LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
spbrfs.f
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00001 *> \brief \b SPBRFS
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SPBRFS + 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 SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
00022 *                          LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          UPLO
00026 *       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       INTEGER            IWORK( * )
00030 *       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00031 *      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00032 *       ..
00033 *  
00034 *
00035 *> \par Purpose:
00036 *  =============
00037 *>
00038 *> \verbatim
00039 *>
00040 *> SPBRFS improves the computed solution to a system of linear
00041 *> equations when the coefficient matrix is symmetric positive definite
00042 *> and banded, and provides error bounds and backward error estimates
00043 *> for the solution.
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] UPLO
00050 *> \verbatim
00051 *>          UPLO is CHARACTER*1
00052 *>          = 'U':  Upper triangle of A is stored;
00053 *>          = 'L':  Lower triangle of A is stored.
00054 *> \endverbatim
00055 *>
00056 *> \param[in] N
00057 *> \verbatim
00058 *>          N is INTEGER
00059 *>          The order of the matrix A.  N >= 0.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] KD
00063 *> \verbatim
00064 *>          KD is INTEGER
00065 *>          The number of superdiagonals of the matrix A if UPLO = 'U',
00066 *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
00067 *> \endverbatim
00068 *>
00069 *> \param[in] NRHS
00070 *> \verbatim
00071 *>          NRHS is INTEGER
00072 *>          The number of right hand sides, i.e., the number of columns
00073 *>          of the matrices B and X.  NRHS >= 0.
00074 *> \endverbatim
00075 *>
00076 *> \param[in] AB
00077 *> \verbatim
00078 *>          AB is REAL array, dimension (LDAB,N)
00079 *>          The upper or lower triangle of the symmetric band matrix A,
00080 *>          stored in the first KD+1 rows of the array.  The j-th column
00081 *>          of A is stored in the j-th column of the array AB as follows:
00082 *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00083 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00084 *> \endverbatim
00085 *>
00086 *> \param[in] LDAB
00087 *> \verbatim
00088 *>          LDAB is INTEGER
00089 *>          The leading dimension of the array AB.  LDAB >= KD+1.
00090 *> \endverbatim
00091 *>
00092 *> \param[in] AFB
00093 *> \verbatim
00094 *>          AFB is REAL array, dimension (LDAFB,N)
00095 *>          The triangular factor U or L from the Cholesky factorization
00096 *>          A = U**T*U or A = L*L**T of the band matrix A as computed by
00097 *>          SPBTRF, in the same storage format as A (see AB).
00098 *> \endverbatim
00099 *>
00100 *> \param[in] LDAFB
00101 *> \verbatim
00102 *>          LDAFB is INTEGER
00103 *>          The leading dimension of the array AFB.  LDAFB >= KD+1.
00104 *> \endverbatim
00105 *>
00106 *> \param[in] B
00107 *> \verbatim
00108 *>          B is REAL array, dimension (LDB,NRHS)
00109 *>          The right hand side matrix B.
00110 *> \endverbatim
00111 *>
00112 *> \param[in] LDB
00113 *> \verbatim
00114 *>          LDB is INTEGER
00115 *>          The leading dimension of the array B.  LDB >= max(1,N).
00116 *> \endverbatim
00117 *>
00118 *> \param[in,out] X
00119 *> \verbatim
00120 *>          X is REAL array, dimension (LDX,NRHS)
00121 *>          On entry, the solution matrix X, as computed by SPBTRS.
00122 *>          On exit, the improved solution matrix X.
00123 *> \endverbatim
00124 *>
00125 *> \param[in] LDX
00126 *> \verbatim
00127 *>          LDX is INTEGER
00128 *>          The leading dimension of the array X.  LDX >= max(1,N).
00129 *> \endverbatim
00130 *>
00131 *> \param[out] FERR
00132 *> \verbatim
00133 *>          FERR is REAL array, dimension (NRHS)
00134 *>          The estimated forward error bound for each solution vector
00135 *>          X(j) (the j-th column of the solution matrix X).
00136 *>          If XTRUE is the true solution corresponding to X(j), FERR(j)
00137 *>          is an estimated upper bound for the magnitude of the largest
00138 *>          element in (X(j) - XTRUE) divided by the magnitude of the
00139 *>          largest element in X(j).  The estimate is as reliable as
00140 *>          the estimate for RCOND, and is almost always a slight
00141 *>          overestimate of the true error.
00142 *> \endverbatim
00143 *>
00144 *> \param[out] BERR
00145 *> \verbatim
00146 *>          BERR is REAL array, dimension (NRHS)
00147 *>          The componentwise relative backward error of each solution
00148 *>          vector X(j) (i.e., the smallest relative change in
00149 *>          any element of A or B that makes X(j) an exact solution).
