*> \brief \b DGEBAL
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOB
* INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DGEBAL balances a general real matrix A. This involves, first,
*> permuting A by a similarity transformation to isolate eigenvalues
*> in the first 1 to ILO-1 and last IHI+1 to N elements on the
*> diagonal; and second, applying a diagonal similarity transformation
*> to rows and columns ILO to IHI to make the rows and columns as
*> close in norm as possible. Both steps are optional.
*>
*> Balancing may reduce the 1-norm of the matrix, and improve the
*> accuracy of the computed eigenvalues and/or eigenvectors.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> Specifies the operations to be performed on A:
*> = 'N': none: simply set ILO = 1, IHI = N, SCALE(I) = 1.0
*> for i = 1,...,N;
*> = 'P': permute only;
*> = 'S': scale only;
*> = 'B': both permute and scale.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the input matrix A.
*> On exit, A is overwritten by the balanced matrix.
*> If JOB = 'N', A is not referenced.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*> \param[out] IHI
*> \verbatim
*> IHI is INTEGER
*> ILO and IHI are set to integers such that on exit
*> A(i,j) = 0 if i > j and j = 1,...,ILO-1 or I = IHI+1,...,N.
*> If JOB = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] SCALE
*> \verbatim
*> SCALE is DOUBLE PRECISION array, dimension (N)
*> Details of the permutations and scaling factors applied to
*> A. If P(j) is the index of the row and column interchanged
*> with row and column j and D(j) is the scaling factor
*> applied to row and column j, then
*> SCALE(j) = P(j) for j = 1,...,ILO-1
*> = D(j) for j = ILO,...,IHI
*> = P(j) for j = IHI+1,...,N.
*> The order in which the interchanges are made is N to IHI+1,
*> then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gebal
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The permutations consist of row and column interchanges which put
*> the matrix in the form
*>
*> ( T1 X Y )
*> P A P = ( 0 B Z )
*> ( 0 0 T2 )
*>
*> where T1 and T2 are upper triangular matrices whose eigenvalues lie
*> along the diagonal. The column indices ILO and IHI mark the starting
*> and ending columns of the submatrix B. Balancing consists of applying
*> a diagonal similarity transformation inv(D) * B * D to make the
*> 1-norms of each row of B and its corresponding column nearly equal.
*> The output matrix is
*>
*> ( T1 X*D Y )
*> ( 0 inv(D)*B*D inv(D)*Z ).
*> ( 0 0 T2 )
*>
*> Information about the permutations P and the diagonal matrix D is
*> returned in the vector SCALE.
*>
*> This subroutine is based on the EISPACK routine BALANC.
*>
*> Modified by Tzu-Yi Chen, Computer Science Division, University of
*> California at Berkeley, USA
*>
*> Refactored by Evert Provoost, Department of Computer Science,
*> KU Leuven, Belgium
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, 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 JOB
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
DOUBLE PRECISION SCLFAC
PARAMETER ( SCLFAC = 2.0D+0 )
DOUBLE PRECISION FACTOR
PARAMETER ( FACTOR = 0.95D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOCONV, CANSWAP
INTEGER I, ICA, IRA, J, K, L
DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
$ SFMIN2
* ..
* .. External Functions ..
LOGICAL DISNAN, LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH, DNRM2
EXTERNAL DISNAN, LSAME, IDAMAX, DLAMCH,
$ DNRM2
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* Test the input parameters
*
INFO = 0
IF( .NOT.LSAME( JOB, 'N' ) .AND.
$ .NOT.LSAME( JOB, 'P' ) .AND.
$ .NOT.LSAME( JOB, 'S' ) .AND.
$ .NOT.LSAME( JOB, 'B' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEBAL', -INFO )
RETURN
END IF
*
* Quick returns.
*
IF( N.EQ.0 ) THEN
ILO = 1
IHI = 0
RETURN
END IF
*
IF( LSAME( JOB, 'N' ) ) THEN
DO I = 1, N
SCALE( I ) = ONE
END DO
ILO = 1
IHI = N
RETURN
END IF
*
* Permutation to isolate eigenvalues if possible.
*
K = 1
L = N
*
IF( .NOT.LSAME( JOB, 'S' ) ) THEN
*
* Row and column exchange.
*
NOCONV = .TRUE.
DO WHILE( NOCONV )
*
* Search for rows isolating an eigenvalue and push them down.
*
NOCONV = .FALSE.
