414 lines
11 KiB
Fortran
414 lines
11 KiB
Fortran
*> \brief \b SGEBAL
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download SGEBAL + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgebal.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgebal.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgebal.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER JOB
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* INTEGER IHI, ILO, INFO, LDA, N
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), SCALE( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> SGEBAL balances a general real matrix A. This involves, first,
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*> permuting A by a similarity transformation to isolate eigenvalues
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*> in the first 1 to ILO-1 and last IHI+1 to N elements on the
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*> diagonal; and second, applying a diagonal similarity transformation
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*> to rows and columns ILO to IHI to make the rows and columns as
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*> close in norm as possible. Both steps are optional.
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*>
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*> Balancing may reduce the 1-norm of the matrix, and improve the
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*> accuracy of the computed eigenvalues and/or eigenvectors.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] JOB
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*> \verbatim
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*> JOB is CHARACTER*1
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*> Specifies the operations to be performed on A:
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*> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
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*> for i = 1,...,N;
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*> = 'P': permute only;
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*> = 'S': scale only;
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*> = 'B': both permute and scale.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The order of the matrix A. N >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is REAL array, dimension (LDA,N)
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*> On entry, the input matrix A.
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*> On exit, A is overwritten by the balanced matrix.
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*> If JOB = 'N', A is not referenced.
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*> See Further Details.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[out] ILO
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*> \verbatim
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*> ILO is INTEGER
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*> \endverbatim
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*> \param[out] IHI
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*> \verbatim
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*> IHI is INTEGER
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*> ILO and IHI are set to integers such that on exit
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*> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
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*> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
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*> \endverbatim
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*>
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*> \param[out] SCALE
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*> \verbatim
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*> SCALE is REAL array, dimension (N)
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*> Details of the permutations and scaling factors applied to
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*> A. If P(j) is the index of the row and column interchanged
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*> with row and column j and D(j) is the scaling factor
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*> applied to row and column j, then
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*> SCALE(j) = P(j) for j = 1,...,ILO-1
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*> = D(j) for j = ILO,...,IHI
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*> = P(j) for j = IHI+1,...,N.
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*> The order in which the interchanges are made is N to IHI+1,
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*> then 1 to ILO-1.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit.
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*> < 0: if INFO = -i, the i-th argument had an illegal value.
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup realGEcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> The permutations consist of row and column interchanges which put
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*> the matrix in the form
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*>
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*> ( T1 X Y )
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*> P A P = ( 0 B Z )
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*> ( 0 0 T2 )
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*>
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*> where T1 and T2 are upper triangular matrices whose eigenvalues lie
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*> along the diagonal. The column indices ILO and IHI mark the starting
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*> and ending columns of the submatrix B. Balancing consists of applying
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*> a diagonal similarity transformation inv(D) * B * D to make the
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*> 1-norms of each row of B and its corresponding column nearly equal.
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*> The output matrix is
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*>
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*> ( T1 X*D Y )
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*> ( 0 inv(D)*B*D inv(D)*Z ).
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*> ( 0 0 T2 )
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*>
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*> Information about the permutations P and the diagonal matrix D is
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*> returned in the vector SCALE.
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*>
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*> This subroutine is based on the EISPACK routine BALANC.
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*>
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*> Modified by Tzu-Yi Chen, Computer Science Division, University of
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*> California at Berkeley, USA
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*>
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*> Refactored by Evert Provoost, Department of Computer Science,
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*> KU Leuven, Belgium
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE SGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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CHARACTER JOB
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INTEGER IHI, ILO, INFO, LDA, N
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), SCALE( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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REAL SCLFAC
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PARAMETER ( SCLFAC = 2.0E+0 )
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REAL FACTOR
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PARAMETER ( FACTOR = 0.95E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL NOCONV, CANSWAP
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INTEGER I, ICA, IRA, J, K, L
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REAL C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
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$ SFMIN2
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* ..
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* .. External Functions ..
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LOGICAL SISNAN, LSAME
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INTEGER ISAMAX
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REAL SLAMCH, SNRM2
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EXTERNAL SISNAN, LSAME, ISAMAX, SLAMCH, SNRM2
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* ..
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* .. External Subroutines ..
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EXTERNAL SSCAL, SSWAP, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, MAX, MIN
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* ..
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* Test the input parameters
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*
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INFO = 0
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IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.LSAME( JOB, 'P' ) .AND.
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$ .NOT.LSAME( JOB, 'S' ) .AND. .NOT.LSAME( JOB, 'B' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -4
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGEBAL', -INFO )
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RETURN
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END IF
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*
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* Quick returns.
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*
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IF( N.EQ.0 ) THEN
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ILO = 1
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IHI = 0
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RETURN
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END IF
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*
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IF( LSAME( JOB, 'N' ) ) THEN
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DO I = 1, N
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SCALE( I ) = ONE
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END DO
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ILO = 1
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IHI = N
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RETURN
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END IF
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*
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* Permutation to isolate eigenvalues if possible.
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*
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K = 1
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L = N
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*
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IF( .NOT.LSAME( JOB, 'S' ) ) THEN
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*
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* Row and column exchange.
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*
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NOCONV = .TRUE.
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DO WHILE( NOCONV )
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*
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* Search for rows isolating an eigenvalue and push them down.
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*
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NOCONV = .FALSE.
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DO I = L, 1, -1
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CANSWAP = .TRUE.
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DO J = 1, L
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IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
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CANSWAP = .FALSE.
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EXIT
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END IF
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END DO
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*
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IF( CANSWAP ) THEN
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SCALE( L ) = I
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IF( I.NE.L ) THEN
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CALL SSWAP( L, A( 1, I ), 1, A( 1, L ), 1 )
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CALL SSWAP( N-K+1, A( I, K ), LDA, A( L, K ), LDA )
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END IF
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NOCONV = .TRUE.
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*
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IF( L.EQ.1 ) THEN
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ILO = 1
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IHI = 1
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RETURN
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END IF
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*
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L = L - 1
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END IF
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END DO
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*
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END DO
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NOCONV = .TRUE.
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DO WHILE( NOCONV )
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*
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* Search for columns isolating an eigenvalue and push them left.
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*
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NOCONV = .FALSE.
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DO J = K, L
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CANSWAP = .TRUE.
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DO I = K, L
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IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
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CANSWAP = .FALSE.
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EXIT
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END IF
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END DO
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*
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IF( CANSWAP ) THEN
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SCALE( K ) = J
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IF( J.NE.K ) THEN
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CALL SSWAP( L, A( 1, J ), 1, A( 1, K ), 1 )
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CALL SSWAP( N-K+1, A( J, K ), LDA, A( K, K ), LDA )
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END IF
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NOCONV = .TRUE.
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*
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K = K + 1
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END IF
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END DO
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*
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END DO
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*
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END IF
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*
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* Initialize SCALE for non-permuted submatrix.
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*
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DO I = K, L
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SCALE( I ) = ONE
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END DO
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*
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* If we only had to permute, we are done.
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*
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IF( LSAME( JOB, 'P' ) ) THEN
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ILO = K
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IHI = L
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RETURN
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END IF
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*
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* Balance the submatrix in rows K to L.
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*
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* Iterative loop for norm reduction.
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*
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SFMIN1 = SLAMCH( 'S' ) / SLAMCH( 'P' )
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SFMAX1 = ONE / SFMIN1
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SFMIN2 = SFMIN1*SCLFAC
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SFMAX2 = ONE / SFMIN2
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*
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NOCONV = .TRUE.
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DO WHILE( NOCONV )
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NOCONV = .FALSE.
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*
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DO I = K, L
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*
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C = SNRM2( L-K+1, A( K, I ), 1 )
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R = SNRM2( L-K+1, A( I, K ), LDA )
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ICA = ISAMAX( L, A( 1, I ), 1 )
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CA = ABS( A( ICA, I ) )
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IRA = ISAMAX( N-K+1, A( I, K ), LDA )
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RA = ABS( A( I, IRA+K-1 ) )
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*
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* Guard against zero C or R due to underflow.
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*
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IF( C.EQ.ZERO .OR. R.EQ.ZERO ) CYCLE
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*
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* Exit if NaN to avoid infinite loop
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*
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IF( SISNAN( C+CA+R+RA ) ) THEN
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INFO = -3
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CALL XERBLA( 'SGEBAL', -INFO )
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RETURN
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END IF
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*
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G = R / SCLFAC
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F = ONE
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S = C + R
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*
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DO WHILE( C.LT.G .AND. MAX( F, C, CA ).LT.SFMAX2 .AND.
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$ MIN( R, G, RA ).GT.SFMIN2 )
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F = F*SCLFAC
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C = C*SCLFAC
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CA = CA*SCLFAC
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R = R / SCLFAC
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G = G / SCLFAC
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RA = RA / SCLFAC
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END DO
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*
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G = C / SCLFAC
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*
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DO WHILE( G.GE.R .AND. MAX( R, RA ).LT.SFMAX2 .AND.
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$ MIN( F, C, G, CA ).GT.SFMIN2 )
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F = F / SCLFAC
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C = C / SCLFAC
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G = G / SCLFAC
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CA = CA / SCLFAC
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R = R*SCLFAC
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RA = RA*SCLFAC
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END DO
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*
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* Now balance.
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*
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IF( ( C+R ).GE.FACTOR*S ) CYCLE
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IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
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IF( F*SCALE( I ).LE.SFMIN1 ) CYCLE
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END IF
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IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
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IF( SCALE( I ).GE.SFMAX1 / F ) CYCLE
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END IF
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G = ONE / F
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SCALE( I ) = SCALE( I )*F
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NOCONV = .TRUE.
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*
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CALL SSCAL( N-K+1, G, A( I, K ), LDA )
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CALL SSCAL( L, F, A( 1, I ), 1 )
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*
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END DO
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*
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END DO
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*
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ILO = K
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IHI = L
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*
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RETURN
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*
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* End of SGEBAL
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*
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END
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