00150 *> \endverbatim
00151 *>
00152 *> \param[out] WORK
00153 *> \verbatim
00154 *>          WORK is REAL array, dimension (3*N)
00155 *> \endverbatim
00156 *>
00157 *> \param[out] IWORK
00158 *> \verbatim
00159 *>          IWORK is INTEGER array, dimension (N)
00160 *> \endverbatim
00161 *>
00162 *> \param[out] INFO
00163 *> \verbatim
00164 *>          INFO is INTEGER
00165 *>          = 0:  successful exit
00166 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00167 *> \endverbatim
00168 *
00169 *> \par Internal Parameters:
00170 *  =========================
00171 *>
00172 *> \verbatim
00173 *>  ITMAX is the maximum number of steps of iterative refinement.
00174 *> \endverbatim
00175 *
00176 *  Authors:
00177 *  ========
00178 *
00179 *> \author Univ. of Tennessee 
00180 *> \author Univ. of California Berkeley 
00181 *> \author Univ. of Colorado Denver 
00182 *> \author NAG Ltd. 
00183 *
00184 *> \date November 2011
00185 *
00186 *> \ingroup realOTHERcomputational
00187 *
00188 *  =====================================================================
00189       SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
00190      $                   LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
00191 *
00192 *  -- LAPACK computational routine (version 3.4.0) --
00193 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00194 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00195 *     November 2011
00196 *
00197 *     .. Scalar Arguments ..
00198       CHARACTER          UPLO
00199       INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
00200 *     ..
00201 *     .. Array Arguments ..
00202       INTEGER            IWORK( * )
00203       REAL               AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
00204      $                   BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
00205 *     ..
00206 *
00207 *  =====================================================================
00208 *
00209 *     .. Parameters ..
00210       INTEGER            ITMAX
00211       PARAMETER          ( ITMAX = 5 )
00212       REAL               ZERO
00213       PARAMETER          ( ZERO = 0.0E+0 )
00214       REAL               ONE
00215       PARAMETER          ( ONE = 1.0E+0 )
00216       REAL               TWO
00217       PARAMETER          ( TWO = 2.0E+0 )
00218       REAL               THREE
00219       PARAMETER          ( THREE = 3.0E+0 )
00220 *     ..
00221 *     .. Local Scalars ..
00222       LOGICAL            UPPER
00223       INTEGER            COUNT, I, J, K, KASE, L, NZ
00224       REAL               EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00225 *     ..
00226 *     .. Local Arrays ..
00227       INTEGER            ISAVE( 3 )
00228 *     ..
00229 *     .. External Subroutines ..
00230       EXTERNAL           SAXPY, SCOPY, SLACN2, SPBTRS, SSBMV, XERBLA
00231 *     ..
00232 *     .. Intrinsic Functions ..
00233       INTRINSIC          ABS, MAX, MIN
00234 *     ..
00235 *     .. External Functions ..
00236       LOGICAL            LSAME
00237       REAL               SLAMCH
00238       EXTERNAL           LSAME, SLAMCH
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( KD.LT.0 ) THEN
00251          INFO = -3
00252       ELSE IF( NRHS.LT.0 ) THEN
00253          INFO = -4
00254       ELSE IF( LDAB.LT.KD+1 ) THEN
00255          INFO = -6
00256       ELSE IF( LDAFB.LT.KD+1 ) THEN
00257          INFO = -8
00258       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00259          INFO = -10
00260       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00261          INFO = -12
00262       END IF
00263       IF( INFO.NE.0 ) THEN
00264          CALL XERBLA( 'SPBRFS', -INFO )
00265          RETURN
00266       END IF
00267 *
00268 *     Quick return if possible
00269 *
00270       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00271          DO 10 J = 1, NRHS
00272             FERR( J ) = ZERO
00273             BERR( J ) = ZERO
00274    10    CONTINUE
00275          RETURN
00276       END IF
00277 *
00278 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00279 *
00280       NZ = MIN( N+1, 2*KD+2 )
00281       EPS = SLAMCH( 'Epsilon' )
00282       SAFMIN = SLAMCH( 'Safe minimum' )
00283       SAFE1 = NZ*SAFMIN
00284       SAFE2 = SAFE1 / EPS
00285 *
00286 *     Do for each right hand side
00287 *
00288       DO 140 J = 1, NRHS
00289 *
00290          COUNT = 1
00291          LSTRES = THREE
00292    20    CONTINUE
00293 *
00294 *        Loop until stopping criterion is satisfied.
