Exemple #1
0
extern "C" magma_int_t
magma_zgeev_m(
    char jobvl, char jobvr, magma_int_t n,
    magmaDoubleComplex *A, magma_int_t lda,
    magmaDoubleComplex *W,
    magmaDoubleComplex *vl, magma_int_t ldvl,
    magmaDoubleComplex *vr, magma_int_t ldvr,
    magmaDoubleComplex *work, magma_int_t lwork,
    double *rwork, magma_int_t *info )
{
/*  -- MAGMA (version 1.4.1) --
       Univ. of Tennessee, Knoxville
       Univ. of California, Berkeley
       Univ. of Colorado, Denver
       December 2013

    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 >= (1+nb)*N.

            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.
    =====================================================================    */

    #define vl(i,j)  (vl + (i) + (j)*ldvl)
    #define vr(i,j)  (vr + (i) + (j)*ldvr)
    
    magma_int_t c_one  = 1;
    magma_int_t c_zero = 0;
    
    double d__1, d__2;
    magmaDoubleComplex z__1, z__2;
    magmaDoubleComplex tmp;
    double scl;
    double dum[1], eps;
    double anrm, cscale, bignum, smlnum;
    magma_int_t i, k, ilo, ihi;
    magma_int_t ibal, ierr, itau, iwrk, nout, liwrk, i__1, i__2, nb;
    magma_int_t scalea, minwrk, irwork, lquery, wantvl, wantvr, select[1];

    char side[2]   = {0, 0};
    char jobvl_[2] = {jobvl, 0};
    char jobvr_[2] = {jobvr, 0};

    irwork = 0;
    *info = 0;
    lquery = lwork == -1;
    wantvl = lapackf77_lsame( jobvl_, "V" );
    wantvr = lapackf77_lsame( jobvr_, "V" );
    if (! wantvl && ! lapackf77_lsame( jobvl_, "N" )) {
        *info = -1;
    } else if (! wantvr && ! lapackf77_lsame( jobvr_, "N" )) {
        *info = -2;
    } else if (n < 0) {
        *info = -3;
    } else if (lda < max(1,n)) {
        *info = -5;
    } else if ( (ldvl < 1) || (wantvl && (ldvl < n))) {
        *info = -8;
    } else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
        *info = -10;
    }

    /* Compute workspace */
    nb = magma_get_zgehrd_nb( n );
    if (*info == 0) {
        minwrk = (1+nb)*n;
        work[0] = MAGMA_Z_MAKE( minwrk, 0 );

        if (lwork < minwrk && ! lquery) {
            *info = -12;
        }
    }

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }
    else if (lquery) {
        return *info;
    }

    /* Quick return if possible */
    if (n == 0) {
        return *info;
    }
    
    #if defined(Version3) || defined(Version4) || defined(Version5)
    magmaDoubleComplex *dT;
    if (MAGMA_SUCCESS != magma_zmalloc( &dT, nb*n )) {
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    #endif
    #if defined(Version4) || defined(Version5)
    magmaDoubleComplex *T;
    if (MAGMA_SUCCESS != magma_zmalloc_cpu( &T, nb*n )) {
        magma_free( dT );
        *info = MAGMA_ERR_HOST_ALLOC;
        return *info;
    }
    #endif

    /* Get machine constants */
    eps    = lapackf77_dlamch( "P" );
    smlnum = lapackf77_dlamch( "S" );
    bignum = 1. / smlnum;
    lapackf77_dlabad( &smlnum, &bignum );
    smlnum = magma_dsqrt( smlnum ) / eps;
    bignum = 1. / smlnum;

    /* Scale A if max element outside range [SMLNUM,BIGNUM] */
    anrm = lapackf77_zlange( "M", &n, &n, A, &lda, dum );
    scalea = 0;
    if (anrm > 0. && anrm < smlnum) {
        scalea = 1;
        cscale = smlnum;
    } else if (anrm > bignum) {
        scalea = 1;
        cscale = bignum;
    }
    if (scalea) {
        lapackf77_zlascl( "G", &c_zero, &c_zero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
    }

    /* Balance the matrix
     * (CWorkspace: none)
     * (RWorkspace: need N) */
    ibal = 0;
    lapackf77_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 = 0;
    iwrk = itau + n;
    liwrk = lwork - iwrk;

    #if defined(Version1)
        // Version 1 - LAPACK
        lapackf77_zgehrd( &n, &ilo, &ihi, A, &lda,
                          &work[itau], &work[iwrk], &liwrk, &ierr );
    #elif defined(Version2)
        // Version 2 - LAPACK consistent HRD
        magma_zgehrd2( n, ilo, ihi, A, lda,
                       &work[itau], &work[iwrk], &liwrk, &ierr );
    #elif defined(Version3)
        // Version 3 - LAPACK consistent MAGMA HRD + matrices T stored,
        magma_zgehrd( n, ilo, ihi, A, lda,
                      &work[itau], &work[iwrk], liwrk, dT, &ierr );
    #elif defined(Version4) || defined(Version5)
        // Version 4 - Multi-GPU, T on host
        magma_zgehrd_m( n, ilo, ihi, A, lda,
                        &work[itau], &work[iwrk], liwrk, T, &ierr );
        magma_zsetmatrix( nb, n, T, nb, dT, nb );
    #endif

    if (wantvl) {
        /* Want left eigenvectors
         * Copy Householder vectors to VL */
        side[0] = 'L';
        lapackf77_zlacpy( MagmaLowerStr, &n, &n, A, &lda, vl, &ldvl );

