LAPACK 3.3.1 Linear Algebra PACKage

# dsprfs.f

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```00001       SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
00002      \$                   FERR, BERR, WORK, IWORK, INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
00010 *
00011 *     .. Scalar Arguments ..
00012       CHARACTER          UPLO
00013       INTEGER            INFO, LDB, LDX, N, NRHS
00014 *     ..
00015 *     .. Array Arguments ..
00016       INTEGER            IPIV( * ), IWORK( * )
00017       DOUBLE PRECISION   AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
00018      \$                   FERR( * ), WORK( * ), X( LDX, * )
00019 *     ..
00020 *
00021 *  Purpose
00022 *  =======
00023 *
00024 *  DSPRFS improves the computed solution to a system of linear
00025 *  equations when the coefficient matrix is symmetric indefinite
00026 *  and packed, and provides error bounds and backward error estimates
00027 *  for the solution.
00028 *
00029 *  Arguments
00030 *  =========
00031 *
00032 *  UPLO    (input) CHARACTER*1
00033 *          = 'U':  Upper triangle of A is stored;
00034 *          = 'L':  Lower triangle of A is stored.
00035 *
00036 *  N       (input) INTEGER
00037 *          The order of the matrix A.  N >= 0.
00038 *
00039 *  NRHS    (input) INTEGER
00040 *          The number of right hand sides, i.e., the number of columns
00041 *          of the matrices B and X.  NRHS >= 0.
00042 *
00043 *  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
00044 *          The upper or lower triangle of the symmetric matrix A, packed
00045 *          columnwise in a linear array.  The j-th column of A is stored
00046 *          in the array AP as follows:
00047 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00048 *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
00049 *
00050 *  AFP     (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
00051 *          The factored form of the matrix A.  AFP contains the block
00052 *          diagonal matrix D and the multipliers used to obtain the
00053 *          factor U or L from the factorization A = U*D*U**T or
00054 *          A = L*D*L**T as computed by DSPTRF, stored as a packed
00055 *          triangular matrix.
00056 *
00057 *  IPIV    (input) INTEGER array, dimension (N)
00058 *          Details of the interchanges and the block structure of D
00059 *          as determined by DSPTRF.
00060 *
00061 *  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
00062 *          The right hand side matrix B.
00063 *
00064 *  LDB     (input) INTEGER
00065 *          The leading dimension of the array B.  LDB >= max(1,N).
00066 *
00067 *  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
00068 *          On entry, the solution matrix X, as computed by DSPTRS.
00069 *          On exit, the improved solution matrix X.
00070 *
00071 *  LDX     (input) INTEGER
00072 *          The leading dimension of the array X.  LDX >= max(1,N).
00073 *
00074 *  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00075 *          The estimated forward error bound for each solution vector
00076 *          X(j) (the j-th column of the solution matrix X).
00077 *          If XTRUE is the true solution corresponding to X(j), FERR(j)
00078 *          is an estimated upper bound for the magnitude of the largest
00079 *          element in (X(j) - XTRUE) divided by the magnitude of the
00080 *          largest element in X(j).  The estimate is as reliable as
00081 *          the estimate for RCOND, and is almost always a slight
00082 *          overestimate of the true error.
00083 *
00084 *  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
00085 *          The componentwise relative backward error of each solution
00086 *          vector X(j) (i.e., the smallest relative change in
00087 *          any element of A or B that makes X(j) an exact solution).
00088 *
00089 *  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
00090 *
00091 *  IWORK   (workspace) INTEGER array, dimension (N)
00092 *
00093 *  INFO    (output) INTEGER
00094 *          = 0:  successful exit
00095 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00096 *
00097 *  Internal Parameters
00098 *  ===================
00099 *
00100 *  ITMAX is the maximum number of steps of iterative refinement.
00101 *
00102 *  =====================================================================
00103 *
00104 *     .. Parameters ..
