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*> \brief \b DPBEQU
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DPBEQU + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpbequ.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dpbequ.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dpbequ.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, KD, LDAB, N
* DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AB( LDAB, * ), S( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DPBEQU 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
*> If INFO = 0, S contains the scale factors for A.
*> \endverbatim
*>
*> \param[out] SCOND
*> \verbatim
*> SCOND is 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.
*> \endverbatim
*>
*> \param[out] AMAX
*> \verbatim
*> AMAX is 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.
*> \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.
*
*> \date December 2016
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
*
* -- LAPACK computational routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION AMAX, SCOND
* ..
* .. Array Arguments ..
DOUBLE PRECISION AB( LDAB, * ), S( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER I, J
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
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( 'DPBEQU', -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 DPBEQU
*
END
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