DO I = L, 1, -1
CANSWAP = .TRUE.
DO J = 1, L
IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
CANSWAP = .FALSE.
EXIT
END IF
END DO
*
IF( CANSWAP ) THEN
SCALE( L ) = I
IF( I.NE.L ) THEN
CALL DSWAP( L, A( 1, I ), 1, A( 1, L ), 1 )
CALL DSWAP( N-K+1, A( I, K ), LDA, A( L, K ),
$ LDA )
END IF
NOCONV = .TRUE.
*
IF( L.EQ.1 ) THEN
ILO = 1
IHI = 1
RETURN
END IF
*
L = L - 1
END IF
END DO
*
END DO
NOCONV = .TRUE.
DO WHILE( NOCONV )
*
* Search for columns isolating an eigenvalue and push them left.
*
NOCONV = .FALSE.
DO J = K, L
CANSWAP = .TRUE.
DO I = K, L
IF( I.NE.J .AND. A( I, J ).NE.ZERO ) THEN
CANSWAP = .FALSE.
EXIT
END IF
END DO
*
IF( CANSWAP ) THEN
SCALE( K ) = J
IF( J.NE.K ) THEN
CALL DSWAP( L, A( 1, J ), 1, A( 1, K ), 1 )
CALL DSWAP( N-K+1, A( J, K ), LDA, A( K, K ),
$ LDA )
END IF
NOCONV = .TRUE.
*
K = K + 1
END IF
END DO
*
END DO
*
END IF
*
* Initialize SCALE for non-permuted submatrix.
*
DO I = K, L
SCALE( I ) = ONE
END DO
*
* If we only had to permute, we are done.
*
IF( LSAME( JOB, 'P' ) ) THEN
ILO = K
IHI = L
RETURN
END IF
*
* Balance the submatrix in rows K to L.
*
* Iterative loop for norm reduction.
*
SFMIN1 = DLAMCH( 'S' ) / DLAMCH( 'P' )
SFMAX1 = ONE / SFMIN1
SFMIN2 = SFMIN1*SCLFAC
SFMAX2 = ONE / SFMIN2
*
NOCONV = .TRUE.
DO WHILE( NOCONV )
NOCONV = .FALSE.
*
DO I = K, L
*
C = DNRM2( L-K+1, A( K, I ), 1 )
R = DNRM2( L-K+1, A( I, K ), LDA )
ICA = IDAMAX( L, A( 1, I ), 1 )
CA = ABS( A( ICA, I ) )
IRA = IDAMAX( N-K+1, A( I, K ), LDA )
RA = ABS( A( I, IRA+K-1 ) )
*
* Guard against zero C or R due to underflow.
*
IF( C.EQ.ZERO .OR. R.EQ.ZERO ) CYCLE
*
* Exit if NaN to avoid infinite loop
*
IF( DISNAN( C+CA+R+RA ) ) THEN
INFO = -3
CALL XERBLA( 'DGEBAL', -INFO )
RETURN
END IF
*
G = R / SCLFAC
F = ONE
S = C + R
*
DO WHILE( C.LT.G .AND. MAX( F, C, CA ).LT.SFMAX2 .AND.
$ MIN( R, G, RA ).GT.SFMIN2 )
F = F*SCLFAC
C = C*SCLFAC
CA = CA*SCLFAC
R = R / SCLFAC
G = G / SCLFAC
RA = RA / SCLFAC
END DO
*
G = C / SCLFAC
*
DO WHILE( G.GE.R .AND. MAX( R, RA ).LT.SFMAX2 .AND.
$ MIN( F, C, G, CA ).GT.SFMIN2 )
F = F / SCLFAC
C = C / SCLFAC
G = G / SCLFAC
CA = CA / SCLFAC
R = R*SCLFAC
RA = RA*SCLFAC
END DO
*
* Now balance.
*
IF( ( C+R ).GE.FACTOR*S ) CYCLE
IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
IF( F*SCALE( I ).LE.SFMIN1 ) CYCLE
END IF
IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
IF( SCALE( I ).GE.SFMAX1 / F ) CYCLE
END IF
G = ONE / F
SCALE( I ) = SCALE( I )*F
NOCONV = .TRUE.
*
CALL DSCAL( N-K+1, G, A( I, K ), LDA )
CALL DSCAL( L, F, A( 1, I ), 1 )
*
END DO
*
END DO
*
ILO = K
IHI = L
*
RETURN
*
* End of DGEBAL
*
END