SUBROUTINE SGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE,
$ RSCALE, WORK, 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, LDB, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), B( LDB, * ), LSCALE( * ),
$ RSCALE( * ), WORK( * )
* ..
*
* Purpose
* =======
*
* SGGBAL balances a pair of general real matrices (A,B). This
* involves, first, permuting A and B by similarity transformations 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 matrices, and improve the
* accuracy of the computed eigenvalues and/or eigenvectors in the
* generalized eigenvalue problem A*x = lambda*B*x.
*
* Arguments
* =========
*
* JOB (input) CHARACTER*1
* Specifies the operations to be performed on A and B:
* = 'N': none: simply set ILO = 1, IHI = N, LSCALE(I) = 1.0
* and RSCALE(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 matrices A and B. N >= 0.
*
* A (input/output) REAL 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.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) REAL array, dimension (LDB,N)
* On entry, the input matrix B.
* On exit, B is overwritten by the balanced matrix.
* If JOB = 'N', B is not referenced.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* ILO (output) INTEGER
* IHI (output) INTEGER
* ILO and IHI are set to integers such that on exit
* A(i,j) = 0 and B(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.
*
* LSCALE (output) REAL array, dimension (N)
* Details of the permutations and scaling factors applied
* to the left side of A and B. If P(j) is the index of the
* row interchanged with row j, and D(j)
* is the scaling factor applied to row j, then
* LSCALE(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.
*
* RSCALE (output) REAL array, dimension (N)
* Details of the permutations and scaling factors applied
* to the right side of A and B. If P(j) is the index of the
* column interchanged with column j, and D(j)
* is the scaling factor applied to column j, then
* LSCALE(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.
*
* WORK (workspace) REAL array, dimension (lwork)
* lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
* at least 1 when JOB = 'N' or 'P'.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* See R.C. WARD, Balancing the generalized eigenvalue problem,
* SIAM J. Sci. Stat. Comp. 2 (1981), 141-152.
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
REAL THREE, SCLFAC
PARAMETER ( THREE = 3.0E+0, SCLFAC = 1.0E+1 )
* ..
* .. Local Scalars ..
INTEGER I, ICAB, IFLOW, IP1, IR, IRAB, IT, J, JC, JP1,
$ K, KOUNT, L, LCAB, LM1, LRAB, LSFMAX, LSFMIN,
$ M, NR, NRP2
REAL ALPHA, BASL, BETA, CAB, CMAX, COEF, COEF2,
$ COEF5, COR, EW, EWC, GAMMA, PGAMMA, RAB, SFMAX,
$ SFMIN, SUM, T, TA, TB, TC
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ISAMAX
REAL SDOT, SLAMCH
EXTERNAL LSAME, ISAMAX, SDOT, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SSCAL, SSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, LOG10, MAX, MIN, REAL, SIGN
* ..
* .. 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
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGGBAL', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
ILO = 1
IHI = N
RETURN
END IF
*
IF( N.EQ.1 ) THEN
ILO = 1
IHI = N
LSCALE( 1 ) = ONE
RSCALE( 1 ) = ONE
RETURN
END IF
*
IF( LSAME( JOB, 'N' ) ) THEN
ILO = 1
IHI = N
DO 10 I = 1, N
LSCALE( I ) = ONE
RSCALE( I ) = ONE
10 CONTINUE
RETURN
END IF
*
K = 1
L = N
IF( LSAME( JOB, 'S' ) )
$ GO TO 190
*
GO TO 30
*
* Permute the matrices A and B to isolate the eigenvalues.
*
* Find row with one nonzero in columns 1 through L
*
20 CONTINUE
L = LM1
IF( L.NE.1 )
$ GO TO 30
*
RSCALE( 1 ) = ONE
LSCALE( 1 ) = ONE
GO TO 190
*
30 CONTINUE
LM1 = L - 1
DO 80 I = L, 1, -1
DO 40 J = 1, LM1
JP1 = J + 1
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 50
40 CONTINUE
J = L
GO TO 70
*
50 CONTINUE
DO 60 J = JP1, L
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 80
60 CONTINUE
J = JP1 - 1
*
70 CONTINUE
M = L
IFLOW = 1
GO TO 160
80 CONTINUE
GO TO 100
*
* Find column with one nonzero in rows K through N
*
90 CONTINUE
K = K + 1
*
100 CONTINUE
DO 150 J = K, L
DO 110 I = K, LM1
IP1 = I + 1
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 120
110 CONTINUE
I = L
GO TO 140
120 CONTINUE
DO 130 I = IP1, L
IF( A( I, J ).NE.ZERO .OR. B( I, J ).NE.ZERO )
$ GO TO 150
130 CONTINUE
I = IP1 - 1
140 CONTINUE
M = K
IFLOW = 2
GO TO 160
150 CONTINUE
GO TO 190
*
* Permute rows M and I
*
160 CONTINUE
LSCALE( M ) = I
IF( I.EQ.M )
$ GO TO 170
CALL SSWAP( N-K+1, A( I, K ), LDA, A( M, K ), LDA )
CALL SSWAP( N-K+1, B( I, K ), LDB, B( M, K ), LDB )
*
* Permute columns M and J
*
170 CONTINUE
RSCALE( M ) = J
IF( J.EQ.M )
$ GO TO 180
CALL SSWAP( L, A( 1, J ), 1, A( 1, M ), 1 )
CALL SSWAP( L, B( 1, J ), 1, B( 1, M ), 1 )
*
180 CONTINUE
GO TO ( 20, 90 )IFLOW
*
190 CONTINUE
ILO = K
IHI = L
*
IF( LSAME( JOB, 'P' ) ) THEN
DO 195 I = ILO, IHI
LSCALE( I ) = ONE
RSCALE( I ) = ONE
195 CONTINUE
RETURN
END IF
*
IF( ILO.EQ.IHI )
$ RETURN
*
* Balance the submatrix in rows ILO to IHI.
*
NR = IHI - ILO + 1
DO 200 I = ILO, IHI
RSCALE( I ) = ZERO
LSCALE( I ) = ZERO
*
WORK( I ) = ZERO
WORK( I+N ) = ZERO
WORK( I+2*N ) = ZERO
WORK( I+3*N ) = ZERO
WORK( I+4*N ) = ZERO
WORK( I+5*N ) = ZERO
200 CONTINUE
*
* Compute right side vector in resulting linear equations
*
BASL = LOG10( SCLFAC )
DO 240 I = ILO, IHI
DO 230 J = ILO, IHI
TB = B( I, J )
TA = A( I, J )
IF( TA.EQ.ZERO )
$ GO TO 210
TA = LOG10( ABS( TA ) ) / BASL
210 CONTINUE
IF( TB.EQ.ZERO )
$ GO TO 220
TB = LOG10( ABS( TB ) ) / BASL
220 CONTINUE
WORK( I+4*N ) = WORK( I+4*N ) - TA - TB
WORK( J+5*N ) = WORK( J+5*N ) - TA - TB
230 CONTINUE
240 CONTINUE
*
COEF = ONE / REAL( 2*NR )
COEF2 = COEF*COEF
COEF5 = HALF*COEF2
NRP2 = NR + 2
BETA = ZERO
IT = 1
*
* Start generalized conjugate gradient iteration
*
250 CONTINUE
*
GAMMA = SDOT( NR, WORK( ILO+4*N ), 1, WORK( ILO+4*N ), 1 ) +
$ SDOT( NR, WORK( ILO+5*N ), 1, WORK( ILO+5*N ), 1 )
*
EW = ZERO
EWC = ZERO
DO 260 I = ILO, IHI
EW = EW + WORK( I+4*N )
EWC = EWC + WORK( I+5*N )
260 CONTINUE
*
GAMMA = COEF*GAMMA - COEF2*( EW**2+EWC**2 ) - COEF5*( EW-EWC )**2
IF( GAMMA.EQ.ZERO )
$ GO TO 350
IF( IT.NE.1 )
$ BETA = GAMMA / PGAMMA
T = COEF5*( EWC-THREE*EW )
TC = COEF5*( EW-THREE*EWC )
*
CALL SSCAL( NR, BETA, WORK( ILO ), 1 )
