SUBROUTINE DGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER JOB
INTEGER IHI, ILO, INFO, LDA, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), SCALE( * )
* ..
*
* Purpose
* =======
*
* 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.
*
* Arguments
* =========
*
* JOB (input) 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.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input/output) 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* ILO (output) INTEGER
* IHI (output) 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.
*
* SCALE (output) 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.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* 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
*
* =====================================================================
*
* .. 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
INTEGER I, ICA, IEXC, IRA, J, K, L, M
DOUBLE PRECISION C, CA, F, G, R, RA, S, SFMAX1, SFMAX2, SFMIN1,
$ SFMIN2
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IDAMAX
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, IDAMAX, DLAMCH
* ..
* .. External Subroutines ..
EXTERNAL DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
* 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
*
K = 1
L = N
*
IF( N.EQ.0 )
$ GO TO 210
*
IF( LSAME( JOB, 'N' ) ) THEN
DO 10 I = 1, N
SCALE( I ) = ONE
10 CONTINUE
GO TO 210
END IF
*
IF( LSAME( JOB, 'S' ) )
$ GO TO 120
*
* Permutation to isolate eigenvalues if possible
*
GO TO 50
*
* Row and column exchange.
*
20 CONTINUE
SCALE( M ) = J
IF( J.EQ.M )
$ GO TO 30
*
CALL DSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
CALL DSWAP( N-K+1, A( J, K ), LDA, A( M, K ), LDA )
*
30 CONTINUE
GO TO ( 40, 80 )IEXC
*
* Search for rows isolating an eigenvalue and push them down.
*
40 CONTINUE
IF( L.EQ.1 )
$ GO TO 210
L = L - 1
*
50 CONTINUE
DO 70 J = L, 1, -1
*
DO 60 I = 1, L
IF( I.EQ.J )
$ GO TO 60
IF( A( J, I ).NE.ZERO )
$ GO TO 70
60 CONTINUE
*
M = L
IEXC = 1
GO TO 20
70 CONTINUE
*
GO TO 90
*
* Search for columns isolating an eigenvalue and push them left.
*
80 CONTINUE
K = K + 1
*
90 CONTINUE
DO 110 J = K, L
*
DO 100 I = K, L
IF( I.EQ.J )
$ GO TO 100
IF( A( I, J ).NE.ZERO )
$ GO TO 110
100 CONTINUE
*
M = K
IEXC = 2
GO TO 20
110 CONTINUE
*
120 CONTINUE
DO 130 I = K, L
SCALE( I ) = ONE
130 CONTINUE
*
IF( LSAME( JOB, 'P' ) )
$ GO TO 210
*
* 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
140 CONTINUE
NOCONV = .FALSE.
*
DO 200 I = K, L
C = ZERO
R = ZERO
*
DO 150 J = K, L
IF( J.EQ.I )
$ GO TO 150
C = C + ABS( A( J, I ) )
R = R + ABS( A( I, J ) )
150 CONTINUE
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 )
$ GO TO 200
G = R / SCLFAC
F = ONE
S = C + R
160 CONTINUE
IF( C.GE.G .OR. MAX( F, C, CA ).GE.SFMAX2 .OR.
$ MIN( R, G, RA ).LE.SFMIN2 )GO TO 170
F = F*SCLFAC
C = C*SCLFAC
CA = CA*SCLFAC
R = R / SCLFAC
G = G / SCLFAC
RA = RA / SCLFAC
GO TO 160
*
170 CONTINUE
G = C / SCLFAC
180 CONTINUE
IF( G.LT.R .OR. MAX( R, RA ).GE.SFMAX2 .OR.
$ MIN( F, C, G, CA ).LE.SFMIN2 )GO TO 190
F = F / SCLFAC
C = C / SCLFAC
G = G / SCLFAC
CA = CA / SCLFAC
R = R*SCLFAC
RA = RA*SCLFAC
GO TO 180
*
* Now balance.
*
190 CONTINUE
IF( ( C+R ).GE.FACTOR*S )
$ GO TO 200
IF( F.LT.ONE .AND. SCALE( I ).LT.ONE ) THEN
IF( F*SCALE( I ).LE.SFMIN1 )
$ GO TO 200
END IF
IF( F.GT.ONE .AND. SCALE( I ).GT.ONE ) THEN
IF( SCALE( I ).GE.SFMAX1 / F )
$ GO TO 200
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 )
*
200 CONTINUE
*
IF( NOCONV )
$ GO TO 140
*
210 CONTINUE
ILO = K
IHI = L
*
RETURN
*
* End of DGEBAL
*
END