Eigen库（2014-01-18 14:30:39）


首先到Eigen官网上下载Eigen源码包，下载后解压完直接放到自己平时软件所在的目录下（例如E:\EigenRoot，并在VS2010的附加包含目录下指定该路径即可），不需要安装。Eigen下载地址为

http://eigen.tuxfamily.org/index.php?title=Main_Page
http://eigen.tuxfamily.org/dox/unsupported/group__MatrixFunctions__Module.html#title9
使用Eigen的第一个例子：

#include <iostream>

#include <vector>

#include <Eigen/Eigen>

using namespace Eigen;

using namespace std;

int main()

{

/*

v1 =

5

6

result:

20 25

24 30

*/

Eigen::Vector2d v1, v2; //Eigen中的变量

v1 << 5, 6; //默认的向量为列向量

cout << "v1 = " << endl << v1 << endl;

v2 << 4, 5 ;

Matrix2d result = v1*v2.transpose();

cout << "result: " << endl << result << endl;

system("pause");

return 0;

}

第二个例子，计算矩阵的基本初等函数：

#include <iostream>

#if 0

#include <vector>

#include <Eigen/Eigen>

#endif

#include <unsupported/Eigen/MatrixFunctions>

using namespace Eigen;

using namespace std;

int main()

{

/*

v1 =

5

6

result:

20 25

24 30

*/

#if 0

Eigen::Vector2d v1, v2; //Eigen中的变量

v1 << 5, 6; //默认的向量为列向量

cout << "v1 = " << endl << v1 << endl;

v2 << 4, 5 ;

Matrix2d result = v1*v2.transpose();

cout << "result: " << endl << result << endl;

#endif

/*

http://eigen.tuxfamily.org/dox/unsupported/group__MatrixFunctions__Module.html

The matrix A is:

0 -0.785398 0

0.785398 0 0

0 0 0

The matrix exponential of A is:

0.707107 -0.707107 0

0.707107 0.707107 0

0 0 1

对上面的矩阵A求正弦的话会出现如下断言失败

Assertion failed: "Taylor series does not converge" && 0, file e:\eigenroot\unsupported\eigen\src\matrixfunctions\matrixfunctionatomic.h, line 89

也就是泰勒级数sinA=A - A^3/3! + A^5/5! -A^7/7!.....对某些矩阵A不收敛，这一点跟实数域和复数域的情况不一样。

cos(A) =

1.32461 0 0

0 1.32461 0

0 0 1

*/

#if 1

{

const double pi = std::acos(-1.0);

MatrixXd A(3,3);

A << 0, -pi/4, 0,

pi/4, 0, 0,

0, 0, 0;

std::cout << "The matrix A is:\n" << A << "\n\n";

std::cout << "The matrix exponential of A is:\n" << A.exp() << "\n\n";

//A = MatrixXd::Random(3,3);

//MatrixXd sinA = A.sin();

//std::cout << "sin(A) = \n" << sinA << "\n\n";

MatrixXd cosA = A.cos();

std::cout << "cos(A) = \n" << cosA << "\n\n";

// The matrix functions satisfy sin^2(A) + cos^2(A) = I,like the scalar functions.

//std::cout << "sin^2(A) + cos^2(A) = \n" << sinA*sinA + cosA*cosA << "\n\n";

}

#endif

system("pause");

return 0;

}

第三个例子：

/*

验证：

{{-3,1,-1},{-7,5,-1},{-6,6,-2}}^2={{8,-4,4},{-8,12,4},{-12,12,4}}

{{-3,1,-1},{-7,5,-1},{-6,6,-2}}^3={{-20,12,-12},{-84,76,-12},{-72,72,-8}}

矩阵代数是不可交换的，然而是结合的，至今研究过的大部分系统都是结合的。约定+只用于可交换的系统，而×不必受此限制。

m1:

-3 1 -1

-7 5 -1

-6 6 -2

m2:

8 -4 4

-8 12 4

-12 12 4

m3:

-20 12 -12

-84 76 -12

-72 72 -8

*/

#if 1

MatrixXf m1(3,3);

m1<<-3,1,-1,-7,5,-1,-6,6,-2;

cout << "m1: " << endl << m1 << endl;

MatrixXf m2 = m1*m1;

cout << "m2: " << endl << m2 << endl;

MatrixXf m3 = m1*m1*m1;

cout << "m3: " << endl << m3 << endl;

#endif

第四个例子：

由矩阵的幂和无穷级数导出的等式：

sin{{4,6,0},{-3,-5,0},{-3,-6,1}}={{-2,0,-1},{1,0,1},{0,1,1}}{{sin1,0,0},{0,sin1,0},{0,0,-sin2}}{{-1,-1,0},{-1,-2,1},{1,2,0}}

cos{{2,0,0},{1,2,-1},{1,0,1}}发散，真的吗？

20131220注意：

{{-2,0,-1},{1,0,1},{0,1,1}}的逆矩阵是{{-1,-1,0},{-1,-2,1},{1,2,0}}，而不是{{-1,-1,1},{-1,-2,2},{0,1,0}}。

A:

4 6 0

-3 -5 0

-3 -6 1

B:

-2 0 -1

1 0 1

0 1 1

C:

0.841471 0 0

0 0.841471 0

0 0 -0.909297

D:

-1 -1 0

-1 -2 1

1 2 0

sin(A) =

2.59224 3.50154 -1.35828e-008

-1.75077 -2.66007 2.92391e-007

-1.75077 -3.50154 0.841471

BCD:

2.59224 3.50154 0

-1.75077 -2.66007 0

-1.75077 -3.50154 0.841471

#if 1

MatrixXf A(3,3);

A<<4,6,0,-3,-5,0,-3,-6,1;

cout << "A: " << endl << A << endl;

MatrixXf B(3,3);

B<<-2,0,-1,1,0,1,0,1,1;

cout << "B: " << endl << B << endl;

MatrixXf C(3,3);

C<<sinf(1),0,0,0,sinf(1),0,0,0,-sinf(2);

cout << "C: " << endl << C << endl;

#if 0

MatrixXf D = B.inverse();

#else

MatrixXf D(3,3);

D<<-1,-1,0,-1,-2,1,1,2,0;

#endif

cout << "D: " << endl << D << endl;

MatrixXf sinA = A.sin();

std::cout << "sin(A) = \n" << sinA << "\n\n";

MatrixXf BCD = B*C*D;

cout << "BCD: " << endl << BCD << endl;

#endif

A:

2 0 0

1 2 -1

1 0 1

cosA:

-0.416147 -9.20451e-008 7.45058e-008

-0.956449 -0.416147 0.956449

-0.956449 -1.97194e-007 0.540302

#if 1

MatrixXf A(3,3);

A<<2,0,0,1,2,-1,1,0,1;

cout << "A: " << endl << A << endl;

MatrixXf cosA = A.cos();

cout << "cosA: " << endl << cosA << endl;

#endif