SUBROUTINE ZGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, $ WORK, LWORK, RWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. Scalar Arguments .. CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), $ W( * ), WORK( * ) * .. * * Purpose * ======= * * ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the * eigenvalues and, optionally, the left and/or right eigenvectors. * * 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. * * Arguments * ========= * * JOBVL (input) CHARACTER*1 * = 'N': left eigenvectors of A are not computed; * = 'V': left eigenvectors of are computed. * * JOBVR (input) CHARACTER*1 * = 'N': right eigenvectors of A are not computed; * = 'V': right eigenvectors of A are computed. * * 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. * * 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. * * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,2*N). * For good performance, LWORK must generally be larger. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * 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 have been computed; * elements and i+1:N of W contain eigenvalues which have * converged. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, SCALEA, WANTVL, WANTVR CHARACTER SIDE INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU, $ IWRK, K, 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, XERBLA, ZDSCAL, ZGEBAK, ZGEBAL, ZGEHRD, $ ZHSEQR, ZLACPY, ZLASCL, ZSCAL, ZTREVC, 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, SQRT * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) WANTVL = LSAME( JOBVL, 'V' ) WANTVR = LSAME( JOBVR, 'V' ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.LSAME( JOBVL, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTVR ) .AND. ( .NOT.LSAME( JOBVR, 'N' ) ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDVL.LT.1 .OR. ( WANTVL .AND. LDVL.LT.N ) ) THEN INFO = -8 ELSE IF( LDVR.LT.1 .OR. ( WANTVR .AND. LDVR.LT.N ) ) THEN INFO = -10 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.) * IF( INFO.EQ.0 ) THEN IF( N.EQ.0 ) THEN MINWRK = 1 MAXWRK = 1 ELSE MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 ) MINWRK = 2*N IF( WANTVL ) THEN MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', $ ' ', N, 1, N, -1 ) ) CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VL, LDVL, $ WORK, -1, INFO ) ELSE IF( WANTVR ) THEN MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', $ ' ', N, 1, N, -1 ) ) CALL ZHSEQR( 'S', 'V', N, 1, N, A, LDA, W, VR, LDVR, $ WORK, -1, INFO ) ELSE CALL ZHSEQR( 'E', 'N', N, 1, N, A, LDA, W, VR, LDVR, $ WORK, -1, INFO ) END IF HSWORK = WORK( 1 ) MAXWRK = MAX( MAXWRK, HSWORK, MINWRK ) END IF WORK( 1 ) = MAXWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -12 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGEEV ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * 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] * 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 * (CWorkspace: none) * (RWorkspace: need N) * IBAL = 1 CALL ZGEBAL( 'B', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR ) * * 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 * (CWorkspace: need 1, prefer HSWORK (see comments) ) * (RWorkspace: none) * IWRK = ITAU CALL ZHSEQR( 'E', '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 2*N) * IRWORK = IBAL + N CALL ZTREVC( SIDE, 'B', SELECT, N, A, LDA, VL, LDVL, VR, LDVR, $ N, NOUT, WORK( IWRK ), RWORK( IRWORK ), IERR ) END IF * IF( WANTVL ) THEN * * Undo balancing of left eigenvectors * (CWorkspace: none) * (RWorkspace: need N) * CALL ZGEBAK( 'B', 'L', N, ILO, IHI, RWORK( IBAL ), 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( IRWORK+K-1 ) = DBLE( VL( K, I ) )**2 + $ DIMAG( VL( K, I ) )**2 10 CONTINUE K = IDAMAX( N, RWORK( IRWORK ), 1 ) TMP = DCONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) 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 * (CWorkspace: none) * (RWorkspace: need N) * CALL ZGEBAK( 'B', 'R', N, ILO, IHI, RWORK( IBAL ), 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( IRWORK+K-1 ) = DBLE( VR( K, I ) )**2 + $ DIMAG( VR( K, I ) )**2 30 CONTINUE K = IDAMAX( N, RWORK( IRWORK ), 1 ) TMP = DCONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) 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.GT.0 ) THEN CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) END IF END IF * WORK( 1 ) = MAXWRK RETURN * * End of ZGEEV * END