        /* Generate unitary matrix in VL
         * (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
         * (RWorkspace: none) */
        #if defined(Version1) || defined(Version2)
            // Version 1 & 2 - LAPACK
            lapackf77_zunghr( &n, &ilo, &ihi, vl, &ldvl, &work[itau],
                              &work[iwrk], &liwrk, &ierr );
        #elif defined(Version3) || defined(Version4)
            // Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
            magma_zunghr( n, ilo, ihi, vl, ldvl, &work[itau], dT, nb, &ierr );
        #elif defined(Version5)
            // Version 5 - Multi-GPU, T on host
            magma_zunghr_m( n, ilo, ihi, vl, ldvl, &work[itau], T, nb, &ierr );
        #endif

        /* Perform QR iteration, accumulating Schur vectors in VL
         * (CWorkspace: need 1, prefer HSWORK (see comments) )
         * (RWorkspace: none) */
        iwrk = itau;
        liwrk = lwork - iwrk;
        lapackf77_zhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, W,
                          vl, &ldvl, &work[iwrk], &liwrk, info );

        if (wantvr) {
            /* Want left and right eigenvectors
             * Copy Schur vectors to VR */
            side[0] = 'B';
            lapackf77_zlacpy( "F", &n, &n, vl, &ldvl, vr, &ldvr );
        }
    }
    else if (wantvr) {
        /* Want right eigenvectors
         * Copy Householder vectors to VR */
        side[0] = 'R';
        lapackf77_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) */
        #if defined(Version1) || defined(Version2)
            // Version 1 & 2 - LAPACK
            lapackf77_zunghr( &n, &ilo, &ihi, vr, &ldvr, &work[itau],
                              &work[iwrk], &liwrk, &ierr );
        #elif defined(Version3) || defined(Version4)
            // Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
            magma_zunghr( n, ilo, ihi, vr, ldvr, &work[itau], dT, nb, &ierr );
        #elif defined(Version5)
            // Version 5 - Multi-GPU, T on host
            magma_zunghr_m( n, ilo, ihi, vr, ldvr, &work[itau], T, nb, &ierr );
        #endif

        /* Perform QR iteration, accumulating Schur vectors in VR
         * (CWorkspace: need 1, prefer HSWORK (see comments) )
         * (RWorkspace: none) */
        iwrk = itau;
        liwrk = lwork - iwrk;
        lapackf77_zhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, W,
                          vr, &ldvr, &work[iwrk], &liwrk, info );
    }
    else {
        /* Compute eigenvalues only
         * (CWorkspace: need 1, prefer HSWORK (see comments) )
         * (RWorkspace: none) */
        iwrk = itau;
        liwrk = lwork - iwrk;
        lapackf77_zhseqr( "E", "N", &n, &ilo, &ihi, A, &lda, W,
                          vr, &ldvr, &work[iwrk], &liwrk, info );
    }

    /* If INFO > 0 from ZHSEQR, then quit */
    if (*info > 0) {
        goto CLEANUP;
    }

    if (wantvl || wantvr) {
        /* Compute left and/or right eigenvectors
         * (CWorkspace: need 2*N)
         * (RWorkspace: need 2*N) */
        irwork = ibal + n;
        lapackf77_ztrevc( side, "B", select, &n, A, &lda, vl, &ldvl,
                          vr, &ldvr, &n, &nout, &work[iwrk], &rwork[irwork], &ierr );
    }

    if (wantvl) {
        /* Undo balancing of left eigenvectors
         * (CWorkspace: none)
         * (RWorkspace: need N) */
        lapackf77_zgebak( "B", "L", &n, &ilo, &ihi, &rwork[ibal], &n,
                          vl, &ldvl, &ierr );

        /* Normalize left eigenvectors and make largest component real */
        for (i = 0; i < n; ++i) {
            scl = 1. / cblas_dznrm2( n, vl(0,i), 1 );
            cblas_zdscal( n, scl, vl(0,i), 1 );
            for (k = 0; k < n; ++k) {
                /* Computing 2nd power */
                d__1 = MAGMA_Z_REAL( *vl(k,i) );
                d__2 = MAGMA_Z_IMAG( *vl(k,i) );
                rwork[irwork + k] = d__1*d__1 + d__2*d__2;
            }
            k = cblas_idamax( n, &rwork[irwork], 1 );
            z__2 = MAGMA_Z_CNJG( *vl(k,i) );
            d__1 = magma_dsqrt( rwork[irwork + k] );
            MAGMA_Z_DSCALE( z__1, z__2, d__1 );
            tmp = z__1;
            cblas_zscal( n, CBLAS_SADDR(tmp), vl(0,i), 1 );
            d__1 = MAGMA_Z_REAL( *vl(k,i) );
            z__1 = MAGMA_Z_MAKE( d__1, 0 );
            *vl(k,i) = z__1;
        }
    }

    if (wantvr) {
        /* Undo balancing of right eigenvectors
         * (CWorkspace: none)
         * (RWorkspace: need N) */
        lapackf77_zgebak( "B", "R", &n, &ilo, &ihi, &rwork[ibal], &n,
                          vr, &ldvr, &ierr );

