LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
ztpt05.f
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00001 *> \brief \b ZTPT05
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *       SUBROUTINE ZTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
00012 *                          XACT, LDXACT, FERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          DIAG, TRANS, UPLO
00016 *       INTEGER            LDB, LDX, LDXACT, N, NRHS
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       DOUBLE PRECISION   BERR( * ), FERR( * ), RESLTS( * )
00020 *       COMPLEX*16         AP( * ), B( LDB, * ), X( LDX, * ),
00021 *      $                   XACT( LDXACT, * )
00022 *       ..
00023 *  
00024 *
00025 *> \par Purpose:
00026 *  =============
00027 *>
00028 *> \verbatim
00029 *>
00030 *> ZTPT05 tests the error bounds from iterative refinement for the
00031 *> computed solution to a system of equations A*X = B, where A is a
00032 *> triangular matrix in packed storage format.
00033 *>
00034 *> RESLTS(1) = test of the error bound
00035 *>           = norm(X - XACT) / ( norm(X) * FERR )
00036 *>
00037 *> A large value is returned if this ratio is not less than one.
00038 *>
00039 *> RESLTS(2) = residual from the iterative refinement routine
00040 *>           = the maximum of BERR / ( (n+1)*EPS + (*) ), where
00041 *>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00042 *> \endverbatim
00043 *
00044 *  Arguments:
00045 *  ==========
00046 *
00047 *> \param[in] UPLO
00048 *> \verbatim
00049 *>          UPLO is CHARACTER*1
00050 *>          Specifies whether the matrix A is upper or lower triangular.
00051 *>          = 'U':  Upper triangular
00052 *>          = 'L':  Lower triangular
00053 *> \endverbatim
00054 *>
00055 *> \param[in] TRANS
00056 *> \verbatim
00057 *>          TRANS is CHARACTER*1
00058 *>          Specifies the form of the system of equations.
00059 *>          = 'N':  A * X = B  (No transpose)
00060 *>          = 'T':  A'* X = B  (Transpose)
00061 *>          = 'C':  A'* X = B  (Conjugate transpose = Transpose)
00062 *> \endverbatim
00063 *>
00064 *> \param[in] DIAG
00065 *> \verbatim
00066 *>          DIAG is CHARACTER*1
00067 *>          Specifies whether or not the matrix A is unit triangular.
00068 *>          = 'N':  Non-unit triangular
00069 *>          = 'U':  Unit triangular
00070 *> \endverbatim
00071 *>
00072 *> \param[in] N
00073 *> \verbatim
00074 *>          N is INTEGER
00075 *>          The number of rows of the matrices X, B, and XACT, and the
00076 *>          order of the matrix A.  N >= 0.
00077 *> \endverbatim
00078 *>
00079 *> \param[in] NRHS
00080 *> \verbatim
00081 *>          NRHS is INTEGER
00082 *>          The number of columns of the matrices X, B, and XACT.
00083 *>          NRHS >= 0.
00084 *> \endverbatim
00085 *>
00086 *> \param[in] AP
00087 *> \verbatim
00088 *>          AP is COMPLEX*16 array, dimension (N*(N+1)/2)
00089 *>          The upper or lower triangular matrix A, packed columnwise in
00090 *>          a linear array.  The j-th column of A is stored in the array
00091 *>          AP as follows:
00092 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00093 *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00094 *>          If DIAG = 'U', the diagonal elements of A are not referenced
00095 *>          and are assumed to be 1.
00096 *> \endverbatim
00097 *>
00098 *> \param[in] B
00099 *> \verbatim
00100 *>          B is COMPLEX*16 array, dimension (LDB,NRHS)
00101 *>          The right hand side vectors for the system of linear
00102 *>          equations.
00103 *> \endverbatim
00104 *>
00105 *> \param[in] LDB
00106 *> \verbatim
00107 *>          LDB is INTEGER
00108 *>          The leading dimension of the array B.  LDB >= max(1,N).
