LAPACK 3.3.1 Linear Algebra PACKage

# sget07.f

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```00001       SUBROUTINE SGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
00002      \$                   LDXACT, FERR, CHKFERR, 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          TRANS
00010       LOGICAL            CHKFERR
00011       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
00012 *     ..
00013 *     .. Array Arguments ..
00014       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
00015      \$                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  SGET07 tests the error bounds from iterative refinement for the
00022 *  computed solution to a system of equations op(A)*X = B, where A is a
00023 *  general n by n matrix and op(A) = A or A**T, depending on TRANS.
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(op(A))*abs(X) +abs(b))_i )
00033 *
00034 *  Arguments
00035 *  =========
00036 *
00037 *  TRANS   (input) CHARACTER*1
00038 *          Specifies the form of the system of equations.
00039 *          = 'N':  A * X = B     (No transpose)
00040 *          = 'T':  A**T * X = B  (Transpose)
00041 *          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
00042 *
00043 *  N       (input) INTEGER
00044 *          The number of rows of the matrices X and XACT.  N >= 0.
00045 *
00046 *  NRHS    (input) INTEGER
00047 *          The number of columns of the matrices X and XACT.  NRHS >= 0.
00048 *
00049 *  A       (input) REAL array, dimension (LDA,N)
00050 *          The original n by n matrix A.
00051 *
00052 *  LDA     (input) INTEGER
00053 *          The leading dimension of the array A.  LDA >= max(1,N).
00054 *
00055 *  B       (input) REAL array, dimension (LDB,NRHS)
00056 *          The right hand side vectors for the system of linear
00057 *          equations.
00058 *
00059 *  LDB     (input) INTEGER
00060 *          The leading dimension of the array B.  LDB >= max(1,N).
00061 *
00062 *  X       (input) REAL array, dimension (LDX,NRHS)
00063 *          The computed solution vectors.  Each vector is stored as a
00064 *          column of the matrix X.
00065 *
00066 *  LDX     (input) INTEGER
00067 *          The leading dimension of the array X.  LDX >= max(1,N).
00068 *
00069 *  XACT    (input) REAL array, dimension (LDX,NRHS)
00070 *          The exact solution vectors.  Each vector is stored as a
00071 *          column of the matrix XACT.
00072 *
00073 *  LDXACT  (input) INTEGER
00074 *          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00075 *
00076 *  FERR    (input) REAL array, dimension (NRHS)
00077 *          The estimated forward error bounds for each solution vector
00078 *          X.  If XTRUE is the true solution, FERR bounds the magnitude
00079 *          of the largest entry in (X - XTRUE) divided by the magnitude
00080 *          of the largest entry in X.
00081 *
00082 *  CHKFERR (input) LOGICAL
00083 *          Set to .TRUE. to check FERR, .FALSE. not to check FERR.
00084 *          When the test system is ill-conditioned, the "true"
00085 *          solution in XACT may be incorrect.
00086 *
00087 *  BERR    (input) REAL array, dimension (NRHS)
00088 *          The componentwise relative backward error of each solution
00089 *          vector (i.e., the smallest relative change in any entry of A
00090 *          or B that makes X an exact solution).
00091 *
00092 *  RESLTS  (output) REAL array, dimension (2)
00093 *          The maximum over the NRHS solution vectors of the ratios:
00094 *          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00095 *          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00096 *
00097 *  =====================================================================
00098 *
00099 *     .. Parameters ..
00100       REAL               ZERO, ONE
00101       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00102 *     ..
00103 *     .. Local Scalars ..
00104       LOGICAL            NOTRAN
00105       INTEGER            I, IMAX, J, K
00106       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00107 *     ..
00108 *     .. External Functions ..
00109       LOGICAL            LSAME
00110       INTEGER            ISAMAX
00111       REAL               SLAMCH
00112       EXTERNAL           LSAME, ISAMAX, SLAMCH
00113 *     ..
00114 *     .. Intrinsic Functions ..
00115       INTRINSIC          ABS, MAX, MIN
00116 *     ..
00117 *     .. Executable Statements ..
00118 *
00119 *     Quick exit if N = 0 or NRHS = 0.
00120 *
00121       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00122          RESLTS( 1 ) = ZERO
00123          RESLTS( 2 ) = ZERO
00124          RETURN
00125       END IF
00126 *
00127       EPS = SLAMCH( 'Epsilon' )
00128       UNFL = SLAMCH( 'Safe minimum' )
00129       OVFL = ONE / UNFL
00130       NOTRAN = LSAME( TRANS, 'N' )
00131 *
00132 *     Test 1:  Compute the maximum of
00133 *        norm(X - XACT) / ( norm(X) * FERR )
00134 *     over all the vectors X and XACT using the infinity-norm.
00135 *
00136       ERRBND = ZERO
00137       IF( CHKFERR ) THEN
00138          DO 30 J = 1, NRHS
00139             IMAX = ISAMAX( N, X( 1, J ), 1 )
00140             XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
00141             DIFF = ZERO
00142             DO 10 I = 1, N
00143                DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
00144  10         CONTINUE
00145 *
00146             IF( XNORM.GT.ONE ) THEN
00147                GO TO 20
00148             ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00149                GO TO 20
00150             ELSE
00151                ERRBND = ONE / EPS
00152                GO TO 30
00153             END IF
00154 *
00155  20         CONTINUE
00156             IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00157                ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00158             ELSE
00159                ERRBND = ONE / EPS
00160             END IF
00161  30      CONTINUE
00162       END IF
00163       RESLTS( 1 ) = ERRBND
00164 *
00165 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00166 *     (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
00167 *
00168       DO 70 K = 1, NRHS
00169          DO 60 I = 1, N
00170             TMP = ABS( B( I, K ) )
00171             IF( NOTRAN ) THEN
00172                DO 40 J = 1, N
00173                   TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
00174    40          CONTINUE
00175             ELSE
00176                DO 50 J = 1, N
00177                   TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
00178    50          CONTINUE
00179             END IF
00180             IF( I.EQ.1 ) THEN
00181                AXBI = TMP
00182             ELSE
00183                AXBI = MIN( AXBI, TMP )
00184             END IF
00185    60    CONTINUE
00186          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00187      \$         MAX( AXBI, ( N+1 )*UNFL ) )
00188          IF( K.EQ.1 ) THEN
00189             RESLTS( 2 ) = TMP
00190          ELSE
00191             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00192          END IF
00193    70 CONTINUE
00194 *
00195       RETURN
00196 *
00197 *     End of SGET07
00198 *
00199       END
```