        /* Normalize right eigenvectors and make largest component real */
        for (i = 0; i < n; ++i) {
            scl = 1. / cblas_dznrm2( n, vr(0,i), 1 );
            cblas_zdscal( n, scl, vr(0,i), 1 );
            for (k = 0; k < n; ++k) {
                /* Computing 2nd power */
                d__1 = MAGMA_Z_REAL( *vr(k,i) );
                d__2 = MAGMA_Z_IMAG( *vr(k,i) );
                rwork[irwork + k] = d__1*d__1 + d__2*d__2;
            }
            k = cblas_idamax( n, &rwork[irwork], 1 );
            z__2 = MAGMA_Z_CNJG( *vr(k,i) );
            d__1 = magma_dsqrt( rwork[irwork + k] );
            MAGMA_Z_DSCALE( z__1, z__2, d__1 );
            tmp = z__1;
            cblas_zscal( n, CBLAS_SADDR(tmp), vr(0,i), 1 );
            d__1 = MAGMA_Z_REAL( *vr(k,i) );
            z__1 = MAGMA_Z_MAKE( d__1, 0 );
            *vr(k,i) = z__1;
        }
    }

CLEANUP:
    /* Undo scaling if necessary */
    if (scalea) {
        i__1 = n - (*info);
        i__2 = max( n - (*info), 1 );
        lapackf77_zlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
                          W + (*info), &i__2, &ierr );
        if (*info > 0) {
            i__1 = ilo - 1;
            lapackf77_zlascl( "G", &c_zero, &c_zero, &cscale, &anrm, &i__1, &c_one,
                              W, &n, &ierr );
        }
    }

    #if defined(Version3) || defined(Version4) || defined(Version5)
    magma_free( dT );
    #endif
    #if defined(Version4) || defined(Version5)
    magma_free_cpu( T );
    #endif
    
    return *info;
} /* magma_zgeev */
extern "C" magma_int_t
magma_zgeev(magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
            magmaDoubleComplex *a, magma_int_t lda,
            magmaDoubleComplex *geev_w_array,
            magmaDoubleComplex *vl, magma_int_t ldvl,
            magmaDoubleComplex *vr, magma_int_t ldvr,
            magmaDoubleComplex *work, magma_int_t lwork,
            double *rwork, magma_int_t *info, magma_queue_t queue)
{
/*  -- clMAGMA (version 1.0.0) --
       Univ. of Tennessee, Knoxville
       Univ. of California, Berkeley
       Univ. of Colorado, Denver
       September 2012

    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 >= (1+nb)*N.   

            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.   
    =====================================================================    */

    magma_int_t c__1 = 1;
    magma_int_t c__0 = 0;
    
    magma_int_t a_dim1, a_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, 
            i__2, i__3;
    double d__1, d__2;
    magmaDoubleComplex z__1, z__2;

    magma_int_t i__, k, ihi;
    double scl;
    magma_int_t ilo;
    double dum[1], eps;
    magmaDoubleComplex tmp;
    magma_int_t ibal;
    double anrm;
    magma_int_t ierr, itau, iwrk, nout;
    magma_int_t scalea;
    double cscale;
    magma_int_t select[1];
    double bignum;
    magma_int_t minwrk;
    magma_int_t wantvl;
    double smlnum;
    magma_int_t irwork;
    magma_int_t lquery, wantvr;
    magma_int_t nb = 0;
    magmaDoubleComplex_ptr dT;

    //magma_timestr_t start, end;

    char side[2] = {0, 0};
    magma_vec_t jobvl_ = jobvl;
    magma_vec_t jobvr_ = jobvr;

    *info = 0;
    lquery = lwork == -1;
    wantvl = lapackf77_lsame(lapack_const(jobvl_), "V");
    wantvr = lapackf77_lsame(lapack_const(jobvr_), "V");
    if (! wantvl && ! lapackf77_lsame(lapack_const(jobvl_), "N")) {
        *info = -1;
    } else if (! wantvr && ! lapackf77_lsame(lapack_const(jobvr_), "N")) {
        *info = -2;
    } else if (n < 0) {
        *info = -3;
    } else if (lda < max(1,n)) {
        *info = -5;
    } else if ( (ldvl < 1) || (wantvl && (ldvl < n))) {
        *info = -8;
    } else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
        *info = -10;
    }

    /*  Compute workspace   */
    if (*info == 0) {
        nb = magma_get_zgehrd_nb(n);
        minwrk = (1+nb)*n;
        work[0] = MAGMA_Z_MAKE((double) minwrk, 0.);

        if (lwork < minwrk && ! lquery) {
            *info = -12;
        }
    }   

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }
    else if (lquery) {
        return *info;
    }

    /* Quick return if possible */
    if (n == 0) {
        return *info;
    }
   
    // if eigenvectors are needed
#if defined(VERSION3)
    if (MAGMA_SUCCESS != magma_malloc(&dT, nb*n*sizeof(magmaDoubleComplex) )) {
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
#endif

    a_dim1 = lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    vl_dim1 = ldvl;
    vl_offset = 1 + vl_dim1;
    vl -= vl_offset;
    vr_dim1 = ldvr;
    vr_offset = 1 + vr_dim1;
    vr -= vr_offset;
    --work;
    --rwork;

    /* Get machine constants */
    eps    = lapackf77_dlamch("P");
    smlnum = lapackf77_dlamch("S");
    bignum = 1. / smlnum;
    lapackf77_dlabad(&smlnum, &bignum);
    smlnum = magma_dsqrt(smlnum) / eps;
    bignum = 1. / smlnum;

    /* Scale A if max element outside range [SMLNUM,BIGNUM] */
    anrm = lapackf77_zlange("M", &n, &n, &a[a_offset], &lda, dum);
    scalea = 0;
    if (anrm > 0. && anrm < smlnum) {
        scalea = 1;
        cscale = smlnum;
    } else if (anrm > bignum) {
        scalea = 1;
        cscale = bignum;
    }
    if (scalea) {
        lapackf77_zlascl("G", &c__0, &c__0, &anrm, &cscale, &n, &n, &a[a_offset], &lda, &
                ierr);
    }

