LAPACK 3.3.1
Linear Algebra PACKage

cppequ.f

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00001       SUBROUTINE CPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.2) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       CHARACTER          UPLO
00010       INTEGER            INFO, N
00011       REAL               AMAX, SCOND
00012 *     ..
00013 *     .. Array Arguments ..
00014       REAL               S( * )
00015       COMPLEX            AP( * )
00016 *     ..
00017 *
00018 *  Purpose
00019 *  =======
00020 *
00021 *  CPPEQU computes row and column scalings intended to equilibrate a
00022 *  Hermitian positive definite matrix A in packed storage and reduce
00023 *  its condition number (with respect to the two-norm).  S contains the
00024 *  scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
00025 *  B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
00026 *  This choice of S puts the condition number of B within a factor N of
00027 *  the smallest possible condition number over all possible diagonal
00028 *  scalings.
00029 *
00030 *  Arguments
00031 *  =========
00032 *
00033 *  UPLO    (input) CHARACTER*1
00034 *          = 'U':  Upper triangle of A is stored;
00035 *          = 'L':  Lower triangle of A is stored.
00036 *
00037 *  N       (input) INTEGER
00038 *          The order of the matrix A.  N >= 0.
00039 *
00040 *  AP      (input) COMPLEX array, dimension (N*(N+1)/2)
00041 *          The upper or lower triangle of the Hermitian matrix A, packed
00042 *          columnwise in a linear array.  The j-th column of A is stored
00043 *          in the array AP as follows:
00044 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00045 *          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00046 *
00047 *  S       (output) REAL array, dimension (N)
00048 *          If INFO = 0, S contains the scale factors for A.
00049 *
00050 *  SCOND   (output) REAL
00051 *          If INFO = 0, S contains the ratio of the smallest S(i) to
00052 *          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too
00053 *          large nor too small, it is not worth scaling by S.
00054 *
00055 *  AMAX    (output) REAL
00056 *          Absolute value of largest matrix element.  If AMAX is very
00057 *          close to overflow or very close to underflow, the matrix
00058 *          should be scaled.
00059 *
00060 *  INFO    (output) INTEGER
00061 *          = 0:  successful exit
00062 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00063 *          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
00064 *
00065 *  =====================================================================
00066 *
00067 *     .. Parameters ..
00068       REAL               ONE, ZERO
00069       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00070 *     ..
00071 *     .. Local Scalars ..
00072       LOGICAL            UPPER
00073       INTEGER            I, JJ
00074       REAL               SMIN
00075 *     ..
00076 *     .. External Functions ..
00077       LOGICAL            LSAME
00078       EXTERNAL           LSAME
00079 *     ..
00080 *     .. External Subroutines ..
00081       EXTERNAL           XERBLA
00082 *     ..
00083 *     .. Intrinsic Functions ..
00084       INTRINSIC          MAX, MIN, REAL, SQRT
00085 *     ..
00086 *     .. Executable Statements ..
00087 *
00088 *     Test the input parameters.
00089 *
00090       INFO = 0
00091       UPPER = LSAME( UPLO, 'U' )
00092       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00093          INFO = -1
00094       ELSE IF( N.LT.0 ) THEN
00095          INFO = -2
00096       END IF
00097       IF( INFO.NE.0 ) THEN
00098          CALL XERBLA( 'CPPEQU', -INFO )
00099          RETURN
00100       END IF
00101 *
00102 *     Quick return if possible
00103 *
00104       IF( N.EQ.0 ) THEN
00105          SCOND = ONE
00106          AMAX = ZERO
00107          RETURN
00108       END IF
00109 *
00110 *     Initialize SMIN and AMAX.
00111 *
00112       S( 1 ) = REAL( AP( 1 ) )
00113       SMIN = S( 1 )
00114       AMAX = S( 1 )
00115 *
00116       IF( UPPER ) THEN
00117 *
00118 *        UPLO = 'U':  Upper triangle of A is stored.
00119 *        Find the minimum and maximum diagonal elements.
00120 *
00121          JJ = 1
00122          DO 10 I = 2, N
00123             JJ = JJ + I
00124             S( I ) = REAL( AP( JJ ) )
00125             SMIN = MIN( SMIN, S( I ) )
00126             AMAX = MAX( AMAX, S( I ) )
00127    10    CONTINUE
00128 *
00129       ELSE
00130 *
00131 *        UPLO = 'L':  Lower triangle of A is stored.
00132 *        Find the minimum and maximum diagonal elements.
00133 *
00134          JJ = 1
00135          DO 20 I = 2, N
00136             JJ = JJ + N - I + 2
00137             S( I ) = REAL( AP( JJ ) )
00138             SMIN = MIN( SMIN, S( I ) )
00139             AMAX = MAX( AMAX, S( I ) )
00140    20    CONTINUE
00141       END IF
00142 *
00143       IF( SMIN.LE.ZERO ) THEN
00144 *
00145 *        Find the first non-positive diagonal element and return.
00146 *
00147          DO 30 I = 1, N
00148             IF( S( I ).LE.ZERO ) THEN
00149                INFO = I
00150                RETURN
00151             END IF
00152    30    CONTINUE
00153       ELSE
00154 *
00155 *        Set the scale factors to the reciprocals
00156 *        of the diagonal elements.
00157 *
00158          DO 40 I = 1, N
00159             S( I ) = ONE / SQRT( S( I ) )
00160    40    CONTINUE
00161 *
00162 *        Compute SCOND = min(S(I)) / max(S(I))
00163 *
00164          SCOND = SQRT( SMIN ) / SQRT( AMAX )
00165       END IF
00166       RETURN
00167 *
00168 *     End of CPPEQU
00169 *
00170       END
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