*> \brief \b SGGBAL * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SGGBAL + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, * RSCALE, WORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOB * INTEGER IHI, ILO, INFO, LDA, LDB, N * .. * .. Array Arguments .. * REAL A( LDA, * ), B( LDB, * ), LSCALE( * ), * \$ RSCALE( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOB *> \verbatim *> JOB is 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. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is 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. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is 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. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= 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 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. *> \endverbatim *> *> \param[out] LSCALE *> \verbatim *> LSCALE is 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. *> \endverbatim *> *> \param[out] RSCALE *> \verbatim *> RSCALE is 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. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is 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'. *> \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. * *> \date December 2016 * *> \ingroup realGBcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> See R.C. WARD, Balancing the generalized eigenvalue problem, *> SIAM J. Sci. Stat. Comp. 2 (1981), 141-152. *> \endverbatim *> * ===================================================================== SUBROUTINE SGGBAL( JOB, N, A, LDA, B, LDB, ILO, IHI, LSCALE, \$ RSCALE, WORK, 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 JOB INTEGER IHI, ILO, INFO, LDA, LDB, N * .. * .. Array Arguments .. REAL A( LDA, * ), B( LDB, * ), LSCALE( * ), \$ RSCALE( * ), WORK( * ) * .. * * ===================================================================== * * .. 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