LAPACK: TESTING/LIN/zppt05.f Source File

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00001       SUBROUTINE ZPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT,
00002      $                   LDXACT, FERR, BERR, RESLTS )
00003 *
00004 *  -- LAPACK test routine (version 3.1) --
00005 *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       CHARACTER          UPLO
00010       INTEGER            LDB, LDX, LDXACT, N, NRHS
00011 *     ..
00012 *     .. Array Arguments ..
00013       DOUBLE PRECISION   BERR( * ), FERR( * ), RESLTS( * )
00014       COMPLEX*16         AP( * ), B( LDB, * ), X( LDX, * ),
00015      $                   XACT( LDXACT, * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  ZPPT05 tests the error bounds from iterative refinement for the
00022 *  computed solution to a system of equations A*X = B, where A is a
00023 *  Hermitian matrix in packed storage format.
00024 *
00025 *  RESLTS(1) = test of the error bound
00026 *            = norm(X - XACT) / ( norm(X) * FERR )
00027 *
00028 *  A large value is returned if this ratio is not less than one.
00029 *
00030 *  RESLTS(2) = residual from the iterative refinement routine
00031 *            = the maximum of BERR / ( (n+1)*EPS + (*) ), where
00032 *              (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00033 *
00034 *  Arguments
00035 *  =========
00036 *
00037 *  UPLO    (input) CHARACTER*1
00038 *          Specifies whether the upper or lower triangular part of the
00039 *          Hermitian matrix A is stored.
00040 *          = 'U':  Upper triangular
00041 *          = 'L':  Lower triangular
00042 *
00043 *  N       (input) INTEGER
00044 *          The number of rows of the matrices X, B, and XACT, and the
00045 *          order of the matrix A.  N >= 0.
00046 *
00047 *  NRHS    (input) INTEGER
00048 *          The number of columns of the matrices X, B, and XACT.
00049 *          NRHS >= 0.
00050 *
00051 *  AP      (input) COMPLEX*16 array, dimension (N*(N+1)/2)
00052 *          The upper or lower triangle of the Hermitian matrix A, packed
00053 *          columnwise in a linear array.  The j-th column of A is stored
00054 *          in the array AP as follows:
00055 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00056 *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00057 *
00058 *  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
00059 *          The right hand side vectors for the system of linear
00060 *          equations.
00061 *
00062 *  LDB     (input) INTEGER
00063 *          The leading dimension of the array B.  LDB >= max(1,N).
00064 *
00065 *  X       (input) COMPLEX*16 array, dimension (LDX,NRHS)
00066 *          The computed solution vectors.  Each vector is stored as a
00067 *          column of the matrix X.
00068 *
00069 *  LDX     (input) INTEGER
00070 *          The leading dimension of the array X.  LDX >= max(1,N).
00071 *
00072 *  XACT    (input) COMPLEX*16 array, dimension (LDX,NRHS)
00073 *          The exact solution vectors.  Each vector is stored as a
00074 *          column of the matrix XACT.
00075 *
00076 *  LDXACT  (input) INTEGER
00077 *          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00078 *
00079 *  FERR    (input) DOUBLE PRECISION array, dimension (NRHS)
00080 *          The estimated forward error bounds for each solution vector
00081 *          X.  If XTRUE is the true solution, FERR bounds the magnitude
00082 *          of the largest entry in (X - XTRUE) divided by the magnitude
00083 *          of the largest entry in X.
00084 *
00085 *  BERR    (input) DOUBLE PRECISION array, dimension (NRHS)
00086 *          The componentwise relative backward error of each solution
00087 *          vector (i.e., the smallest relative change in any entry of A
00088 *          or B that makes X an exact solution).
00089 *
00090 *  RESLTS  (output) DOUBLE PRECISION array, dimension (2)
00091 *          The maximum over the NRHS solution vectors of the ratios:
00092 *          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00093 *          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00094 *
00095 *  =====================================================================
00096 *
00097 *     .. Parameters ..
00098       DOUBLE PRECISION   ZERO, ONE
00099       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00100 *     ..