    /* Balance the matrix   
       (CWorkspace: none)   
       (RWorkspace: need N) */
    ibal = 1;
    lapackf77_zgebal("B", &n, &a[a_offset], &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;
    i__1 = lwork - iwrk + 1;

    //start = get_current_time();
#if defined(VERSION1)
    /*
     * Version 1 - LAPACK
     */
    lapackf77_zgehrd(&n, &ilo, &ihi, &a[a_offset], &lda,
                     &work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION2)
    /*
     *  Version 2 - LAPACK consistent HRD
     */
    magma_zgehrd2(n, ilo, ihi, &a[a_offset], lda,
                  &work[itau], &work[iwrk], &i__1, &ierr);    
#elif defined(VERSION3)
    /*  
     * Version 3 - LAPACK consistent MAGMA HRD + matrices T stored, 
     */
    magma_zgehrd(n, ilo, ihi, &a[a_offset], lda,
                 &work[itau], &work[iwrk], i__1, dT, 0, &ierr, queue);
#endif
    //end = get_current_time();
    //printf("    Time for zgehrd = %5.2f sec\n", GetTimerValue(start,end)/1000.);

    if (wantvl) {
      /*        Want left eigenvectors   
                Copy Householder vectors to VL */
        side[0] = 'L';
        lapackf77_zlacpy(MagmaLowerStr, &n, &n, 
                         &a[a_offset], &lda, &vl[vl_offset], &ldvl);

        /* Generate unitary matrix in VL   
           (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)   
           (RWorkspace: none) */
        i__1 = lwork - iwrk + 1;

        //start = get_current_time();
#if defined(VERSION1) || defined(VERSION2)
        /*
         * Version 1 & 2 - LAPACK
         */
        lapackf77_zunghr(&n, &ilo, &ihi, &vl[vl_offset], &ldvl, 
                         &work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
        /*
         * Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
         */
        magma_zunghr(n, ilo, ihi, &vl[vl_offset], ldvl, &work[itau], 
                     dT, 0, nb, &ierr, queue);
#endif
        //end = get_current_time();
        //printf("    Time for zunghr = %5.2f sec\n", GetTimerValue(start,end)/1000.);

        /* Perform QR iteration, accumulating Schur vectors in VL   
           (CWorkspace: need 1, prefer HSWORK (see comments) )   
           (RWorkspace: none) */
        iwrk = itau;
        i__1 = lwork - iwrk + 1;
        lapackf77_zhseqr("S", "V", &n, &ilo, &ihi, &a[a_offset], &lda, geev_w_array,
                &vl[vl_offset], &ldvl, &work[iwrk], &i__1, info);

        if (wantvr) 
          {
            /* Want left and right eigenvectors   
               Copy Schur vectors to VR */
            side[0] = 'B';
            lapackf77_zlacpy("F", &n, &n, &vl[vl_offset], &ldvl, &vr[vr_offset], &ldvr);
          }

    } else if (wantvr) {
        /*  Want right eigenvectors   
            Copy Householder vectors to VR */
        side[0] = 'R';
        lapackf77_zlacpy("L", &n, &n, &a[a_offset], &lda, &vr[vr_offset], &ldvr);

        /* Generate unitary matrix in VR   
           (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)   
           (RWorkspace: none) */
        i__1 = lwork - iwrk + 1;
        //start = get_current_time();
#if defined(VERSION1) || defined(VERSION2)
        /*
         * Version 1 & 2 - LAPACK
         */
        lapackf77_zunghr(&n, &ilo, &ihi, &vr[vr_offset], &ldvr, 
                         &work[itau], &work[iwrk], &i__1, &ierr);
#elif defined(VERSION3)
        /*
         * Version 3 - LAPACK consistent MAGMA HRD + matrices T stored
         */
        magma_zunghr(n, ilo, ihi, &vr[vr_offset], ldvr, 
                     &work[itau], dT, 0, nb, &ierr, queue);
#endif
        //end = get_current_time();
        //printf("    Time for zunghr = %5.2f sec\n", GetTimerValue(start,end)/1000.);

        /* Perform QR iteration, accumulating Schur vectors in VR   
           (CWorkspace: need 1, prefer HSWORK (see comments) )   
           (RWorkspace: none) */
        iwrk = itau;
        i__1 = lwork - iwrk + 1;
        lapackf77_zhseqr("S", "V", &n, &ilo, &ihi, &a[a_offset], &lda, geev_w_array, 
                &vr[vr_offset], &ldvr, &work[iwrk], &i__1, info);
    } else {
      /*  Compute eigenvalues only   
          (CWorkspace: need 1, prefer HSWORK (see comments) )   
          (RWorkspace: none) */
        iwrk = itau;
        i__1 = lwork - iwrk + 1;
        lapackf77_zhseqr("E", "N", &n, &ilo, &ihi, &a[a_offset], &lda, geev_w_array,
                &vr[vr_offset], &ldvr, &work[iwrk], &i__1, info);
    }

    /* If INFO > 0 from ZHSEQR, then quit */
    if (*info > 0) {
        goto L50;
    }

    if (wantvl || wantvr) {
        /*  Compute left and/or right eigenvectors   
            (CWorkspace: need 2*N)   
            (RWorkspace: need 2*N) */
        irwork = ibal + n;
        lapackf77_ztrevc(side, "B", select, &n, &a[a_offset], &lda, &vl[vl_offset], &ldvl,
                &vr[vr_offset], &ldvr, &n, &nout, &work[iwrk], &rwork[irwork], 
                &ierr);
    }

    if (wantvl) {
        /*  Undo balancing of left eigenvectors   
            (CWorkspace: none)   
            (RWorkspace: need N) */
        lapackf77_zgebak("B", "L", &n, &ilo, &ihi, &rwork[ibal], &n, 
                         &vl[vl_offset], &ldvl, &ierr);