CALL SSCAL( NR, BETA, WORK( ILO+N ), 1 )
*
CALL SAXPY( NR, COEF, WORK( ILO+4*N ), 1, WORK( ILO+N ), 1 )
CALL SAXPY( NR, COEF, WORK( ILO+5*N ), 1, WORK( ILO ), 1 )
*
DO 270 I = ILO, IHI
WORK( I ) = WORK( I ) + TC
WORK( I+N ) = WORK( I+N ) + T
270 CONTINUE
*
* Apply matrix to vector
*
DO 300 I = ILO, IHI
KOUNT = 0
SUM = ZERO
DO 290 J = ILO, IHI
IF( A( I, J ).EQ.ZERO )
$ GO TO 280
KOUNT = KOUNT + 1
SUM = SUM + WORK( J )
280 CONTINUE
IF( B( I, J ).EQ.ZERO )
$ GO TO 290
KOUNT = KOUNT + 1
SUM = SUM + WORK( J )
290 CONTINUE
WORK( I+2*N ) = REAL( KOUNT )*WORK( I+N ) + SUM
300 CONTINUE
*
DO 330 J = ILO, IHI
KOUNT = 0
SUM = ZERO
DO 320 I = ILO, IHI
IF( A( I, J ).EQ.ZERO )
$ GO TO 310
KOUNT = KOUNT + 1
SUM = SUM + WORK( I+N )
310 CONTINUE
IF( B( I, J ).EQ.ZERO )
$ GO TO 320
KOUNT = KOUNT + 1
SUM = SUM + WORK( I+N )
320 CONTINUE
WORK( J+3*N ) = REAL( KOUNT )*WORK( J ) + SUM
330 CONTINUE
*
SUM = SDOT( NR, WORK( ILO+N ), 1, WORK( ILO+2*N ), 1 ) +
$ SDOT( NR, WORK( ILO ), 1, WORK( ILO+3*N ), 1 )
ALPHA = GAMMA / SUM
*
* Determine correction to current iteration
*
CMAX = ZERO
DO 340 I = ILO, IHI
COR = ALPHA*WORK( I+N )
IF( ABS( COR ).GT.CMAX )
$ CMAX = ABS( COR )
LSCALE( I ) = LSCALE( I ) + COR
COR = ALPHA*WORK( I )
IF( ABS( COR ).GT.CMAX )
$ CMAX = ABS( COR )
RSCALE( I ) = RSCALE( I ) + COR
340 CONTINUE
IF( CMAX.LT.HALF )
$ GO TO 350
*
CALL SAXPY( NR, -ALPHA, WORK( ILO+2*N ), 1, WORK( ILO+4*N ), 1 )
CALL SAXPY( NR, -ALPHA, WORK( ILO+3*N ), 1, WORK( ILO+5*N ), 1 )
*
PGAMMA = GAMMA
IT = IT + 1
IF( IT.LE.NRP2 )
$ GO TO 250
*
* End generalized conjugate gradient iteration
*
350 CONTINUE
SFMIN = SLAMCH( 'S' )
SFMAX = ONE / SFMIN
LSFMIN = INT( LOG10( SFMIN ) / BASL+ONE )
LSFMAX = INT( LOG10( SFMAX ) / BASL )
DO 360 I = ILO, IHI
IRAB = ISAMAX( N-ILO+1, A( I, ILO ), LDA )
RAB = ABS( A( I, IRAB+ILO-1 ) )
IRAB = ISAMAX( N-ILO+1, B( I, ILO ), LDB )
RAB = MAX( RAB, ABS( B( I, IRAB+ILO-1 ) ) )
LRAB = INT( LOG10( RAB+SFMIN ) / BASL+ONE )
IR = LSCALE( I ) + SIGN( HALF, LSCALE( I ) )
IR = MIN( MAX( IR, LSFMIN ), LSFMAX, LSFMAX-LRAB )
LSCALE( I ) = SCLFAC**IR
ICAB = ISAMAX( IHI, A( 1, I ), 1 )
CAB = ABS( A( ICAB, I ) )
ICAB = ISAMAX( IHI, B( 1, I ), 1 )
CAB = MAX( CAB, ABS( B( ICAB, I ) ) )
LCAB = INT( LOG10( CAB+SFMIN ) / BASL+ONE )
JC = RSCALE( I ) + SIGN( HALF, RSCALE( I ) )
JC = MIN( MAX( JC, LSFMIN ), LSFMAX, LSFMAX-LCAB )
RSCALE( I ) = SCLFAC**JC
360 CONTINUE
*
* Row scaling of matrices A and B
*
DO 370 I = ILO, IHI
CALL SSCAL( N-ILO+1, LSCALE( I ), A( I, ILO ), LDA )
CALL SSCAL( N-ILO+1, LSCALE( I ), B( I, ILO ), LDB )
370 CONTINUE
*
* Column scaling of matrices A and B
*
DO 380 J = ILO, IHI
CALL SSCAL( IHI, RSCALE( J ), A( 1, J ), 1 )
CALL SSCAL( IHI, RSCALE( J ), B( 1, J ), 1 )
380 CONTINUE
*
RETURN
*
* End of SGGBAL
*
END