LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cppequ.f
Go to the documentation of this file.
00001 *> \brief \b CPPEQU
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CPPEQU + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cppequ.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cppequ.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cppequ.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, N
00026 *       REAL               AMAX, SCOND
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       REAL               S( * )
00030 *       COMPLEX            AP( * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *> CPPEQU computes row and column scalings intended to equilibrate a
00040 *> Hermitian positive definite matrix A in packed storage and reduce
00041 *> its condition number (with respect to the two-norm).  S contains the
00042 *> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
00043 *> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
00044 *> This choice of S puts the condition number of B within a factor N of
00045 *> the smallest possible condition number over all possible diagonal
00046 *> scalings.
00047 *> \endverbatim
00048 *
00049 *  Arguments:
00050 *  ==========
00051 *
00052 *> \param[in] UPLO
00053 *> \verbatim
00054 *>          UPLO is CHARACTER*1
00055 *>          = 'U':  Upper triangle of A is stored;
00056 *>          = 'L':  Lower triangle of A is stored.
00057 *> \endverbatim
00058 *>
00059 *> \param[in] N
00060 *> \verbatim
00061 *>          N is INTEGER
00062 *>          The order of the matrix A.  N >= 0.
00063 *> \endverbatim
00064 *>
00065 *> \param[in] AP
00066 *> \verbatim
00067 *>          AP is COMPLEX array, dimension (N*(N+1)/2)
00068 *>          The upper or lower triangle of the Hermitian matrix A, packed
00069 *>          columnwise in a linear array.  The j-th column of A is stored
00070 *>          in the array AP as follows:
00071 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00072 *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00073 *> \endverbatim
00074 *>
00075 *> \param[out] S
00076 *> \verbatim
00077 *>          S is REAL array, dimension (N)
00078 *>          If INFO = 0, S contains the scale factors for A.
00079 *> \endverbatim
00080 *>
00081 *> \param[out] SCOND
00082 *> \verbatim
00083 *>          SCOND is REAL
00084 *>          If INFO = 0, S contains the ratio of the smallest S(i) to
00085 *>          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
00086 *>          large nor too small, it is not worth scaling by S.
00087 *> \endverbatim
00088 *>
00089 *> \param[out] AMAX
00090 *> \verbatim
00091 *>          AMAX is REAL
00092 *>          Absolute value of largest matrix element.  If AMAX is very
00093 *>          close to overflow or very close to underflow, the matrix
00094 *>          should be scaled.
00095 *> \endverbatim
00096 *>
00097 *> \param[out] INFO
00098 *> \verbatim
00099 *>          INFO is INTEGER
00100 *>          = 0:  successful exit
00101 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00102 *>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
00103 *> \endverbatim
00104 *
00105 *  Authors:
00106 *  ========
00107 *
00108 *> \author Univ. of Tennessee 
00109 *> \author Univ. of California Berkeley 
00110 *> \author Univ. of Colorado Denver 
00111 *> \author NAG Ltd. 
00112 *
00113 *> \date November 2011
00114 *
00115 *> \ingroup complexOTHERcomputational
00116 *
00117 *  =====================================================================
00118       SUBROUTINE CPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
00119 *
00120 *  -- LAPACK computational routine (version 3.4.0) --
00121 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00122 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00123 *     November 2011
00124 *
00125 *     .. Scalar Arguments ..
00126       CHARACTER          UPLO
00127       INTEGER            INFO, N
00128       REAL               AMAX, SCOND
00129 *     ..
00130 *     .. Array Arguments ..
00131       REAL               S( * )
00132       COMPLEX            AP( * )
00133 *     ..
00134 *
00135 *  =====================================================================
00136 *
00137 *     .. Parameters ..
00138       REAL               ONE, ZERO
00139       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00140 *     ..
00141 *     .. Local Scalars ..
00142       LOGICAL            UPPER
00143       INTEGER            I, JJ
00144       REAL               SMIN
00145 *     ..
00146 *     .. External Functions ..
00147       LOGICAL            LSAME
00148       EXTERNAL           LSAME
00149 *     ..
00150 *     .. External Subroutines ..
00151       EXTERNAL           XERBLA
00152 *     ..
00153 *     .. Intrinsic Functions ..
00154       INTRINSIC          MAX, MIN, REAL, SQRT
00155 *     ..
00156 *     .. Executable Statements ..
00157 *
00158 *     Test the input parameters.
00159 *
00160       INFO = 0
00161       UPPER = LSAME( UPLO, 'U' )
00162       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00163          INFO = -1
00164       ELSE IF( N.LT.0 ) THEN
00165          INFO = -2
00166       END IF
00167       IF( INFO.NE.0 ) THEN
00168          CALL XERBLA( 'CPPEQU', -INFO )
00169          RETURN
00170       END IF
00171 *
00172 *     Quick return if possible
00173 *
00174       IF( N.EQ.0 ) THEN
00175          SCOND = ONE
00176          AMAX = ZERO
00177          RETURN
00178       END IF
00179 *
00180 *     Initialize SMIN and AMAX.
00181 *
00182       S( 1 ) = REAL( AP( 1 ) )
00183       SMIN = S( 1 )
00184       AMAX = S( 1 )
00185 *
00186       IF( UPPER ) THEN
00187 *
00188 *        UPLO = 'U':  Upper triangle of A is stored.
00189 *        Find the minimum and maximum diagonal elements.
00190 *
00191          JJ = 1
00192          DO 10 I = 2, N
00193             JJ = JJ + I
00194             S( I ) = REAL( AP( JJ ) )
00195             SMIN = MIN( SMIN, S( I ) )
00196             AMAX = MAX( AMAX, S( I ) )
00197    10    CONTINUE
00198 *
00199       ELSE
00200 *
00201 *        UPLO = 'L':  Lower triangle of A is stored.
00202 *        Find the minimum and maximum diagonal elements.
00203 *
00204          JJ = 1
00205          DO 20 I = 2, N
00206             JJ = JJ + N - I + 2
00207             S( I ) = REAL( AP( JJ ) )
00208             SMIN = MIN( SMIN, S( I ) )
00209             AMAX = MAX( AMAX, S( I ) )
00210    20    CONTINUE
00211       END IF
00212 *
00213       IF( SMIN.LE.ZERO ) THEN
00214 *
00215 *        Find the first non-positive diagonal element and return.
00216 *
00217          DO 30 I = 1, N
00218             IF( S( I ).LE.ZERO ) THEN
00219                INFO = I
00220                RETURN
00221             END IF
00222    30    CONTINUE
00223       ELSE
00224 *
00225 *        Set the scale factors to the reciprocals
00226 *        of the diagonal elements.
00227 *
00228          DO 40 I = 1, N
00229             S( I ) = ONE / SQRT( S( I ) )
00230    40    CONTINUE
00231 *
00232 *        Compute SCOND = min(S(I)) / max(S(I))
00233 *
00234          SCOND = SQRT( SMIN ) / SQRT( AMAX )
00235       END IF
00236       RETURN
00237 *
00238 *     End of CPPEQU
00239 *
00240       END
 All Files Functions