LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
stpt05.f
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00001 *> \brief \b STPT05
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 STPT05( 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 *       REAL               AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
00020 *      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> STPT05 tests the error bounds from iterative refinement for the
00030 *> computed solution to a system of equations A*X = B, where A is a
00031 *> triangular matrix in packed storage format.
00032 *>
00033 *> RESLTS(1) = test of the error bound
00034 *>           = norm(X - XACT) / ( norm(X) * FERR )
00035 *>
00036 *> A large value is returned if this ratio is not less than one.
00037 *>
00038 *> RESLTS(2) = residual from the iterative refinement routine
00039 *>           = the maximum of BERR / ( (n+1)*EPS + (*) ), where
00040 *>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00041 *> \endverbatim
00042 *
00043 *  Arguments:
00044 *  ==========
00045 *
00046 *> \param[in] UPLO
00047 *> \verbatim
00048 *>          UPLO is CHARACTER*1
00049 *>          Specifies whether the matrix A is upper or lower triangular.
00050 *>          = 'U':  Upper triangular
00051 *>          = 'L':  Lower triangular
00052 *> \endverbatim
00053 *>
00054 *> \param[in] TRANS
00055 *> \verbatim
00056 *>          TRANS is CHARACTER*1
00057 *>          Specifies the form of the system of equations.
00058 *>          = 'N':  A * X = B  (No transpose)
00059 *>          = 'T':  A'* X = B  (Transpose)
00060 *>          = 'C':  A'* X = B  (Conjugate transpose = Transpose)
00061 *> \endverbatim
00062 *>
00063 *> \param[in] DIAG
00064 *> \verbatim
00065 *>          DIAG is CHARACTER*1
00066 *>          Specifies whether or not the matrix A is unit triangular.
00067 *>          = 'N':  Non-unit triangular
00068 *>          = 'U':  Unit triangular
00069 *> \endverbatim
00070 *>
00071 *> \param[in] N
00072 *> \verbatim
00073 *>          N is INTEGER
00074 *>          The number of rows of the matrices X, B, and XACT, and the
00075 *>          order of the matrix A.  N >= 0.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] NRHS
00079 *> \verbatim
00080 *>          NRHS is INTEGER
00081 *>          The number of columns of the matrices X, B, and XACT.
00082 *>          NRHS >= 0.
00083 *> \endverbatim
00084 *>
00085 *> \param[in] AP
00086 *> \verbatim
00087 *>          AP is REAL array, dimension (N*(N+1)/2)
00088 *>          The upper or lower triangular matrix A, packed columnwise in
00089 *>          a linear array.  The j-th column of A is stored in the array
00090 *>          AP as follows:
00091 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00092 *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00093 *>          If DIAG = 'U', the diagonal elements of A are not referenced
00094 *>          and are assumed to be 1.
00095 *> \endverbatim
00096 *>
00097 *> \param[in] B
00098 *> \verbatim
00099 *>          B is REAL array, dimension (LDB,NRHS)
00100 *>          The right hand side vectors for the system of linear
00101 *>          equations.
00102 *> \endverbatim
00103 *>
00104 *> \param[in] LDB
00105 *> \verbatim
00106 *>          LDB is INTEGER
00107 *>          The leading dimension of the array B.  LDB >= max(1,N).
00108 *> \endverbatim
00109 *>
00110 *> \param[in] X
00111 *> \verbatim
00112 *>          X is REAL array, dimension (LDX,NRHS)
00113 *>          The computed solution vectors.  Each vector is stored as a
00114 *>          column of the matrix X.
00115 *> \endverbatim
00116 *>
00117 *> \param[in] LDX
00118 *> \verbatim
00119 *>          LDX is INTEGER
00120 *>          The leading dimension of the array X.  LDX >= max(1,N).
00121 *> \endverbatim
00122 *>
00123 *> \param[in] XACT
00124 *> \verbatim
00125 *>          XACT is REAL array, dimension (LDX,NRHS)
00126 *>          The exact solution vectors.  Each vector is stored as a
00127 *>          column of the matrix XACT.
00128 *> \endverbatim
00129 *>
00130 *> \param[in] LDXACT
00131 *> \verbatim
00132 *>          LDXACT is INTEGER
00133 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00134 *> \endverbatim
00135 *>
00136 *> \param[in] FERR
00137 *> \verbatim
00138 *>          FERR is REAL array, dimension (NRHS)
00139 *>          The estimated forward error bounds for each solution vector
00140 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00141 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00142 *>          of the largest entry in X.
00143 *> \endverbatim
00144 *>
00145 *> \param[in] BERR
00146 *> \verbatim
00147 *>          BERR is REAL array, dimension (NRHS)
00148 *>          The componentwise relative backward error of each solution
00149 *>          vector (i.e., the smallest relative change in any entry of A
00150 *>          or B that makes X an exact solution).
00151 *> \endverbatim
00152 *>
00153 *> \param[out] RESLTS
00154 *> \verbatim
00155 *>          RESLTS is REAL array, dimension (2)
00156 *>          The maximum over the NRHS solution vectors of the ratios:
00157 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00158 *>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00159 *> \endverbatim
00160 *
00161 *  Authors:
00162 *  ========
00163 *
00164 *> \author Univ. of Tennessee 
00165 *> \author Univ. of California Berkeley 
00166 *> \author Univ. of Colorado Denver 
00167 *> \author NAG Ltd. 