00101 *     .. Local Scalars ..
00102       LOGICAL            UPPER
00103       INTEGER            I, IMAX, J, JC, K
00104       DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00105       COMPLEX*16         ZDUM
00106 *     ..
00107 *     .. External Functions ..
00108       LOGICAL            LSAME
00109       INTEGER            IZAMAX
00110       DOUBLE PRECISION   DLAMCH
00111       EXTERNAL           LSAME, IZAMAX, DLAMCH
00112 *     ..
00113 *     .. Intrinsic Functions ..
00114       INTRINSIC          ABS, DBLE, DIMAG, MAX, MIN
00115 *     ..
00116 *     .. Statement Functions ..
00117       DOUBLE PRECISION   CABS1
00118 *     ..
00119 *     .. Statement Function definitions ..
00120       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00121 *     ..
00122 *     .. Executable Statements ..
00123 *
00124 *     Quick exit if N = 0 or NRHS = 0.
00125 *
00126       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00127          RESLTS( 1 ) = ZERO
00128          RESLTS( 2 ) = ZERO
00129          RETURN
00130       END IF
00131 *
00132       EPS = DLAMCH( 'Epsilon' )
00133       UNFL = DLAMCH( 'Safe minimum' )
00134       OVFL = ONE / UNFL
00135       UPPER = LSAME( UPLO, 'U' )
00136 *
00137 *     Test 1:  Compute the maximum of
00138 *        norm(X - XACT) / ( norm(X) * FERR )
00139 *     over all the vectors X and XACT using the infinity-norm.
00140 *
00141       ERRBND = ZERO
00142       DO 30 J = 1, NRHS
00143          IMAX = IZAMAX( N, X( 1, J ), 1 )
00144          XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
00145          DIFF = ZERO
00146          DO 10 I = 1, N
00147             DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
00148    10    CONTINUE
00149 *
00150          IF( XNORM.GT.ONE ) THEN
00151             GO TO 20
00152          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00153             GO TO 20
00154          ELSE
00155             ERRBND = ONE / EPS
00156             GO TO 30
00157          END IF
00158 *
00159    20    CONTINUE
00160          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00161             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00162          ELSE
00163             ERRBND = ONE / EPS
00164          END IF
00165    30 CONTINUE
00166       RESLTS( 1 ) = ERRBND
00167 *
00168 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00169 *     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00170 *
00171       DO 90 K = 1, NRHS
00172          DO 80 I = 1, N
00173             TMP = CABS1( B( I, K ) )
00174             IF( UPPER ) THEN
00175                JC = ( ( I-1 )*I ) / 2
00176                DO 40 J = 1, I - 1
00177                   TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
00178    40          CONTINUE
00179                TMP = TMP + ABS( DBLE( AP( JC+I ) ) )*CABS1( X( I, K ) )
00180                JC = JC + I + I
00181                DO 50 J = I + 1, N
00182                   TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
00183                   JC = JC + J
00184    50          CONTINUE
00185             ELSE
00186                JC = I
00187                DO 60 J = 1, I - 1
00188                   TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
00189                   JC = JC + N - J
00190    60          CONTINUE
00191                TMP = TMP + ABS( DBLE( AP( JC ) ) )*CABS1( X( I, K ) )
00192                DO 70 J = I + 1, N
00193                   TMP = TMP + CABS1( AP( JC+J-I ) )*CABS1( X( J, K ) )
00194    70          CONTINUE
00195             END IF
00196             IF( I.EQ.1 ) THEN
00197                AXBI = TMP
00198             ELSE
00199                AXBI = MIN( AXBI, TMP )
00200             END IF
00201    80    CONTINUE
00202          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00203      $         MAX( AXBI, ( N+1 )*UNFL ) )
00204          IF( K.EQ.1 ) THEN
00205             RESLTS( 2 ) = TMP
00206          ELSE
00207             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00208          END IF
00209    90 CONTINUE
00210 *
00211       RETURN
00212 *
00213 *     End of ZPPT05
00214 *
00215       END