00105       INTEGER            ITMAX
00106       PARAMETER          ( ITMAX = 5 )
00107       DOUBLE PRECISION   ZERO
00108       PARAMETER          ( ZERO = 0.0D+0 )
00109       DOUBLE PRECISION   ONE
00110       PARAMETER          ( ONE = 1.0D+0 )
00111       DOUBLE PRECISION   TWO
00112       PARAMETER          ( TWO = 2.0D+0 )
00113       DOUBLE PRECISION   THREE
00114       PARAMETER          ( THREE = 3.0D+0 )
00115 *     ..
00116 *     .. Local Scalars ..
00117       LOGICAL            UPPER
00118       INTEGER            COUNT, I, IK, J, K, KASE, KK, NZ
00119       DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
00120 *     ..
00121 *     .. Local Arrays ..
00122       INTEGER            ISAVE( 3 )
00123 *     ..
00124 *     .. External Subroutines ..
00125       EXTERNAL           DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
00126 *     ..
00127 *     .. Intrinsic Functions ..
00128       INTRINSIC          ABS, MAX
00129 *     ..
00130 *     .. External Functions ..
00131       LOGICAL            LSAME
00132       DOUBLE PRECISION   DLAMCH
00133       EXTERNAL           LSAME, DLAMCH
00134 *     ..
00135 *     .. Executable Statements ..
00136 *
00137 *     Test the input parameters.
00138 *
00139       INFO = 0
00140       UPPER = LSAME( UPLO, 'U' )
00141       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00142          INFO = -1
00143       ELSE IF( N.LT.0 ) THEN
00144          INFO = -2
00145       ELSE IF( NRHS.LT.0 ) THEN
00146          INFO = -3
00147       ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
00148          INFO = -8
00149       ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
00150          INFO = -10
00151       END IF
00152       IF( INFO.NE.0 ) THEN
00153          CALL XERBLA( 'DSPRFS', -INFO )
00154          RETURN
00155       END IF
00156 *
00157 *     Quick return if possible
00158 *
00159       IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
00160          DO 10 J = 1, NRHS
00161             FERR( J ) = ZERO
00162             BERR( J ) = ZERO
00163    10    CONTINUE
00164          RETURN
00165       END IF
00166 *
00167 *     NZ = maximum number of nonzero elements in each row of A, plus 1
00168 *
00169       NZ = N + 1
00170       EPS = DLAMCH( 'Epsilon' )
00171       SAFMIN = DLAMCH( 'Safe minimum' )
00172       SAFE1 = NZ*SAFMIN
00173       SAFE2 = SAFE1 / EPS
00174 *
00175 *     Do for each right hand side
00176 *
00177       DO 140 J = 1, NRHS
00178 *
00179          COUNT = 1
00180          LSTRES = THREE
00181    20    CONTINUE
00182 *
00183 *        Loop until stopping criterion is satisfied.
00184 *
00185 *        Compute residual R = B - A * X
00186 *
00187          CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
00188          CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
00189      \$               1 )
00190 *
00191 *        Compute componentwise relative backward error from formula
00192 *
00193 *        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
00194 *
00195 *        where abs(Z) is the componentwise absolute value of the matrix
00196 *        or vector Z.  If the i-th component of the denominator is less
00197 *        than SAFE2, then SAFE1 is added to the i-th components of the
00198 *        numerator and denominator before dividing.
00199 *
00200          DO 30 I = 1, N
00201             WORK( I ) = ABS( B( I, J ) )
00202    30    CONTINUE
00203 *
00204 *        Compute abs(A)*abs(X) + abs(B).
