*> \brief \b SPOEQUB * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SPOEQUB + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, N * REAL AMAX, SCOND * .. * .. Array Arguments .. * REAL A( LDA, * ), S( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SPOEQUB computes row and column scalings intended to equilibrate a *> symmetric positive definite matrix A 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. *> *> This routine differs from SPOEQU by restricting the scaling factors *> to a power of the radix. Barring over- and underflow, scaling by *> these factors introduces no additional rounding errors. However, the *> scaled diagonal entries are no longer approximately 1 but lie *> between sqrt(radix) and 1/sqrt(radix). *> \endverbatim * * Arguments: * ========== * *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is REAL array, dimension (LDA,N) *> The N-by-N symmetric positive definite matrix whose scaling *> factors are to be computed. Only the diagonal elements of A *> are referenced. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] S *> \verbatim *> S is REAL array, dimension (N) *> If INFO = 0, S contains the scale factors for A. *> \endverbatim *> *> \param[out] SCOND *> \verbatim *> SCOND is REAL *> 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. *> \endverbatim *> *> \param[out] AMAX *> \verbatim *> AMAX is REAL *> Absolute value of largest matrix element. If AMAX is very *> close to overflow or very close to underflow, the matrix *> should be scaled. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup realPOcomputational * * ===================================================================== SUBROUTINE SPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER INFO, LDA, N REAL AMAX, SCOND * .. * .. Array Arguments .. REAL A( LDA, * ), S( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. INTEGER I REAL SMIN, BASE, TMP * .. * .. External Functions .. REAL SLAMCH EXTERNAL SLAMCH * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, SQRT, LOG, INT * .. * .. Executable Statements .. * * Test the input parameters. * * Positive definite only performs 1 pass of equilibration. * INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -3 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SPOEQUB', -INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF BASE = SLAMCH( 'B' ) TMP = -0.5 / LOG ( BASE ) * * Find the minimum and maximum diagonal elements. * S( 1 ) = A( 1, 1 ) SMIN = S( 1 ) AMAX = S( 1 ) DO 10 I = 2, N S( I ) = A( I, I ) SMIN = MIN( SMIN, S( I ) ) AMAX = MAX( AMAX, S( I ) ) 10 CONTINUE * IF( SMIN.LE.ZERO ) THEN * * Find the first non-positive diagonal element and return. * DO 20 I = 1, N IF( S( I ).LE.ZERO ) THEN INFO = I RETURN END IF 20 CONTINUE ELSE * * Set the scale factors to the reciprocals * of the diagonal elements. * DO 30 I = 1, N S( I ) = BASE ** INT( TMP * LOG( S( I ) ) ) 30 CONTINUE * * Compute SCOND = min(S(I)) / max(S(I)). * SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF * RETURN * * End of SPOEQUB * END