00109 *> \endverbatim
00110 *>
00111 *> \param[in] X
00112 *> \verbatim
00113 *>          X is COMPLEX*16 array, dimension (LDX,NRHS)
00114 *>          The computed solution vectors.  Each vector is stored as a
00115 *>          column of the matrix X.
00116 *> \endverbatim
00117 *>
00118 *> \param[in] LDX
00119 *> \verbatim
00120 *>          LDX is INTEGER
00121 *>          The leading dimension of the array X.  LDX >= max(1,N).
00122 *> \endverbatim
00123 *>
00124 *> \param[in] XACT
00125 *> \verbatim
00126 *>          XACT is COMPLEX*16 array, dimension (LDX,NRHS)
00127 *>          The exact solution vectors.  Each vector is stored as a
00128 *>          column of the matrix XACT.
00129 *> \endverbatim
00130 *>
00131 *> \param[in] LDXACT
00132 *> \verbatim
00133 *>          LDXACT is INTEGER
00134 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00135 *> \endverbatim
00136 *>
00137 *> \param[in] FERR
00138 *> \verbatim
00139 *>          FERR is DOUBLE PRECISION array, dimension (NRHS)
00140 *>          The estimated forward error bounds for each solution vector
00141 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00142 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00143 *>          of the largest entry in X.
00144 *> \endverbatim
00145 *>
00146 *> \param[in] BERR
00147 *> \verbatim
00148 *>          BERR is DOUBLE PRECISION array, dimension (NRHS)
00149 *>          The componentwise relative backward error of each solution
00150 *>          vector (i.e., the smallest relative change in any entry of A
00151 *>          or B that makes X an exact solution).
00152 *> \endverbatim
00153 *>
00154 *> \param[out] RESLTS
00155 *> \verbatim
00156 *>          RESLTS is DOUBLE PRECISION array, dimension (2)
00157 *>          The maximum over the NRHS solution vectors of the ratios:
00158 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00159 *>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00160 *> \endverbatim
00161 *
00162 *  Authors:
00163 *  ========
00164 *
00165 *> \author Univ. of Tennessee 
00166 *> \author Univ. of California Berkeley 
00167 *> \author Univ. of Colorado Denver 
00168 *> \author NAG Ltd. 
00169 *
00170 *> \date November 2011
00171 *
00172 *> \ingroup complex16_lin
00173 *
00174 *  =====================================================================
00175       SUBROUTINE ZTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
00176      $                   XACT, LDXACT, FERR, BERR, RESLTS )
00177 *
00178 *  -- LAPACK test routine (version 3.4.0) --
00179 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00180 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00181 *     November 2011
00182 *
00183 *     .. Scalar Arguments ..
00184       CHARACTER          DIAG, TRANS, UPLO
00185       INTEGER            LDB, LDX, LDXACT, N, NRHS
00186 *     ..
00187 *     .. Array Arguments ..
00188       DOUBLE PRECISION   BERR( * ), FERR( * ), RESLTS( * )
00189       COMPLEX*16         AP( * ), B( LDB, * ), X( LDX, * ),
00190      $                   XACT( LDXACT, * )
00191 *     ..
00192 *
00193 *  =====================================================================
00194 *
00195 *     .. Parameters ..
00196       DOUBLE PRECISION   ZERO, ONE
00197       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00198 *     ..
00199 *     .. Local Scalars ..
00200       LOGICAL            NOTRAN, UNIT, UPPER
00201       INTEGER            I, IFU, IMAX, J, JC, K
00202       DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00203       COMPLEX*16         ZDUM
00204 *     ..
00205 *     .. External Functions ..
00206       LOGICAL            LSAME
00207       INTEGER            IZAMAX
00208       DOUBLE PRECISION   DLAMCH
00209       EXTERNAL           LSAME, IZAMAX, DLAMCH
00210 *     ..
00211 *     .. Intrinsic Functions ..
00212       INTRINSIC          ABS, DBLE, DIMAG, MAX, MIN
00213 *     ..
00214 *     .. Statement Functions ..