00168 *
00169 *> \date November 2011
00170 *
00171 *> \ingroup single_lin
00172 *
00173 *  =====================================================================
00174       SUBROUTINE STPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
00175      $                   XACT, LDXACT, FERR, BERR, RESLTS )
00176 *
00177 *  -- LAPACK test routine (version 3.4.0) --
00178 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00179 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00180 *     November 2011
00181 *
00182 *     .. Scalar Arguments ..
00183       CHARACTER          DIAG, TRANS, UPLO
00184       INTEGER            LDB, LDX, LDXACT, N, NRHS
00185 *     ..
00186 *     .. Array Arguments ..
00187       REAL               AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
00188      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
00189 *     ..
00190 *
00191 *  =====================================================================
00192 *
00193 *     .. Parameters ..
00194       REAL               ZERO, ONE
00195       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00196 *     ..
00197 *     .. Local Scalars ..
00198       LOGICAL            NOTRAN, UNIT, UPPER
00199       INTEGER            I, IFU, IMAX, J, JC, K
00200       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00201 *     ..
00202 *     .. External Functions ..
00203       LOGICAL            LSAME
00204       INTEGER            ISAMAX
00205       REAL               SLAMCH
00206       EXTERNAL           LSAME, ISAMAX, SLAMCH
00207 *     ..
00208 *     .. Intrinsic Functions ..
00209       INTRINSIC          ABS, MAX, MIN
00210 *     ..
00211 *     .. Executable Statements ..
00212 *
00213 *     Quick exit if N = 0 or NRHS = 0.
00214 *
00215       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00216          RESLTS( 1 ) = ZERO
00217          RESLTS( 2 ) = ZERO
00218          RETURN
00219       END IF
00220 *
00221       EPS = SLAMCH( 'Epsilon' )
00222       UNFL = SLAMCH( 'Safe minimum' )
00223       OVFL = ONE / UNFL
00224       UPPER = LSAME( UPLO, 'U' )
00225       NOTRAN = LSAME( TRANS, 'N' )
00226       UNIT = LSAME( DIAG, 'U' )
00227 *
00228 *     Test 1:  Compute the maximum of
00229 *        norm(X - XACT) / ( norm(X) * FERR )
00230 *     over all the vectors X and XACT using the infinity-norm.
00231 *
00232       ERRBND = ZERO
00233       DO 30 J = 1, NRHS
00234          IMAX = ISAMAX( N, X( 1, J ), 1 )
00235          XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
00236          DIFF = ZERO
00237          DO 10 I = 1, N
00238             DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
00239    10    CONTINUE
00240 *
00241          IF( XNORM.GT.ONE ) THEN
00242             GO TO 20
00243          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00244             GO TO 20
00245          ELSE
00246             ERRBND = ONE / EPS
00247             GO TO 30
00248          END IF
00249 *
00250    20    CONTINUE
00251          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00252             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00253          ELSE
00254             ERRBND = ONE / EPS
00255          END IF
00256    30 CONTINUE
00257       RESLTS( 1 ) = ERRBND
00258 *
00259 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00260 *     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00261 *
00262       IFU = 0
00263       IF( UNIT )
00264      $   IFU = 1
00265       DO 90 K = 1, NRHS
00266          DO 80 I = 1, N
00267             TMP = ABS( B( I, K ) )
00268             IF( UPPER ) THEN
00269                JC = ( ( I-1 )*I ) / 2
00270                IF( .NOT.NOTRAN ) THEN
00271                   DO 40 J = 1, I - IFU
00272                      TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) )
00273    40             CONTINUE
00274                   IF( UNIT )
00275      $               TMP = TMP + ABS( X( I, K ) )
00276                ELSE
00277                   JC = JC + I
00278                   IF( UNIT ) THEN
00279                      TMP = TMP + ABS( X( I, K ) )
00280                      JC = JC + I
00281                   END IF
00282                   DO 50 J = I + IFU, N
00283                      TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
00284                      JC = JC + J
00285    50             CONTINUE
00286                END IF
00287             ELSE
00288                IF( NOTRAN ) THEN
00289                   JC = I
00290                   DO 60 J = 1, I - IFU
00291                      TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
00292                      JC = JC + N - J
00293    60             CONTINUE
00294                   IF( UNIT )
00295      $               TMP = TMP + ABS( X( I, K ) )
00296                ELSE
00297                   JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
00298                   IF( UNIT )
00299      $               TMP = TMP + ABS( X( I, K ) )
00300                   DO 70 J = I + IFU, N
00301                      TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) )
00302    70             CONTINUE
00303                END IF
00304             END IF
00305             IF( I.EQ.1 ) THEN
00306                AXBI = TMP
00307             ELSE
00308                AXBI = MIN( AXBI, TMP )
00309             END IF
00310    80    CONTINUE
00311          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00312      $         MAX( AXBI, ( N+1 )*UNFL ) )
00313          IF( K.EQ.1 ) THEN
00314             RESLTS( 2 ) = TMP
00315          ELSE
00316             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00317          END IF
00318    90 CONTINUE
00319 *
00320       RETURN
00321 *
00322 *     End of STPT05
00323 *
00324       END
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