Question

我是研究生，试图使用Eigen和LAPACK将我的MATLAB原型代码重写为C ++代码。广义特征值求解器（A * x = lamba * B * x）在该程序中占有一定的份额。因为Eigen的广义特征求解器可能导致错误的特征值（根据我的经验），我决定使用LAPACK（在OS X中使用Accelerate框架）

下面的代码是我写的tgevc示例http://www.nag.com/numeric/fl/manual/pdf/F08/f08ykf.pdf的C ++翻译。除非我指定计算右特征向量的特征值，否则一切看起来都很好。

当我设置

char howmny = 'B';

它计算每个特征值的每个特征向量。它的工作很精细。对于较大的问题，这些值也是正确的。

然而，当我设置

char howmny = 'S';

结果没有意义。它看起来没什么。

这样做的原因是首先可能加快速度。但是，分析显示它占整个过程的不到4％（对于较大的问题）。所以它不会有太大变化。无论如何，我仍然想知道我是否理解错误。

这是我的示例代码。

#include <iostream>
#include <cmath>
#include <Eigen/Core>
#include <Accelerate/Accelerate.h>

using namespace std;
using namespace Eigen;

int main(int argc, const char * argv[])
{
    // parameters
    int nmax = 5, lda = nmax, ldb = nmax, ldq = nmax, ldz = nmax, lwork=ldq*6, N=5, M;

    // Local scalars
    int ihi, ilo, info, irows, icols, jwork;
    char compq, compz, job, job1, job2, side;

    // Local arrays
    MatrixXd A(lda,nmax), B(ldb, nmax), Q(ldq,ldq), Z(ldz, ldz);
    VectorXd alphai(nmax), alphar(nmax), beta(nmax), lscale(nmax), rscale(nmax), tau(nmax), work(lwork);

    cout << " F08XEF Example Program Results " << endl;

    for (int i = 0; i < N ; i++) {
        A(i,0) = (i+1)*1.0;
    }
    for (int j = 1; j < N ; j++) {
        A.col(j) = A.col(0).array()*A.col(j-1).array();
    }

    B.block(0,0,N,N) = A.block(0, 0, N, N).transpose();
    cout << "Matrix A is" << endl;
    cout << A.block(0, 0, N, N) << endl;
    cout << "Matrix B is" << endl;
    cout << B.block(0,0,N,N) << endl;

    // Balance matrix pair (A, B)
    job = 'B';

    dggbal_(&job, &N, A.data(), &lda, B.data(), &ldb, &ilo, &ihi, lscale.data(), rscale.data(), work.data(), &info);
    cout << "Matrix A after balacing" << endl;
    cout << A.block(0, 0, N, N) << endl;
    cout << "Matrix B after balacing" << endl;
    cout << B.block(0,0,N,N) << endl;

    // Reduce B to triangular form using QR
    irows = ihi + 1 - ilo;
    icols = N + 1 - ilo;
    dgeqrf_(&irows, &icols, B.block(ilo-1,ilo-1,irows,icols).data(), &ldb, tau.data(), work.data(), &lwork, &info);

    // Apply the orthogonal transformation t matrix A
    job1 = 'L';
    job2 = 'T';
    dormqr_(&job1, &job2, &irows, &icols, &irows, B.block(ilo-1,ilo-1,irows,icols).data(), &ldb, tau.data(), A.block(ilo-1,ilo-1,irows,irows).data(), &lda, work.data(), &lwork, &info);
    Z.block(0, 0, N, N) = MatrixXd::Identity(N, N);
    // Compute the generalized Hassenberg form of (A, B)
    compq = 'V';
    compz = 'V';
    dgghrd_(&compq,&compz,&N,&ilo,&ihi,A.block(ilo-1,ilo-1,irows,irows).data(),&lda, B.block(ilo-1,ilo-1,irows,irows).data(),&ldb,Q.data(),&ldq,Z.data(),&ldz,&info);
    cout << "Matrix A in Hessenber form" << endl;
    cout << A.block(0, 0, N, N) << endl;
    cout << "Matrix B is triangular" << endl;
    cout << B.block(0, 0, N, N) << endl;

    // Routine dhgeqz
    // Worspace query : jwork = -1
    jwork = -1;
    job = 'S';
    dhgeqz_(&job, &compq, &compz, &N, &ilo, &ihi, A.data(), &lda, B.data(), &ldb, alphar.data(), alphai.data(), beta.data(), Q.data(), &ldq, Z.data(), &ldz, work.data(), &jwork, &info);
    cout << "Minimal required lwork = " << round(work(0)) << endl;
    cout << "Actual value of lwork = " << lwork << endl;
    // Compute the generalized Schur form if the workspace lwork is adequate
    if (round(work(0)) <= lwork)
    {
        dhgeqz_(&job, &compq, &compz, &N, &ilo, &ihi, A.data(), &lda, B.data(), &ldb, alphar.data(), alphai.data(), beta.data(), Q.data(), &ldq, Z.data(), &ldz, work.data(), &lwork, &info);
    }

    // Print the generalized eigenvalue

    for (int i = 0; i < N ; i++) {
        if (beta(i) != 0.0)
        {
            cout << alphar(i)/beta(i) << "," << alphai(i)/beta(i)<< endl;
        }
    }


    // Compute right generalized eigenvectors of the balaced matrix

    // !This is where the problem shows up!
    char howmny = 'S';
    side = 'R';
    __CLPK_logical select[5]; // int selec[5] works same.

    select[0]=false;
    select[1]=false;
    select[2]=false;
    select[3]=false;
    select[4]=false;


    dtgevc_(&side, &howmny, select, &N, A.data(), &lda, B.data(), &ldb, Q.data(), &ldq, Z.data(), &ldz, &N, &M, work.data(), &info);

    job = 'B';
    dggbak_(&job, &side, &N, &ilo, &ihi, lscale.data(), rscale.data(), &N, Z.data(), &ldz, &info);

    cout << "Right eigenvectors" << endl;
    cout << Z << endl;

    return 0;
}

如何使用LAPACK从矩阵对的广义Schur分解中获得指定的特征向量？

0 个答案: