SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, \$ LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, \$ RCONDV, WORK, LWORK, RWORK, INFO ) * * -- LAPACK driver routine (version 3.0) -- * Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., * Courant Institute, Argonne National Lab, and Rice University * June 30, 1999 * 8-15-00: Improve consistency of WS calculations (eca) * * .. Scalar Arguments .. CHARACTER BALANC, JOBVL, JOBVR, SENSE INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N DOUBLE PRECISION ABNRM * .. * .. Array Arguments .. DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ), \$ SCALE( * ) COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), \$ W( * ), WORK( * ) * .. * * Purpose * ======= * * ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the * eigenvalues and, optionally, the left and/or right eigenvectors. * * Optionally also, it computes a balancing transformation to improve * the conditioning of the eigenvalues and eigenvectors (ILO, IHI, * SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues * (RCONDE), and reciprocal condition numbers for the right * eigenvectors (RCONDV). * * The right eigenvector v(j) of A satisfies * A * v(j) = lambda(j) * v(j) * where lambda(j) is its eigenvalue. * The left eigenvector u(j) of A satisfies * u(j)**H * A = lambda(j) * u(j)**H * where u(j)**H denotes the conjugate transpose of u(j). * * The computed eigenvectors are normalized to have Euclidean norm * equal to 1 and largest component real. * * Balancing a matrix means permuting the rows and columns to make it * more nearly upper triangular, and applying a diagonal similarity * transformation D * A * D**(-1), where D is a diagonal matrix, to * make its rows and columns closer in norm and the condition numbers * of its eigenvalues and eigenvectors smaller. The computed * reciprocal condition numbers correspond to the balanced matrix. * Permuting rows and columns will not change the condition numbers * (in exact arithmetic) but diagonal scaling will. For further * explanation of balancing, see section 4.10.2 of the LAPACK * Users' Guide. * * Arguments * ========= * * BALANC (input) CHARACTER*1 * Indicates how the input matrix should be diagonally scaled * and/or permuted to improve the conditioning of its * eigenvalues. * = 'N': Do not diagonally scale or permute; * = 'P': Perform permutations to make the matrix more nearly * upper triangular. Do not diagonally scale; * = 'S': Diagonally scale the matrix, ie. replace A by * D*A*D**(-1), where D is a diagonal matrix chosen * to make the rows and columns of A more equal in * norm. Do not permute; * = 'B': Both diagonally scale and permute A. * * Computed reciprocal condition numbers will be for the matrix * after balancing and/or permuting. Permuting does not change * condition numbers (in exact arithmetic), but balancing does. * * JOBVL (input) CHARACTER*1 * = 'N': left eigenvectors of A are not computed; * = 'V': left eigenvectors of A are computed. * If SENSE = 'E' or 'B', JOBVL must = 'V'. * * JOBVR (input) CHARACTER*1 * = 'N': right eigenvectors of A are not computed; * = 'V': right eigenvectors of A are computed. * If SENSE = 'E' or 'B', JOBVR must = 'V'. * * SENSE (input) CHARACTER*1 * Determines which reciprocal condition numbers are computed. * = 'N': None are computed; * = 'E': Computed for eigenvalues only; * = 'V': Computed for right eigenvectors only; * = 'B': Computed for eigenvalues and right eigenvectors. * * If SENSE = 'E' or 'B', both left and right eigenvectors * must also be computed (JOBVL = 'V' and JOBVR = 'V'). * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) COMPLEX*16 array, dimension (LDA,N) * On entry, the N-by-N matrix A. * On exit, A has been overwritten. If JOBVL = 'V' or * JOBVR = 'V', A contains the Schur form of the balanced * version of the matrix A. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * W (output) COMPLEX*16 array, dimension (N) * W contains the computed eigenvalues. * * VL (output) COMPLEX*16 array, dimension (LDVL,N) * If JOBVL = 'V', the left eigenvectors u(j) are stored one * after another in the columns of VL, in the same order * as their eigenvalues. * If JOBVL = 'N', VL is not referenced. * u(j) = VL(:,j), the j-th column of VL. * * LDVL (input) INTEGER * The leading dimension of the array VL. LDVL >= 1; if * JOBVL = 'V', LDVL >= N. * * VR (output) COMPLEX*16 array, dimension (LDVR,N) * If JOBVR = 'V', the right eigenvectors v(j) are stored one * after another in the columns of VR, in the same order * as their eigenvalues. * If JOBVR = 'N', VR is not referenced. * v(j) = VR(:,j), the j-th column of VR. * * LDVR (input) INTEGER * The leading dimension of the array VR. LDVR >= 1; if * JOBVR = 'V', LDVR >= N. * * ILO,IHI (output) INTEGER * ILO and IHI are integer values determined when A was * balanced. The balanced A(i,j) = 0 if I > J and * J = 1,...,ILO-1 or I = IHI+1,...,N. * * SCALE (output) DOUBLE PRECISION array, dimension (N) * Details of the permutations and scaling factors applied * when balancing 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. * * ABNRM (output) DOUBLE PRECISION * The one-norm of the balanced matrix (the maximum * of the sum of absolute values of elements of any column). * * RCONDE (output) DOUBLE PRECISION array, dimension (N) * RCONDE(j) is the reciprocal condition number of the j-th * eigenvalue. * * RCONDV (output) DOUBLE PRECISION array, dimension (N) * RCONDV(j) is the reciprocal condition number of the j-th * right eigenvector. * * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. If SENSE = 'N' or 'E', * LWORK >= max(1,2*N), and if SENSE = 'V' or 'B', * LWORK >= N*N+2*N. * For good performance, LWORK must generally be larger. * * If LWORK = -1, a workspace query is assumed. The optimal * size for the WORK array is calculated and stored in WORK(1), * and no other work except argument checking is performed. * * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = i, the QR algorithm failed to compute all the * eigenvalues, and no eigenvectors or condition numbers * have been computed; elements 1:ILO-1 and i+1:N of W * contain eigenvalues which have converged. * * ===================================================================== * * .. Parameters .. INTEGER LQUERV PARAMETER ( LQUERV = -1 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL SCALEA, WANTVL, WANTVR, WNTSNB, WNTSNE, WNTSNN, \$ WNTSNV CHARACTER JOB, SIDE INTEGER HSWORK, I, ICOND, IERR, ITAU, IWRK, K, MAXB, \$ MAXWRK, MINWRK, NOUT DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM COMPLEX*16 TMP * .. * .. Local Arrays .. LOGICAL SELECT( 1 ) DOUBLE PRECISION DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, DLASCL, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, \$ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC, \$ ZTRSNA, ZUNGHR * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX, ILAENV DOUBLE PRECISION DLAMCH, DZNRM2, ZLANGE EXTERNAL LSAME, IDAMAX, ILAENV, DLAMCH, DZNRM2, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN, SQRT * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 WANTVL = LSAME( JOBVL, 'V' ) WANTVR = LSAME( JOBVR, 'V' ) WNTSNN = LSAME( SENSE, 'N' ) WNTSNE = LSAME( SENSE, 'E' ) WNTSNV = LSAME( SENSE, 'V' ) WNTSNB = LSAME( SENSE, 'B' ) IF( .NOT.( LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, \$ 'S' ) .OR. LSAME( BALANC, 'P' ) .OR. LSAME( BALANC, 'B' ) ) ) \$ THEN INFO = -1 ELSE IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN INFO = -2 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN INFO = -3 ELSE IF( .NOT.( WNTSNN .OR. WNTSNE .OR. WNTSNB .OR. WNTSNV ) .OR. \$ ( ( WNTSNE .OR. WNTSNB ) .AND. .NOT.( WANTVL .AND. \$ WANTVR ) ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN INFO = -10 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN INFO = -12 END IF * * Compute workspace * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of workspace needed at that point in the code, * as well as the preferred amount for good performance. * CWorkspace refers to complex workspace, and RWorkspace to real * workspace. NB refers to the optimal block size for the * immediately following subroutine, as returned by ILAENV. * HSWORK refers to the workspace preferred by ZHSEQR, as * calculated below. HSWORK is computed assuming ILO=1 and IHI=N, * the worst case.) * MINWRK = 1 IF( INFO.EQ.0 ) THEN MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN MINWRK = MAX( 1, 2*N ) IF( .NOT.( WNTSNN .OR. WNTSNE ) ) \$ MINWRK = MAX( MINWRK, N*N+2*N ) MAXB = MAX( ILAENV( 8, 'ZHSEQR', 'SN', N, 1, N, -1 ), 2 ) IF( WNTSNN ) THEN K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'ZHSEQR', 'EN', N, \$ 1, N, -1 ) ) ) ELSE K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'ZHSEQR', 'SN', N, \$ 1, N, -1 ) ) ) END IF HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, 1, HSWORK ) IF( .NOT.( WNTSNN .OR. WNTSNE ) ) \$ MAXWRK = MAX( MAXWRK, N*N+2*N ) ELSE MINWRK = MAX( 1, 2*N ) IF( .NOT.( WNTSNN .OR. WNTSNE ) ) \$ MINWRK = MAX( MINWRK, N*N+2*N ) MAXB = MAX( ILAENV( 8, 'ZHSEQR', 'SN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'ZHSEQR', 'EN', N, 1, \$ N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, 1, HSWORK ) MAXWRK = MAX( MAXWRK, N+( N-1 )* \$ ILAENV( 1, 'ZUNGHR', ' ', N, 1, N, -1 ) ) IF( .NOT.( WNTSNN .OR. WNTSNE ) ) \$ MAXWRK = MAX( MAXWRK, N*N+2*N ) MAXWRK = MAX( MAXWRK, 2*N, 1 ) END IF WORK( 1 ) = MAXWRK IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV ) \$ INFO = -20 END IF * * Quick returns * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGEEVX', -INFO ) RETURN END IF IF( LWORK.EQ.LQUERV ) \$ RETURN IF( N.EQ.0 ) \$ RETURN * * Get machine constants * EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM * * Scale A if max element outside range [SMLNUM,BIGNUM] * ICOND = 0 ANRM = ZLANGE( 'M', N, N, A, LDA, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) \$ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) * * Balance the matrix and compute ABNRM * CALL ZGEBAL( BALANC, N, A, LDA, ILO, IHI, SCALE, IERR ) ABNRM = ZLANGE( '1', N, N, A, LDA, DUM ) IF( SCALEA ) THEN DUM( 1 ) = ABNRM CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) ABNRM = DUM( 1 ) END IF * * Reduce to upper Hessenberg form * (CWorkspace: need 2*N, prefer N+N*NB) * (RWorkspace: none) * ITAU = 1 IWRK = ITAU + N CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * IF( WANTVL ) THEN * * Want left eigenvectors * Copy Householder vectors to VL * SIDE = 'L' CALL ZLACPY( 'L', N, N, A, LDA, VL, LDVL ) * * Generate unitary matrix in VL * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) * (RWorkspace: none) * CALL