO TO 20
00393          END IF
00394 *
00395 *        Bound error from formula
00396 *
00397 *        norm(X - XTRUE) / norm(X) .le. FERR =
00398 *        norm( abs(inv(A))*
00399 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00400 *
00401 *        where
00402 *          norm(Z) is the magnitude of the largest component of Z
00403 *          inv(A) is the inverse of A
00404 *          abs(Z) is the componentwise absolute value of the matrix or
00405 *             vector Z
00406 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00407 *          EPS is machine epsilon
00408 *
00409 *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
00410 *        is incremented by SAFE1 if the i-th component of
00411 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00412 *
00413          DO 60 I = 1, N
00414             IF( RWORK( I ).GT.SAFE2 ) THEN
00415                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
00416             ELSE
00417                RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
00418      $                      SAFE1
00419             END IF
00420    60    CONTINUE
00421          IX = ISAMAX( N, RWORK, 1 )
00422          FERR( J ) = RWORK( IX )
00423 *
00424 *        Estimate the norm of inv(A).
00425 *
00426 *        Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
00427 *
00428 *           m(i,j) =  abs(A(i,j)), i = j,
00429 *           m(i,j) = -abs(A(i,j)), i .ne. j,
00430 *
00431 *        and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**H.
00432 *
00433 *        Solve M(L) * x = e.
00434 *
00435          RWORK( 1 ) = ONE
00436          DO 70 I = 2, N
00437             RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) )
00438    70    CONTINUE
00439 *
00440 *        Solve D * M(L)**H * x = b.
00441 *
00442          RWORK( N ) = RWORK( N ) / DF( N )
00443          DO 80 I = N - 1, 1, -1
00444             RWORK( I ) = RWORK( I ) / DF( I ) +
00445      $                   RWORK( I+1 )*ABS( EF( I ) )
00446    80    CONTINUE
00447 *
00448 *        Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
00449 *
00450          IX = ISAMAX( N, RWORK, 1 )
00451          FERR( J ) = FERR( J )*ABS( RWORK( IX ) )
00452 *
00453 *        Normalize error.
00454 *
00455          LSTRES = ZERO
00456          DO 90 I = 1, N
00457             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
00458    90    CONTINUE
00459          IF( LSTRES.NE.ZERO )
00460      $      FERR( J ) = FERR( J ) / LSTRES
00461 *
00462   100 CONTINUE
00463 *
00464       RETURN
00465 *
00466 *     End of CPTRFS
00467 *
00468       END
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