        /* Normalize left eigenvectors and make largest component real */
        for (i__ = 1; i__ <= n; ++i__) {
            scl = 1. / cblas_dznrm2(n, &vl[i__ * vl_dim1 + 1], 1);
            cblas_zdscal(n, scl, &vl[i__ * vl_dim1 + 1], 1);
            i__2 = n;
            for (k = 1; k <= i__2; ++k) 
            {
                i__3 = k + i__ * vl_dim1;
                /* Computing 2nd power */
                d__1 = MAGMA_Z_REAL(vl[i__3]);
                /* Computing 2nd power */
                d__2 = MAGMA_Z_IMAG(vl[k + i__ * vl_dim1]);
                rwork[irwork + k - 1] = d__1 * d__1 + d__2 * d__2;
            }
            /* Comment:
                   Fortran BLAS does not have to add 1
                   C       BLAS must add one to cblas_idamax */
            k = cblas_idamax(n, &rwork[irwork], 1)+1;
            z__2 = MAGMA_Z_CNJG(vl[k + i__ * vl_dim1]);
            d__1 = magma_dsqrt(rwork[irwork + k - 1]);
            MAGMA_Z_DSCALE(z__1, z__2, d__1);
            tmp = z__1;
            cblas_zscal(n, CBLAS_SADDR(tmp), &vl[i__ * vl_dim1 + 1], 1);
            i__2 = k + i__ * vl_dim1;
            i__3 = k + i__ * vl_dim1;
            d__1 = MAGMA_Z_REAL(vl[i__3]);
            MAGMA_Z_SET2REAL(z__1, d__1);
            vl[i__2] = z__1;
        }
    }

    if (wantvr) {
      /*  Undo balancing of right eigenvectors   
          (CWorkspace: none)   
          (RWorkspace: need N) */
        lapackf77_zgebak("B", "R", &n, &ilo, &ihi, &rwork[ibal], &n, 
                         &vr[vr_offset], &ldvr, &ierr);

        /* Normalize right eigenvectors and make largest component real */
        for (i__ = 1; i__ <= n; ++i__) {
            scl = 1. / cblas_dznrm2(n, &vr[i__ * vr_dim1 + 1], 1);
            cblas_zdscal(n, scl, &vr[i__ * vr_dim1 + 1], 1);
            i__2 = n;
            for (k = 1; k <= i__2; ++k) {
                i__3 = k + i__ * vr_dim1;
                /* Computing 2nd power */
                d__1 = MAGMA_Z_REAL(vr[i__3]);
                /* Computing 2nd power */
                d__2 = MAGMA_Z_IMAG(vr[k + i__ * vr_dim1]);
                rwork[irwork + k - 1] = d__1 * d__1 + d__2 * d__2;
            }
            /* Comment:
                   Fortran BLAS does not have to add 1
                   C       BLAS must add one to cblas_idamax */
            k = cblas_idamax(n, &rwork[irwork], 1)+1;
            z__2 = MAGMA_Z_CNJG(vr[k + i__ * vr_dim1]);
            d__1 = magma_dsqrt(rwork[irwork + k - 1]);
            MAGMA_Z_DSCALE(z__1, z__2, d__1);
            tmp = z__1;
            cblas_zscal(n, CBLAS_SADDR(tmp), &vr[i__ * vr_dim1 + 1], 1);
            i__2 = k + i__ * vr_dim1;
            i__3 = k + i__ * vr_dim1;
            d__1 = MAGMA_Z_REAL(vr[i__3]);
            MAGMA_Z_SET2REAL(z__1, d__1);
            vr[i__2] = z__1;
        }
    }

    /*  Undo scaling if necessary */
L50:
    if (scalea) {
        i__1 = n - *info;
        /* Computing MAX */
        i__3 = n - *info;
        i__2 = max(i__3,1);
        lapackf77_zlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, 
                geev_w_array + *info, &i__2, &ierr);
        if (*info > 0) {
            i__1 = ilo - 1;
            lapackf77_zlascl("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, 
                    geev_w_array, &n, &ierr);
        }
    }

#if defined(VERSION3)
    magma_free( dT );
#endif
    return *info;
} /* magma_zgeev */
Exemple #3
0
/**
    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
    ---------
    @param[in]
    jobvl   magma_vec_t
      -     = MagmaNoVec:        left eigenvectors of A are not computed;
      -     = MagmaVec:          left eigenvectors of are computed.

    @param[in]
    jobvr   magma_vec_t
      -     = MagmaNoVec:        right eigenvectors of A are not computed;
      -     = MagmaVec:          right eigenvectors of A are computed.

    @param[in]
    n       INTEGER
            The order of the matrix A. N >= 0.

    @param[in,out]
    A       COMPLEX_16 array, dimension (LDA,N)
            On entry, the N-by-N matrix A.
            On exit, A has been overwritten.

    @param[in]
    lda     INTEGER
            The leading dimension of the array A.  LDA >= max(1,N).

    @param[out]
    w       COMPLEX_16 array, dimension (N)
            w contains the computed eigenvalues.

    @param[out]
    VL      COMPLEX_16 array, dimension (LDVL,N)
            If JOBVL = MagmaVec, the left eigenvectors u(j) are stored one
            after another in the columns of VL, in the same order
            as their eigenvalues.
            If JOBVL = MagmaNoVec, VL is not referenced.
            u(j) = VL(:,j), the j-th column of VL.

