*> \brief \b SPBEQU * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SPBEQU + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, KD, LDAB, N * REAL AMAX, SCOND * .. * .. Array Arguments .. * REAL AB( LDAB, * ), S( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SPBEQU computes row and column scalings intended to equilibrate a *> symmetric positive definite band 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': Upper triangular of A is stored; *> = 'L': Lower triangular of A is stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] KD *> \verbatim *> KD is INTEGER *> The number of superdiagonals of the matrix A if UPLO = 'U', *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. *> \endverbatim *> *> \param[in] AB *> \verbatim *> AB is REAL array, dimension (LDAB,N) *> The upper or lower triangle of the symmetric band matrix A, *> stored in the first KD+1 rows of the array. The j-th column *> of A is stored in the j-th column of the array AB as follows: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array A. LDAB >= KD+1. *> \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 realOTHERcomputational * * ===================================================================== SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, 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 .. CHARACTER UPLO INTEGER INFO, KD, LDAB, N REAL AMAX, SCOND * .. * .. Array Arguments .. REAL AB( LDAB, * ), S( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) * .. * .. Local Scalars .. LOGICAL UPPER INTEGER I, J REAL 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 ELSE IF( KD.LT.0 ) THEN INFO = -3 ELSE IF( LDAB.LT.KD+1 ) THEN INFO = -5 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SPBEQU', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SCOND = ONE AMAX = ZERO RETURN END IF * IF( UPPER ) THEN J = KD + 1 ELSE J = 1 END IF * * Initialize SMIN and AMAX. * S( 1 ) = AB( J, 1 ) SMIN = S( 1 ) AMAX = S( 1 ) * * Find the minimum and maximum diagonal elements. * DO 10 I = 2, N S( I ) = AB( J, 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 ) = ONE / SQRT( S( I ) ) 30 CONTINUE * * Compute SCOND = min(S(I)) / max(S(I)) * SCOND = SQRT( SMIN ) / SQRT( AMAX ) END IF RETURN * * End of SPBEQU * END