ZUNGHR( N, ILO, IHI, VL, LDVL, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * * Perform QR iteration, accumulating Schur vectors in VL * (CWorkspace: need 1, prefer HSWORK (see comments) ) * (RWorkspace: none) * IWRK = ITAU CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VL, LDVL, \$ WORK( IWRK ), LWORK-IWRK+1, INFO ) * IF( WANTVR ) THEN * * Want left and right eigenvectors * Copy Schur vectors to VR * SIDE = 'B' CALL ZLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) END IF * ELSE IF( WANTVR ) THEN * * Want right eigenvectors * Copy Householder vectors to VR * SIDE = 'R' CALL ZLACPY( 'L', N, N, A, LDA, VR, LDVR ) * * Generate unitary matrix in VR * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) * (RWorkspace: none) * CALL ZUNGHR( N, ILO, IHI, VR, LDVR, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * * Perform QR iteration, accumulating Schur vectors in VR * (CWorkspace: need 1, prefer HSWORK (see comments) ) * (RWorkspace: none) * IWRK = ITAU CALL ZHSEQR( 'S', 'V', N, ILO, IHI, A, LDA, W, VR, LDVR, \$ WORK( IWRK ), LWORK-IWRK+1, INFO ) * ELSE * * Compute eigenvalues only * If condition numbers desired, compute Schur form * IF( WNTSNN ) THEN JOB = 'E' ELSE JOB = 'S' END IF * * (CWorkspace: need 1, prefer HSWORK (see comments) ) * (RWorkspace: none) * IWRK = ITAU CALL ZHSEQR( JOB, 'N', N, ILO, IHI, A, LDA, W, VR, LDVR, \$ WORK( IWRK ), LWORK-IWRK+1, INFO ) END IF * * If INFO > 0 from ZHSEQR, then quit * IF( INFO.GT.0 ) \$ GO TO 50 * IF( WANTVL .OR. WANTVR ) THEN * * Compute left and/or right eigenvectors * (CWorkspace: need 2*N) * (RWorkspace: need N) * CALL ZTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, \$ N, NOUT, WORK( IWRK ), RWORK, IERR ) END IF * * Compute condition numbers if desired * (CWorkspace: need N*N+2*N unless SENSE = 'E') * (RWorkspace: need 2*N unless SENSE = 'E') * IF( .NOT.WNTSNN ) THEN CALL ZTRSNA( SENSE, 'A', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, \$ RCONDE, RCONDV, N, NOUT, WORK( IWRK ), N, RWORK, \$ ICOND ) END IF * IF( WANTVL ) THEN * * Undo balancing of left eigenvectors * CALL ZGEBAK( BALANC, 'L', N, ILO, IHI, SCALE, N, VL, LDVL, \$ IERR ) * * Normalize left eigenvectors and make largest component real * DO 20 I = 1, N SCL = ONE / DZNRM2( N, VL( 1, I ), 1 ) CALL ZDSCAL( N, SCL, VL( 1, I ), 1 ) DO 10 K = 1, N RWORK( K ) = DBLE( VL( K, I ) )**2 + \$ DIMAG( VL( K, I ) )**2 10 CONTINUE K = IDAMAX( N, RWORK, 1 ) TMP = DCONJG( VL( K, I ) ) / SQRT( RWORK( K ) ) CALL ZSCAL( N, TMP, VL( 1, I ), 1 ) VL( K, I ) = DCMPLX( DBLE( VL( K, I ) ), ZERO ) 20 CONTINUE END IF * IF( WANTVR ) THEN * * Undo balancing of right eigenvectors * CALL ZGEBAK( BALANC, 'R', N, ILO, IHI, SCALE, N, VR, LDVR, \$ IERR ) * * Normalize right eigenvectors and make largest component real * DO 40 I = 1, N SCL = ONE / DZNRM2( N, VR( 1, I ), 1 ) CALL ZDSCAL( N, SCL, VR( 1, I ), 1 ) DO 30 K = 1, N RWORK( K ) = DBLE( VR( K, I ) )**2 + \$ DIMAG( VR( K, I ) )**2 30 CONTINUE K = IDAMAX( N, RWORK, 1 ) TMP = DCONJG( VR( K, I ) ) / SQRT( RWORK( K ) ) CALL ZSCAL( N, TMP, VR( 1, I ), 1 ) VR( K, I ) = DCMPLX( DBLE( VR( K, I ) ), ZERO ) 40 CONTINUE END IF * * Undo scaling if necessary * 50 CONTINUE IF( SCALEA ) THEN CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ), \$ MAX( N-INFO, 1 ), IERR ) IF( INFO.EQ.0 ) THEN IF( ( WNTSNV .OR. WNTSNB ) .AND. ICOND.EQ.0 ) \$ CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, RCONDV, N, \$ IERR ) ELSE CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) END IF END IF * WORK( 1 ) = MAXWRK RETURN * * End of ZGEEVX * END