    @param[in]
    ldvl    INTEGER
            The leading dimension of the array VL.  LDVL >= 1; if
            JOBVL = MagmaVec, LDVL >= N.

    @param[out]
    VR      COMPLEX_16 array, dimension (LDVR,N)
            If JOBVR = MagmaVec, the right eigenvectors v(j) are stored one
            after another in the columns of VR, in the same order
            as their eigenvalues.
            If JOBVR = MagmaNoVec, VR is not referenced.
            v(j) = VR(:,j), the j-th column of VR.

    @param[in]
    ldvr    INTEGER
            The leading dimension of the array VR.  LDVR >= 1; if
            JOBVR = MagmaVec, LDVR >= N.

    @param[out]
    work    (workspace) COMPLEX_16 array, dimension (MAX(1,LWORK))
            On exit, if INFO = 0, WORK[0] returns the optimal LWORK.

    @param[in]
    lwork   INTEGER
            The dimension of the array WORK.  LWORK >= (1+nb)*N.
            For optimal performance, LWORK >= (1+2*nb)*N.
    \n
            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.

    @param
    rwork   (workspace) DOUBLE PRECISION array, dimension (2*N)

    @param[out]
    info    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.

    @ingroup magma_zgeev_driver
    ********************************************************************/
extern "C" magma_int_t
magma_zgeev(
    magma_vec_t jobvl, magma_vec_t jobvr, magma_int_t n,
    magmaDoubleComplex *A, magma_int_t lda,
    #ifdef COMPLEX
    magmaDoubleComplex *w,
    #else
    double *wr, double *wi,
    #endif
    magmaDoubleComplex *VL, magma_int_t ldvl,
    magmaDoubleComplex *VR, magma_int_t ldvr,
    magmaDoubleComplex *work, magma_int_t lwork,
    #ifdef COMPLEX
    double *rwork,
    #endif
    magma_int_t *info )
{
    #define VL(i,j)  (VL + (i) + (j)*ldvl)
    #define VR(i,j)  (VR + (i) + (j)*ldvr)
    
    const magma_int_t ione  = 1;
    const magma_int_t izero = 0;
    
    double d__1, d__2;
    magmaDoubleComplex tmp;
    double scl;
    double dum[1], eps;
    double anrm, cscale, bignum, smlnum;
    magma_int_t i, k, ilo, ihi;
    magma_int_t ibal, ierr, itau, iwrk, nout, liwrk, nb;
    magma_int_t scalea, minwrk, optwrk, irwork, lquery, wantvl, wantvr, select[1];

    magma_side_t side = MagmaRight;

    magma_timer_t time_total=0, time_gehrd=0, time_unghr=0, time_hseqr=0, time_trevc=0, time_sum=0;
    magma_flops_t flop_total=0, flop_gehrd=0, flop_unghr=0, flop_hseqr=0, flop_trevc=0, flop_sum=0;
    timer_start( time_total );
    flops_start( flop_total );
    
    irwork = 0;
    *info = 0;
    lquery = (lwork == -1);
    wantvl = (jobvl == MagmaVec);
    wantvr = (jobvr == MagmaVec);
    if (! wantvl && jobvl != MagmaNoVec) {
        *info = -1;
    } else if (! wantvr && jobvr != MagmaNoVec) {
        *info = -2;
    } else if (n < 0) {
        *info = -3;
    } else if (lda < max(1,n)) {
        *info = -5;
    } else if ( (ldvl < 1) || (wantvl && (ldvl < n))) {
        *info = -8;
    } else if ( (ldvr < 1) || (wantvr && (ldvr < n))) {
        *info = -10;
    }

    /* Compute workspace */
    nb = magma_get_zgehrd_nb( n );
    if (*info == 0) {
        minwrk = (1+  nb)*n;
        optwrk = (1+2*nb)*n;
        work[0] = MAGMA_Z_MAKE( optwrk, 0 );

        if (lwork < minwrk && ! lquery) {
            *info = -12;
        }
    }

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }
    else if (lquery) {
        return *info;
    }

    /* Quick return if possible */
    if (n == 0) {
        return *info;
    }
    
    #if defined(VERSION3)
    magmaDoubleComplex_ptr dT;
    if (MAGMA_SUCCESS != magma_zmalloc( &dT, nb*n )) {
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    #endif

    /* Get machine constants */
    eps    = lapackf77_dlamch( "P" );
    smlnum = lapackf77_dlamch( "S" );
    bignum = 1. / smlnum;
    lapackf77_dlabad( &smlnum, &bignum );
    smlnum = magma_dsqrt( smlnum ) / eps;
    bignum = 1. / smlnum;

    /* Scale A if max element outside range [SMLNUM,BIGNUM] */
    anrm = lapackf77_zlange( "M", &n, &n, A, &lda, dum );
    scalea = 0;
    if (anrm > 0. && anrm < smlnum) {
        scalea = 1;
        cscale = smlnum;
    } else if (anrm > bignum) {
        scalea = 1;
        cscale = bignum;
    }
    if (scalea) {
        lapackf77_zlascl( "G", &izero, &izero, &anrm, &cscale, &n, &n, A, &lda, &ierr );
    }

    /* Balance the matrix
     * (CWorkspace: none)
     * (RWorkspace: need N)
     *  - this space is reserved until after gebak */
    ibal = 0;
    lapackf77_zgebal( "B", &n, A, &lda, &ilo, &ihi, &rwork[ibal], &ierr );

