矩陣、向量初始化

#include <iostream>
#include "Eigen/Dense"
using namespace Eigen;
int main()
{
    MatrixXf m1(3,4);   //動態矩陣，建立3行4列。
    MatrixXf m2(4,3);   //4行3列，依此類推。
    MatrixXf m3(3,3);

    Vector3f v1;        //若是靜態陣列，則不用指定行或者列
    /* 初始化 */
    Matrix3d m = Matrix3d::Random();
    m1 = MatrixXf::Zero
 
(3,4);       //用0矩陣初始化,要指定行列數
    m2 = MatrixXf::Zero(4,3);
    m3 = MatrixXf::Identity(3,3);   //用單位矩陣初始化
    v1 = Vector3f::Zero();          //同理，若是靜態的，不用指定行列數

    m1 << 1,0,0,1,      //也可以以這種方式初始化
        1,5,0,1,
        0,0,9,1;
    m2 << 1,0,0,
        0,4,0,
        0,0,7,
        1,1,1;
    //向量初始化，與矩陣類似
 

    Vector3d v3(1,2,3);
    VectorXf vx(30);
}

C++陣列和矩陣轉換

使用Map函式，可以實現Eigen的矩陣和c++中的陣列直接轉換,語法如下：

//@param MatrixType 矩陣型別
//@param MapOptions 可選引數，指的是指標是否對齊，Aligned, or Unaligned. The default is Unaligned.
//@param StrideType 可選引數，步長
/*
    Map<typename MatrixType,
        int MapOptions,
        typename StrideType>
*/
 

    int i;
    //陣列轉矩陣
    double *aMat = new double[20];
    for(i =0;i<20;i++)
    {
        aMat[i] = rand()%11;
    }
    //靜態矩陣，編譯時確定維數 Matrix<double,4,5> 
    Eigen:Map<Matrix<double,4,5> > staMat(aMat);


    //輸出
    for (int i = 0; i < staMat.size(); i++)
        std::cout << *(staMat.data() + i) << " ";
    std::cout << std::endl << std::endl;


    //動態矩陣，執行時確定 MatrixXd
    Map<MatrixXd> dymMat(aMat,4,5);


    //輸出，應該和上面一致
    for (int i = 0; i < dymMat.size(); i++)
        std::cout << *(dymMat.data() + i) << " ";
    std::cout << std::endl << std::endl;

    //Matrix中的資料存在一維陣列中，預設是行優先的格式，即一行行的存
    //data()返回Matrix中的指標
    dymMat.data();

矩陣基礎操作

eigen過載了基礎的+ - * / += -= = /= 可以表示標量和矩陣或者矩陣和矩陣

#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
int main()
{
    //單個取值，單個賦值
    double value00 = staMat(0,0);
    double value10 = staMat(1,0);
    staMat(0,0) = 100;
    std::cout << value00 <<value10<<std::endl;
    std::cout <<staMat<<std::endl<<std::endl;
    //加減乘除示例 Matrix2d 等同於 Matrix<double,2,2>
    Matrix2d a;
     a << 1, 2,
     3, 4;
    MatrixXd b(2,2);
     b << 2, 3,
     1, 4;

    Matrix2d c = a + b;
    std::cout<< c<<std::endl<<std::endl;

    c = a - b;
    std::cout<<c<<std::endl<<std::endl;

    c = a * 2;
    std::cout<<c<<std::endl<<std::endl;

    c = 2.5 * a;
    std::cout<<c<<std::endl<<std::endl;

    c = a / 2;
    std::cout<<c<<std::endl<<std::endl;

    c = a * b;
    std::cout<<c<<std::endl<<std::endl;

點積和叉積

#include <iostream>
#include <Eigen/Dense>
using namespace Eigen;
using namespace std;
int main()
{
    //點積、叉積(針對向量的)
    Vector3d v(1,2,3);
    Vector3d w(0,1,2);
    std::cout<<v.dot(w)<<std::endl<<std::endl;
    std::cout<<w.cross(v)<<std::endl<<std::endl;
}
*/

