LAPACK 3.3.1
Linear Algebra PACKage

sppequ.f

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