00295 *
00296 *        Compute residual R = B - A * X
00297 *
00298          CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
00299          CALL SSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
00300      $               WORK( N+1 ), 1 )
00301 *
00302 *        Compute componentwise relative backward error from formula
00303 *
00304 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00305 *
00306 *        where abs(Z) is the componentwise absolute value of the matrix
00307 *        or vector Z.  If the i-th component of the denominator is less
00308 *        than SAFE2, then SAFE1 is added to the i-th components of the
00309 *        numerator and denominator before dividing.
00310 *
00311          DO 30 I = 1, N
00312             WORK( I ) = ABS( B( I, J ) )
00313    30    CONTINUE
00314 *
00315 *        Compute abs(A)*abs(X) + abs(B).
00316 *
00317          IF( UPPER ) THEN
00318             DO 50 K = 1, N
00319                S = ZERO
00320                XK = ABS( X( K, J ) )
00321                L = KD + 1 - K
00322                DO 40 I = MAX( 1, K-KD ), K - 1
00323                   WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
00324                   S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
00325    40          CONTINUE
00326                WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
00327    50       CONTINUE
00328          ELSE
00329             DO 70 K = 1, N
00330                S = ZERO
00331                XK = ABS( X( K, J ) )
00332                WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
00333                L = 1 - K
00334                DO 60 I = K + 1, MIN( N, K+KD )
00335                   WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
00336                   S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
00337    60          CONTINUE
00338                WORK( K ) = WORK( K ) + S
00339    70       CONTINUE
00340          END IF
00341          S = ZERO
00342          DO 80 I = 1, N
00343             IF( WORK( I ).GT.SAFE2 ) THEN
00344                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
00345             ELSE
00346                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
00347      $             ( WORK( I )+SAFE1 ) )
00348             END IF
00349    80    CONTINUE
00350          BERR( J ) = S
00351 *
00352 *        Test stopping criterion. Continue iterating if
00353 *           1) The residual BERR(J) is larger than machine epsilon, and
00354 *           2) BERR(J) decreased by at least a factor of 2 during the
00355 *              last iteration, and
00356 *           3) At most ITMAX iterations tried.
00357 *
00358          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00359      $       COUNT.LE.ITMAX ) THEN
00360 *
00361 *           Update solution and try again.
00362 *
00363             CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
00364      $                   INFO )
00365             CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
00366             LSTRES = BERR( J )
00367             COUNT = COUNT + 1
00368             GO TO 20
00369          END IF
00370 *
00371 *        Bound error from formula
00372 *
00373 *        norm(X - XTRUE) / norm(X) .le. FERR =
00374 *        norm( abs(inv(A))*
00375 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00376 *
00377 *        where
00378 *          norm(Z) is the magnitude of the largest component of Z
00379 *          inv(A) is the inverse of A
00380 *          abs(Z) is the componentwise absolute value of the matrix or
00381 *             vector Z
00382 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00383 *          EPS is machine epsilon
00384 *
00385 *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
00386 *        is incremented by SAFE1 if the i-th component of
00387 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00388 *
00389 *        Use SLACN2 to estimate the infinity-norm of the matrix
00390 *           inv(A) * diag(W),
00391 *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
00392 *
00393          DO 90 I = 1, N
00394             IF( WORK( I ).GT.SAFE2 ) THEN
00395                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
00396             ELSE
00397                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
00398             END IF
00399    90    CONTINUE
00400 *
00401          KASE = 0
00402   100    CONTINUE
00403          CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
00404      $                KASE, ISAVE )
00405          IF( KASE.NE.0 ) THEN
00406             IF( KASE.EQ.1 ) THEN
00407 *
00408 *              Multiply by diag(W)*inv(A**T).
00409 *
00410                CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
00411      $                      INFO )
00412                DO 110 I = 1, N
00413                   WORK( N+I ) = WORK( N+I )*WORK( I )
00414   110          CONTINUE
00415             ELSE IF( KASE.EQ.2 ) THEN
00416 *
00417 *              Multiply by inv(A)*diag(W).
00418 *
00419                DO 120 I = 1, N
00420                   WORK( N+I ) = WORK( N+I )*WORK( I )
00421   120          CONTINUE
00422                CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
00423      $                      INFO )
00424             END IF
00425             GO TO 100
00426          END IF
00427 *
00428 *        Normalize error.
00429 *
00430          LSTRES = ZERO
00431          DO 130 I = 1, N
00432             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
00433   130    CONTINUE
00434          IF( LSTRES.NE.ZERO )
00435      $      FERR( J ) = FERR( J ) / LSTRES
00436 *
00437   140 CONTINUE
00438 *
00439       RETURN
00440 *
00441 *     End of SPBRFS
00442 *
00443       END
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