LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sla_porpvgrw.f
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00001 *> \brief \b SLA_PORPVGRW
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SLA_PORPVGRW + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       REAL FUNCTION SLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF, LDAF, WORK )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER*1        UPLO
00025 *       INTEGER            NCOLS, LDA, LDAF
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       REAL               A( LDA, * ), AF( LDAF, * ), WORK( * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> 
00038 *> SLA_PORPVGRW computes the reciprocal pivot growth factor
00039 *> norm(A)/norm(U). The "max absolute element" norm is used. If this is
00040 *> much less than 1, the stability of the LU factorization of the
00041 *> (equilibrated) matrix A could be poor. This also means that the
00042 *> solution X, estimated condition numbers, and error bounds could be
00043 *> unreliable.
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] UPLO
00050 *> \verbatim
00051 *>          UPLO is CHARACTER*1
00052 *>       = 'U':  Upper triangle of A is stored;
00053 *>       = 'L':  Lower triangle of A is stored.
00054 *> \endverbatim
00055 *>
00056 *> \param[in] NCOLS
00057 *> \verbatim
00058 *>          NCOLS is INTEGER
00059 *>     The number of columns of the matrix A. NCOLS >= 0.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] A
00063 *> \verbatim
00064 *>          A is REAL array, dimension (LDA,N)
00065 *>     On entry, the N-by-N matrix A.
00066 *> \endverbatim
00067 *>
00068 *> \param[in] LDA
00069 *> \verbatim
00070 *>          LDA is INTEGER
00071 *>     The leading dimension of the array A.  LDA >= max(1,N).
00072 *> \endverbatim
00073 *>
00074 *> \param[in] AF
00075 *> \verbatim
00076 *>          AF is REAL array, dimension (LDAF,N)
00077 *>     The triangular factor U or L from the Cholesky factorization
00078 *>     A = U**T*U or A = L*L**T, as computed by SPOTRF.
00079 *> \endverbatim
00080 *>
00081 *> \param[in] LDAF
00082 *> \verbatim
00083 *>          LDAF is INTEGER
00084 *>     The leading dimension of the array AF.  LDAF >= max(1,N).
00085 *> \endverbatim
00086 *>
00087 *> \param[in] WORK
00088 *> \verbatim
00089 *>          WORK is REAL array, dimension (2*N)
00090 *> \endverbatim
00091 *
00092 *  Authors:
00093 *  ========
00094 *
00095 *> \author Univ. of Tennessee 
00096 *> \author Univ. of California Berkeley 
00097 *> \author Univ. of Colorado Denver 
00098 *> \author NAG Ltd. 
00099 *
00100 *> \date November 2011
00101 *
00102 *> \ingroup realPOcomputational
00103 *
00104 *  =====================================================================
00105       REAL FUNCTION SLA_PORPVGRW( UPLO, NCOLS, A, LDA, AF, LDAF, WORK )
00106 *
00107 *  -- LAPACK computational routine (version 3.4.0) --
00108 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00109 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00110 *     November 2011
00111 *
00112 *     .. Scalar Arguments ..
00113       CHARACTER*1        UPLO
00114       INTEGER            NCOLS, LDA, LDAF
00115 *     ..
00116 *     .. Array Arguments ..
00117       REAL               A( LDA, * ), AF( LDAF, * ), WORK( * )
00118 *     ..
00119 *
00120 *  =====================================================================
00121 *
00122 *     .. Local Scalars ..
00123       INTEGER            I, J
00124       REAL               AMAX, UMAX, RPVGRW
00125       LOGICAL            UPPER
00126 *     ..
00127 *     .. Intrinsic Functions ..
00128       INTRINSIC          ABS, MAX, MIN
00129 *     ..
00130 *     .. External Functions ..
00131       EXTERNAL           LSAME, SLASET
00132       LOGICAL            LSAME
00133 *     ..
00134 *     .. Executable Statements ..
00135 *
00136       UPPER = LSAME( 'Upper', UPLO )
00137 *
00138 *     SPOTRF will have factored only the NCOLSxNCOLS leading minor, so
00139 *     we restrict the growth search to that minor and use only the first
00140 *     2*NCOLS workspace entries.
00141 *
00142       RPVGRW = 1.0
00143       DO I = 1, 2*NCOLS
00144          WORK( I ) = 0.0
00145       END DO
00146 *
00147 *     Find the max magnitude entry of each column.
00148 *
00149       IF ( UPPER ) THEN
00150          DO J = 1, NCOLS
00151             DO I = 1, J
00152                WORK( NCOLS+J ) =
00153      $              MAX( ABS( A( I, J ) ), WORK( NCOLS+J ) )
00154             END DO
00155          END DO
00156       ELSE
00157          DO J = 1, NCOLS
00158             DO I = J, NCOLS
00159                WORK( NCOLS+J ) =
00160      $              MAX( ABS( A( I, J ) ), WORK( NCOLS+J ) )
00161             END DO
00162          END DO
00163       END IF
00164 *
00165 *     Now find the max magnitude entry of each column of the factor in
00166 *     AF.  No pivoting, so no permutations.
00167 *
00168       IF ( LSAME( 'Upper', UPLO ) ) THEN
00169          DO J = 1, NCOLS
00170             DO I = 1, J
00171                WORK( J ) = MAX( ABS( AF( I, J ) ), WORK( J ) )
00172             END DO
00173          END DO
00174       ELSE
00175          DO J = 1, NCOLS
00176             DO I = J, NCOLS
00177                WORK( J ) = MAX( ABS( AF( I, J ) ), WORK( J ) )
00178             END DO
00179          END DO
00180       END IF
00181 *
00182 *     Compute the *inverse* of the max element growth factor.  Dividing
00183 *     by zero would imply the largest entry of the factor's column is
00184 *     zero.  Than can happen when either the column of A is zero or
00185 *     massive pivots made the factor underflow to zero.  Neither counts
00186 *     as growth in itself, so simply ignore terms with zero
00187 *     denominators.
00188 *
00189       IF ( LSAME( 'Upper', UPLO ) ) THEN
00190          DO I = 1, NCOLS
00191             UMAX = WORK( I )
00192             AMAX = WORK( NCOLS+I )
00193             IF ( UMAX /= 0.0 ) THEN
00194                RPVGRW = MIN( AMAX / UMAX, RPVGRW )
00195             END IF
00196          END DO
00197       ELSE
00198          DO I = 1, NCOLS
00199             UMAX = WORK( I )
00200             AMAX = WORK( NCOLS+I )
00201             IF ( UMAX /= 0.0 ) THEN
00202                RPVGRW = MIN( AMAX / UMAX, RPVGRW )
00203             END IF
00204          END DO
00205       END IF
00206 
00207       SLA_PORPVGRW = RPVGRW
00208       END
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