SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, N DOUBLE PRECISION AMAX, SCOND * .. * .. Array Arguments .. DOUBLE PRECISION AP( * ), S( * ) * .. * * Purpose * ======= * * DPPEQU computes row and column scalings intended to equilibrate a * symmetric positive definite matrix A in packed storage and reduce * its condition number (with respect to the two-norm). S contains the * scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix * B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal. * This choice of S puts the condition number of B within a factor N of * the smallest possible condition number over all possible diagonal * scalings. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) * The upper or lower triangle of the symmetric matrix A, packed * columnwise in a linear array. The j-th column of A is stored * in the array AP as follows: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. * * S (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, S contains the scale factors for A. * * SCOND (output) DOUBLE PRECISION * If INFO = 0, S contains the ratio of the smallest S(i) to * the largest S(i). If SCOND >= 0.1 and AMAX is neither too * large nor too small, it is not worth scaling by S. * * AMAX (output) DOUBLE PRECISION * Absolute value of largest matrix element. If AMAX is very * close to overflow or very close to underflow, the matrix * should be scaled. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the i-th diagonal element is nonpositive. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I, JJ DOUBLE PRECISION SMIN * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 UPPER = LSAME( UPLO, 'U' ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DPPEQU', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF * * Initialize SMIN and AMAX. * S( 1 ) = AP( 1 ) SMIN = S( 1 ) AMAX = S( 1 ) * IF( UPPER ) THEN * * UPLO = 'U': Upper triangle of A is stored. * Find the minimum and maximum diagonal elements. * JJ = 1 DO 10 I = 2, N JJ = JJ + I S( I ) = AP( JJ ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 10 CONTINUE * ELSE * * UPLO = 'L': Lower triangle of A is stored. * Find the minimum and maximum diagonal elements. * JJ = 1 DO 20 I = 2, N JJ = JJ + N - I + 2 S( I ) = AP( JJ ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 20 CONTINUE END IF * IF( SMIN.LE.ZERO ) THEN * * Find the first non-positive diagonal element and return. * DO 30 I = 1, N IF( S( I ).LE.ZERO ) THEN INFO = I RETURN END IF 30 CONTINUE ELSE * * Set the scale factors to the reciprocals * of the diagonal elements. * DO 40 I = 1, N S( I ) = ONE / SQRT( S( I ) ) 40 CONTINUE * * Compute SCOND = min(S(I)) / max(S(I)) * SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF RETURN * * End of DPPEQU * END