00215       DOUBLE PRECISION   CABS1
00216 *     ..
00217 *     .. Statement Function definitions ..
00218       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00219 *     ..
00220 *     .. Executable Statements ..
00221 *
00222 *     Quick exit if N = 0 or NRHS = 0.
00223 *
00224       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00225          RESLTS( 1 ) = ZERO
00226          RESLTS( 2 ) = ZERO
00227          RETURN
00228       END IF
00229 *
00230       EPS = DLAMCH( 'Epsilon' )
00231       UNFL = DLAMCH( 'Safe minimum' )
00232       OVFL = ONE / UNFL
00233       UPPER = LSAME( UPLO, 'U' )
00234       NOTRAN = LSAME( TRANS, 'N' )
00235       UNIT = LSAME( DIAG, 'U' )
00236 *
00237 *     Test 1:  Compute the maximum of
00238 *        norm(X - XACT) / ( norm(X) * FERR )
00239 *     over all the vectors X and XACT using the infinity-norm.
00240 *
00241       ERRBND = ZERO
00242       DO 30 J = 1, NRHS
00243          IMAX = IZAMAX( N, X( 1, J ), 1 )
00244          XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
00245          DIFF = ZERO
00246          DO 10 I = 1, N
00247             DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
00248    10    CONTINUE
00249 *
00250          IF( XNORM.GT.ONE ) THEN
00251             GO TO 20
00252          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00253             GO TO 20
00254          ELSE
00255             ERRBND = ONE / EPS
00256             GO TO 30
00257          END IF
00258 *
00259    20    CONTINUE
00260          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00261             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00262          ELSE
00263             ERRBND = ONE / EPS
00264          END IF
00265    30 CONTINUE
00266       RESLTS( 1 ) = ERRBND
00267 *
00268 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00269 *     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00270 *
00271       IFU = 0
00272       IF( UNIT )
00273      $   IFU = 1
00274       DO 90 K = 1, NRHS
00275          DO 80 I = 1, N
00276             TMP = CABS1( B( I, K ) )
00277             IF( UPPER ) THEN
00278                JC = ( ( I-1 )*I ) / 2
00279                IF( .NOT.NOTRAN ) THEN
00280                   DO 40 J = 1, I - IFU
00281                      TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
00282    40             CONTINUE
00283                   IF( UNIT )
00284      $               TMP = TMP + CABS1( X( I, K ) )
00285                ELSE
00286                   JC = JC + I
00287                   IF( UNIT ) THEN
00288                      TMP = TMP + CABS1( X( I, K ) )
00289                      JC = JC + I
00290                   END IF
00291                   DO 50 J = I + IFU, N
00292                      TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
00293                      JC = JC + J
00294    50             CONTINUE
00295                END IF
00296             ELSE
00297                IF( NOTRAN ) THEN
00298                   JC = I
00299                   DO 60 J = 1, I - IFU
00300                      TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
00301                      JC = JC + N - J
00302    60             CONTINUE
00303                   IF( UNIT )
00304      $               TMP = TMP + CABS1( X( I, K ) )
00305                ELSE
00306                   JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
00307                   IF( UNIT )
00308      $               TMP = TMP + CABS1( X( I, K ) )
00309                   DO 70 J = I + IFU, N
00310                      TMP = TMP + CABS1( AP( JC+J-I ) )*
00311      $                     CABS1( X( J, K ) )
00312    70             CONTINUE
00313                END IF
00314             END IF
00315             IF( I.EQ.1 ) THEN
00316                AXBI = TMP
00317             ELSE
00318                AXBI = MIN( AXBI, TMP )
00319             END IF
00320    80    CONTINUE
00321          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00322      $         MAX( AXBI, ( N+1 )*UNFL ) )
00323          IF( K.EQ.1 ) THEN
00324             RESLTS( 2 ) = TMP
00325          ELSE
00326             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00327          END IF
00328    90 CONTINUE
00329 *
00330       RETURN
00331 *
00332 *     End of ZTPT05
00333 *
00334       END
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