轉置、伴隨、行列式、逆矩陣

小矩陣（4 * 4及以下）eigen會自動優化，預設採用LU分解，效率不高

#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main()
{
    Matrix2d c;
     c << 1, 2,
     3, 4;
    //轉置、伴隨
    std::cout<<c<<std::endl<<std::endl;
    std::cout<<"轉置\n"<<c.transpose()<<std::endl<<std::endl;
    std::cout<<"伴隨\n"<<c.adjoint()<<std::endl<<std::endl;
    //逆矩陣、行列式
    std::cout << "行列式： " << c.determinant() << std::endl;
    std::cout << "逆矩陣\n" << c.inverse() << std::endl;
}

計算特徵值和特徵向量

#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main()
{
    //特徵向量、特徵值
    std::cout << "Here is the matrix A:\n" << a << std::endl;
    SelfAdjointEigenSolver<Matrix2d> eigensolver(a);
    if (eigensolver.info() != Success) abort();
     std::cout << "特徵值:\n" << eigensolver.eigenvalues() << std::endl;
     std::cout << "Here's a matrix whose columns are eigenvectors of A \n"
     << "corresponding to these eigenvalues:\n"
     << eigensolver.eigenvectors() << std::endl;
}

解線性方程

#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main()
{
    //線性方程求解 Ax =B;
    Matrix4d A;
    A << 2,-1,-1,1,
        1,1,-2,1,
        4,-6,2,-2,
        3,6,-9,7;

    Vector4d B(2,4,4,9);

    Vector4d x = A.colPivHouseholderQr().solve(B);
    Vector4d x2 = A.llt().solve(B);
    Vector4d x3 = A.ldlt().solve(B);    


    std::cout << "The solution is:\n" << x <<"\n\n"<<x2<<"\n\n"<<x3 <<std::endl;
}

除了colPivHouseholderQr、LLT、LDLT，還有以下的函式可以求解線性方程組，請注意精度和速度： 解小矩陣（4*4）基本沒有速度差別

最小二乘求解

最小二乘求解有兩種方式，jacobiSvd或者colPivHouseholderQr，4*4以下的小矩陣速度沒有區別，jacobiSvd可能更快，大矩陣最好用colPivHouseholderQr

#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
int main()
{
    MatrixXf A1 = MatrixXf::Random(3, 2);
    std::cout << "Here is the matrix A:\n" << A1 << std::endl;
    VectorXf b1 = VectorXf::Random(3);
    std::cout << "Here is the right hand side b:\n" << b1 << std::endl;
    //jacobiSvd 方式:Slow (but fast for small matrices)
    std::cout << "The least-squares solution is:\n"
    << A1.jacobiSvd(ComputeThinU | ComputeThinV).solve(b1) << std::endl;
    //colPivHouseholderQr方法:fast
    std::cout << "The least-squares solution is:\n"
    << A1.colPivHouseholderQr().solve(b1) << std::endl;
}

稀疏矩陣

稀疏矩陣的標頭檔案包括：

#include

typedef Eigen::Triplet<double> T;
std::vector<T> tripletList;
triplets.reserve(estimation_of_entries); //estimation_of_entries是預估的條目
for(...)
{
    tripletList.push_back(T(i,j,v_ij));//第 i,j個有值的位置的值
}
SparseMatrixType mat(rows,cols);
mat.setFromTriplets(tripletList.begin(), tripletList.end());
// mat is ready to go!

2.直接將已知的非0值插入

SparseMatrix<double> mat(rows,cols);
mat.reserve(VectorXi::Constant(cols,6));
for(...)
{
    // i,j 個非零值 v_ij != 0
    mat.insert(i,j) = v_ij;
}
mat.makeCompressed(); // optional

稀疏矩陣支援大部分一元和二元運算:

sm1.real() sm1.imag() -sm1 0.5*sm1
sm1+sm2 sm1-sm2 sm1.cwiseProduct(sm2)
二元運算中，稀疏矩陣和普通矩陣可以混合使用

//dm表示普通矩陣
dm2 = sm1 + dm1;
也支援計算轉置矩陣和伴隨矩陣

參考以下連結