SUBROUTINE CGEEV( JOBVL, JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, \$ 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 JOBVL, JOBVR INTEGER INFO, LDA, LDVL, LDVR, LWORK, N * .. * .. Array Arguments .. REAL RWORK( * ) COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), \$ W( * ), WORK( * ) * .. * * Purpose * ======= * * CGEEV 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 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 array, dimension (N) * W contains the computed eigenvalues. * * VL (output) COMPLEX 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 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 array, dimension (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, 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) REAL 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 .. INTEGER LQUERV PARAMETER ( LQUERV = -1 ) REAL ZERO, ONE PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) * .. * .. Local Scalars .. LOGICAL SCALEA, WANTVL, WANTVR CHARACTER SIDE INTEGER HSWORK, I, IBAL, IERR, IHI, ILO, IRWORK, ITAU, \$ IWRK, K, MAXB, MAXWRK, MINWRK, NOUT REAL ANRM, BIGNUM, CSCALE, EPS, SCL, SMLNUM COMPLEX TMP * .. * .. Local Arrays .. LOGICAL SELECT( 1 ) REAL DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL CGEBAK, CGEBAL, CGEHRD, CHSEQR, CLACPY, CLASCL, \$ CSCAL, CSSCAL, CTREVC, CUNGHR, SLABAD, XERBLA * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV, ISAMAX REAL CLANGE, SCNRM2, SLAMCH EXTERNAL LSAME, ILAENV, ISAMAX, CLANGE, SCNRM2, SLAMCH * .. * .. Intrinsic Functions .. INTRINSIC AIMAG, CMPLX, CONJG, MAX, MIN, REAL, SQRT * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 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 CHSEQR, 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, 'CGEHRD', ' ', N, 1, N, 0 ) IF( ( .NOT.WANTVL ) .AND. ( .NOT.WANTVR ) ) THEN MINWRK = MAX( 1, 2*N ) MAXB = MAX( ILAENV( 8, 'CHSEQR', 'EN', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'EN', N, 1, \$ N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, HSWORK ) ELSE MINWRK = MAX( 1, 2*N ) MAXWRK = MAX( MAXWRK, N+( N-1 )* \$ ILAENV( 1, 'CUNGHR', ' ', N, 1, N, -1 ) ) MAXB = MAX( ILAENV( 8, 'CHSEQR', 'SV', N, 1, N, -1 ), 2 ) K = MIN( MAXB, N, MAX( 2, ILAENV( 4, 'CHSEQR', 'SV', N, 1, \$ N, -1 ) ) ) HSWORK = MAX( K*( K+2 ), 2*N ) MAXWRK = MAX( MAXWRK, HSWORK, 2*N ) END IF WORK( 1 ) = MAXWRK IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV ) \$ INFO = -12 END IF * * Quick returns * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CGEEV ', -INFO ) RETURN END IF IF( LWORK.EQ.LQUERV ) RETURN IF( N.EQ.0 ) \$ RETURN * * Get machine constants * EPS = SLAMCH( 'P' ) SMLNUM = SLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM * * Scale A if max element outside range [SMLNUM,BIGNUM] * ANRM = CLANGE( '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 CLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) * * Balance the matrix * (CWorkspace: none) * (RWorkspace: need N) * IBAL = 1 CALL CGEBAL( '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 CGEHRD( 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 CLACPY( '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 CUNGHR( 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 CHSEQR( '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 CLACPY( 'F', N, N, VL, LDVL, VR, LDVR ) END IF * ELSE IF( WANTVR ) THEN * * Want right eigenvectors * Copy Householder vectors to VR * SIDE = 'R' CALL CLACPY( '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 CUNGHR( 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 CHSEQR( '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 CHSEQR( 'E', 'N', N, ILO, IHI, A, LDA, W, VR, LDVR, \$ WORK( IWRK ), LWORK-IWRK+1, INFO ) END IF * * If INFO > 0 from CHSEQR, 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 CTREVC( 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 CGEBAK( '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 / SCNRM2( N, VL( 1, I ), 1 ) CALL CSSCAL( N, SCL, VL( 1, I ), 1 ) DO 10 K = 1, N RWORK( IRWORK+K-1 ) = REAL( VL( K, I ) )**2 + \$ AIMAG( VL( K, I ) )**2 10 CONTINUE K = ISAMAX( N, RWORK( IRWORK ), 1 ) TMP = CONJG( VL( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) CALL CSCAL( N, TMP, VL( 1, I ), 1 ) VL( K, I ) = CMPLX( REAL( VL( K, I ) ), ZERO ) 20 CONTINUE END IF * IF( WANTVR ) THEN * * Undo balancing of right eigenvectors * (CWorkspace: none) * (RWorkspace: need N) * CALL CGEBAK( '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 / SCNRM2( N, VR( 1, I ), 1 ) CALL CSSCAL( N, SCL, VR( 1, I ), 1 ) DO 30 K = 1, N RWORK( IRWORK+K-1 ) = REAL( VR( K, I ) )**2 + \$ AIMAG( VR( K, I ) )**2 30 CONTINUE K = ISAMAX( N, RWORK( IRWORK ), 1 ) TMP = CONJG( VR( K, I ) ) / SQRT( RWORK( IRWORK+K-1 ) ) CALL CSCAL( N, TMP, VR( 1, I ), 1 ) VR( K, I ) = CMPLX( REAL( VR( K, I ) ), ZERO ) 40 CONTINUE END IF * * Undo scaling if necessary * 50 CONTINUE IF( SCALEA ) THEN CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, N-INFO, 1, W( INFO+1 ), \$ MAX( N-INFO, 1 ), IERR ) IF( INFO.GT.0 ) THEN CALL CLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, W, N, IERR ) END IF END IF * WORK( 1 ) = MAXWRK RETURN * * End of CGEEV * END