00205 *
00206          KK = 1
00207          IF( UPPER ) THEN
00208             DO 50 K = 1, N
00209                S = ZERO
00210                XK = ABS( X( K, J ) )
00211                IK = KK
00212                DO 40 I = 1, K - 1
00213                   WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
00214                   S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
00215                   IK = IK + 1
00216    40          CONTINUE
00217                WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
00218                KK = KK + K
00219    50       CONTINUE
00220          ELSE
00221             DO 70 K = 1, N
00222                S = ZERO
00223                XK = ABS( X( K, J ) )
00224                WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
00225                IK = KK + 1
00226                DO 60 I = K + 1, N
00227                   WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
00228                   S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
00229                   IK = IK + 1
00230    60          CONTINUE
00231                WORK( K ) = WORK( K ) + S
00232                KK = KK + ( N-K+1 )
00233    70       CONTINUE
00234          END IF
00235          S = ZERO
00236          DO 80 I = 1, N
00237             IF( WORK( I ).GT.SAFE2 ) THEN
00238                S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
00239             ELSE
00240                S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
00241      \$             ( WORK( I )+SAFE1 ) )
00242             END IF
00243    80    CONTINUE
00244          BERR( J ) = S
00245 *
00246 *        Test stopping criterion. Continue iterating if
00247 *           1) The residual BERR(J) is larger than machine epsilon, and
00248 *           2) BERR(J) decreased by at least a factor of 2 during the
00249 *              last iteration, and
00250 *           3) At most ITMAX iterations tried.
00251 *
00252          IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
00253      \$       COUNT.LE.ITMAX ) THEN
00254 *
00255 *           Update solution and try again.
00256 *
00257             CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
00258             CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
00259             LSTRES = BERR( J )
00260             COUNT = COUNT + 1
00261             GO TO 20
00262          END IF
00263 *
00264 *        Bound error from formula
00265 *
00266 *        norm(X - XTRUE) / norm(X) .le. FERR =
00267 *        norm( abs(inv(A))*
00268 *           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
00269 *
00270 *        where
00271 *          norm(Z) is the magnitude of the largest component of Z
00272 *          inv(A) is the inverse of A
00273 *          abs(Z) is the componentwise absolute value of the matrix or
00274 *             vector Z
00275 *          NZ is the maximum number of nonzeros in any row of A, plus 1
00276 *          EPS is machine epsilon
00277 *
00278 *        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
00279 *        is incremented by SAFE1 if the i-th component of
00280 *        abs(A)*abs(X) + abs(B) is less than SAFE2.
00281 *
00282 *        Use DLACN2 to estimate the infinity-norm of the matrix
00283 *           inv(A) * diag(W),
00284 *        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
00285 *
00286          DO 90 I = 1, N
00287             IF( WORK( I ).GT.SAFE2 ) THEN
00288                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
00289             ELSE
00290                WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
00291             END IF
00292    90    CONTINUE
00293 *
00294          KASE = 0
00295   100    CONTINUE
00296          CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
00297      \$                KASE, ISAVE )
00298          IF( KASE.NE.0 ) THEN
00299             IF( KASE.EQ.1 ) THEN
00300 *
00301 *              Multiply by diag(W)*inv(A**T).
00302 *
00303                CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
00304      \$                      INFO )
00305                DO 110 I = 1, N
00306                   WORK( N+I ) = WORK( I )*WORK( N+I )
00307   110          CONTINUE
00308             ELSE IF( KASE.EQ.2 ) THEN
00309 *
00310 *              Multiply by inv(A)*diag(W).
00311 *
00312                DO 120 I = 1, N
00313                   WORK( N+I ) = WORK( I )*WORK( N+I )
00314   120          CONTINUE
00315                CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
00316      \$                      INFO )
00317             END IF
00318             GO TO 100
00319          END IF
00320 *
00321 *        Normalize error.
00322 *
00323          LSTRES = ZERO
00324          DO 130 I = 1, N
00325             LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
00326   130    CONTINUE
00327          IF( LSTRES.NE.ZERO )
00328      \$      FERR( J ) = FERR( J ) / LSTRES
00329 *
00330   140 CONTINUE
00331 *
00332       RETURN
00333 *
00334 *     End of DSPRFS
00335 *
00336       END
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