*> \brief \b ZGEBAL * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGEBAL + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGEBAL( JOB, N, A, LDA, ILO, IHI, SCALE, INFO ) * * .. Scalar Arguments .. * CHARACTER JOB * INTEGER IHI, ILO, INFO, LDA, N * .. * .. Array Arguments .. * DOUBLE PRECISION SCALE( * ) * COMPLEX*16 A( LDA, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGEBAL balances a general complex 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 COMPLEX*16 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 INTEGER 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 complex16GEcomputational * *> \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 CBAL. *> *> Modified by Tzu-Yi Chen, Computer Science Division, University of *> California at Berkeley, USA *> \endverbatim *> * ===================================================================== SUBROUTINE ZGEBAL( 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 SCALE( * ) COMPLEX*16 A( LDA, * ) * .. * * ===================================================================== * * .. 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 DISNAN, LSAME INTEGER IZAMAX DOUBLE PRECISION DLAMCH, DZNRM2 EXTERNAL DISNAN, LSAME, IZAMAX, DLAMCH, DZNRM2 * .. * .. External Subroutines .. EXTERNAL XERBLA, ZDSCAL, ZSWAP * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DIMAG, 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( 'ZGEBAL', -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 ZSWAP( L, A( 1, J ), 1, A( 1, M ), 1 ) CALL ZSWAP( 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( DBLE( A( J, I ) ).NE.ZERO .OR. DIMAG( 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( DBLE( A( I, J ) ).NE.ZERO .OR. DIMAG( 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 = DZNRM2( L-K+1, A( K, I ), 1 ) R = DZNRM2( L-K+1, A( I, K ), LDA ) ICA = IZAMAX( L, A( 1, I ), 1 ) CA = ABS( A( ICA, I ) ) IRA = IZAMAX( 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 IF( DISNAN( C+F+CA+R+G+RA ) ) THEN * * Exit if NaN to avoid infinite loop * INFO = -3 CALL XERBLA( 'ZGEBAL', -INFO ) RETURN END IF 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 ZDSCAL( N-K+1, G, A( I, K ), LDA ) CALL ZDSCAL( L, F, A( 1, I ), 1 ) * 200 CONTINUE * IF( NOCONV ) \$ GO TO 140 * 210 CONTINUE ILO = K IHI = L * RETURN * * End of ZGEBAL * END