    /* Reduce to upper Hessenberg form
     * (CWorkspace: need 2*N, prefer N + N*NB)
     * (RWorkspace: N)
     *  - including N reserved for gebal/gebak, unused by zgehrd */
    itau = 0;
    iwrk = itau + n;
    liwrk = lwork - iwrk;

    timer_start( time_gehrd );
    flops_start( flop_gehrd );
    #if defined(VERSION1)
        // Version 1 - LAPACK
        lapackf77_zgehrd( &n, &ilo, &ihi, A, &lda,
                          &work[itau], &work[iwrk], &liwrk, &ierr );
    #elif defined(VERSION2)
        // Version 2 - LAPACK consistent HRD
        magma_zgehrd2( n, ilo, ihi, A, lda,
                       &work[itau], &work[iwrk], liwrk, &ierr );
    #elif defined(VERSION3)
        // Version 3 - LAPACK consistent MAGMA HRD + T matrices stored,
        magma_zgehrd( n, ilo, ihi, A, lda,
                      &work[itau], &work[iwrk], liwrk, dT, &ierr );
    #endif
    time_sum += timer_stop( time_gehrd );
    flop_sum += flops_stop( flop_gehrd );

    if (wantvl) {
        /* Want left eigenvectors
         * Copy Householder vectors to VL */
        side = MagmaLeft;
        lapackf77_zlacpy( MagmaLowerStr, &n, &n, A, &lda, VL, &ldvl );

        /* Generate unitary matrix in VL
         * (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
         * (RWorkspace: N)
         *  - including N reserved for gebal/gebak, unused by zunghr */
        timer_start( time_unghr );
        flops_start( flop_unghr );
        #if defined(VERSION1) || defined(VERSION2)
            // Version 1 & 2 - LAPACK
            lapackf77_zunghr( &n, &ilo, &ihi, VL, &ldvl, &work[itau],
                              &work[iwrk], &liwrk, &ierr );
        #elif defined(VERSION3)
            // Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
            magma_zunghr( n, ilo, ihi, VL, ldvl, &work[itau], dT, nb, &ierr );
        #endif
        time_sum += timer_stop( time_unghr );
        flop_sum += flops_stop( flop_unghr );
        
        timer_start( time_hseqr );
        flops_start( flop_hseqr );
        /* Perform QR iteration, accumulating Schur vectors in VL
         * (CWorkspace: need 1, prefer HSWORK (see comments) )
         * (RWorkspace: N)
         *  - including N reserved for gebal/gebak, unused by zhseqr */
        iwrk = itau;
        liwrk = lwork - iwrk;
        lapackf77_zhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, w,
                          VL, &ldvl, &work[iwrk], &liwrk, info );
        time_sum += timer_stop( time_hseqr );
        flop_sum += flops_stop( flop_hseqr );

        if (wantvr) {
            /* Want left and right eigenvectors
             * Copy Schur vectors to VR */
            side = MagmaBothSides;
            lapackf77_zlacpy( "F", &n, &n, VL, &ldvl, VR, &ldvr );
        }
    }
    else if (wantvr) {
        /* Want right eigenvectors
         * Copy Householder vectors to VR */
        side = MagmaRight;
        lapackf77_zlacpy( "L", &n, &n, A, &lda, VR, &ldvr );

        /* Generate unitary matrix in VR
         * (CWorkspace: need 2*N-1, prefer N + (N-1)*NB)
         * (RWorkspace: N)
         *  - including N reserved for gebal/gebak, unused by zunghr */
        timer_start( time_unghr );
        flops_start( flop_unghr );
        #if defined(VERSION1) || defined(VERSION2)
            // Version 1 & 2 - LAPACK
            lapackf77_zunghr( &n, &ilo, &ihi, VR, &ldvr, &work[itau],
                              &work[iwrk], &liwrk, &ierr );
        #elif defined(VERSION3)
            // Version 3 - LAPACK consistent MAGMA HRD + T matrices stored
            magma_zunghr( n, ilo, ihi, VR, ldvr, &work[itau], dT, nb, &ierr );
        #endif
        time_sum += timer_stop( time_unghr );
        flop_sum += flops_stop( flop_unghr );

        /* Perform QR iteration, accumulating Schur vectors in VR
         * (CWorkspace: need 1, prefer HSWORK (see comments) )
         * (RWorkspace: N)
         *  - including N reserved for gebal/gebak, unused by zhseqr */
        timer_start( time_hseqr );
        flops_start( flop_hseqr );
        iwrk = itau;
        liwrk = lwork - iwrk;
        lapackf77_zhseqr( "S", "V", &n, &ilo, &ihi, A, &lda, w,
                          VR, &ldvr, &work[iwrk], &liwrk, info );
        time_sum += timer_stop( time_hseqr );
        flop_sum += flops_stop( flop_hseqr );
    }
    else {
        /* Compute eigenvalues only
         * (CWorkspace: need 1, prefer HSWORK (see comments) )
         * (RWorkspace: N)
         *  - including N reserved for gebal/gebak, unused by zhseqr */
        timer_start( time_hseqr );
        flops_start( flop_hseqr );
        iwrk = itau;
        liwrk = lwork - iwrk;
        lapackf77_zhseqr( "E", "N", &n, &ilo, &ihi, A, &lda, w,
                          VR, &ldvr, &work[iwrk], &liwrk, info );
        time_sum += timer_stop( time_hseqr );
        flop_sum += flops_stop( flop_hseqr );
    }

    /* If INFO > 0 from ZHSEQR, then quit */
    if (*info > 0) {
        goto CLEANUP;
    }

    timer_start( time_trevc );
    flops_start( flop_trevc );
    if (wantvl || wantvr) {
        /* Compute left and/or right eigenvectors
         * (CWorkspace: need 2*N)
         * (RWorkspace: need 2*N)
         *  - including N reserved for gebal/gebak, unused by ztrevc */
        irwork = ibal + n;
        #if TREVC_VERSION == 1
        lapackf77_ztrevc( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
                          VR, &ldvr, &n, &nout, &work[iwrk], &rwork[irwork], &ierr );
        #elif TREVC_VERSION == 2
        liwrk = lwork - iwrk;
        lapackf77_ztrevc3( lapack_side_const(side), "B", select, &n, A, &lda, VL, &ldvl,
                           VR, &ldvr, &n, &nout, &work[iwrk], &liwrk, &rwork[irwork], &ierr );
        #elif TREVC_VERSION == 3
        magma_ztrevc3( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
                       VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
        #elif TREVC_VERSION == 4
        magma_ztrevc3_mt( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
                          VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
        #elif TREVC_VERSION == 5
        magma_ztrevc3_mt_gpu( side, MagmaBacktransVec, select, n, A, lda, VL, ldvl,
                              VR, ldvr, n, &nout, &work[iwrk], liwrk, &rwork[irwork], &ierr );
        #else
        #error Unknown TREVC_VERSION
        #endif
    }
    time_sum += timer_stop( time_trevc );
    flop_sum += flops_stop( flop_trevc );

    if (wantvl) {
        /* Undo balancing of left eigenvectors
         * (CWorkspace: none)
         * (RWorkspace: need N) */
        lapackf77_zgebak( "B", "L", &n, &ilo, &ihi, &rwork[ibal], &n,
                          VL, &ldvl, &ierr );

        /* Normalize left eigenvectors and make largest component real */
        for (i = 0; i < n; ++i) {
            scl = 1. / magma_cblas_dznrm2( n, VL(0,i), 1 );
            blasf77_zdscal( &n, &scl, VL(0,i), &ione );
            for (k = 0; k < n; ++k) {
                /* Computing 2nd power */
                d__1 = MAGMA_Z_REAL( *VL(k,i) );
                d__2 = MAGMA_Z_IMAG( *VL(k,i) );
                rwork[irwork + k] = d__1*d__1 + d__2*d__2;
            }
            k = blasf77_idamax( &n, &rwork[irwork], &ione ) - 1;  // subtract 1; k is 0-based
            tmp = MAGMA_Z_CNJG( *VL(k,i) ) / magma_dsqrt( rwork[irwork + k] );
            blasf77_zscal( &n, &tmp, VL(0,i), &ione );
            *VL(k,i) = MAGMA_Z_MAKE( MAGMA_Z_REAL( *VL(k,i) ), 0 );
        }
    }

    if (wantvr) {
        /* Undo balancing of right eigenvectors
         * (CWorkspace: none)
         * (RWorkspace: need N) */
        lapackf77_zgebak( "B", "R", &n, &ilo, &ihi, &rwork[ibal], &n,
                          VR, &ldvr, &ierr );

        /* Normalize right eigenvectors and make largest component real */
        for (i = 0; i < n; ++i) {
            scl = 1. / magma_cblas_dznrm2( n, VR(0,i), 1 );
            blasf77_zdscal( &n, &scl, VR(0,i), &ione );
            for (k = 0; k < n; ++k) {
                /* Computing 2nd power */
                d__1 = MAGMA_Z_REAL( *VR(k,i) );
                d__2 = MAGMA_Z_IMAG( *VR(k,i) );
                rwork[irwork + k] = d__1*d__1 + d__2*d__2;
            }
            k = blasf77_idamax( &n, &rwork[irwork], &ione ) - 1;  // subtract 1; k is 0-based
            tmp = MAGMA_Z_CNJG( *VR(k,i) ) / magma_dsqrt( rwork[irwork + k] );
            blasf77_zscal( &n, &tmp, VR(0,i), &ione );
            *VR(k,i) = MAGMA_Z_MAKE( MAGMA_Z_REAL( *VR(k,i) ), 0 );
        }
    }

CLEANUP:
    /* Undo scaling if necessary */
    if (scalea) {
        // converged eigenvalues, stored in WR[i+1:n] and WI[i+1:n] for i = INFO
        magma_int_t nval = n - (*info);
        magma_int_t ld   = max( nval, 1 );
        lapackf77_zlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, w + (*info), &ld, &ierr );
        if (*info > 0) {
            // first ilo columns were already upper triangular,
            // so the corresponding eigenvalues are also valid.
            nval = ilo - 1;
            lapackf77_zlascl( "G", &izero, &izero, &cscale, &anrm, &nval, &ione, w, &n, &ierr );
        }
    }

    #if defined(VERSION3)
    magma_free( dT );
    #endif
    
    timer_stop( time_total );
    flops_stop( flop_total );
    timer_printf( "dgeev times n %5d, gehrd %7.3f, unghr %7.3f, hseqr %7.3f, trevc %7.3f, total %7.3f, sum %7.3f\n",
                  (int) n, time_gehrd, time_unghr, time_hseqr, time_trevc, time_total, time_sum );
    timer_printf( "dgeev flops n %5d, gehrd %7lld, unghr %7lld, hseqr %7lld, trevc %7lld, total %7lld, sum %7lld\n",
                  (int) n, flop_gehrd, flop_unghr, flop_hseqr, flop_trevc, flop_total, flop_sum );

    work[0] = MAGMA_Z_MAKE( (double) optwrk, 0. );

    return *info;
} /* magma_zgeev */