整理下Eigen库的教程，参考：http://eigen.tuxfamily.org/dox/index.html
归约、迭代器和广播
归约
在Eigen中，有些函数可以统计matrix/array的某类特征，返回一个标量。
int main()
{
Eigen::Matrix2d mat;
mat << 1, 2,
3, 4;
cout << "Here is mat.sum(): " << mat.sum() << endl;
cout << "Here is mat.prod(): " << mat.prod() << endl;
cout << "Here is mat.mean(): " << mat.mean() << endl;
cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;
cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;
cout << "Here is mat.trace(): " << mat.trace() << endl;
}
范数计算
L2范数 squareNorm()，等价于计算vector的自身点积，norm()返回squareNorm的开方根。
这些操作应用于matrix，norm() 会返回Frobenius或Hilbert-Schmidt范数。
如果你想使用其他Lp范数，可以使用lpNorm< p >()方法。p可以取Infinity，表示L∞范数。
int main()
{
VectorXf v(2);
MatrixXf m(2,2), n(2,2);
v << -1,
2;
m << 1,-2,
-3,4;
cout << "v.squaredNorm() = " << v.squaredNorm() << endl;
cout << "v.norm() = " << v.norm() << endl;
cout << "v.lpNorm<1>() = " << v.lpNorm<1>() << endl;
cout << "v.lpNorm<Infinity>() = " << v.lpNorm<Infinity>() << endl;
cout << endl;
cout << "m.squaredNorm() = " << m.squaredNorm() << endl;
cout << "m.norm() = " << m.norm() << endl;
cout << "m.lpNorm<1>() = " << m.lpNorm<1>() << endl;
cout << "m.lpNorm<Infinity>() = " << m.lpNorm<Infinity>() << endl;
}
输出
v.squaredNorm() = 5
v.norm() = 2.23607
v.lpNorm<1>() = 3
v.lpNorm<Infinity>() = 2
m.squaredNorm() = 30
m.norm() = 5.47723
m.lpNorm<1>() = 10
m.lpNorm<Infinity>() = 4
Operator norm: 1-norm和∞-norm可以通过其他方式得到。
int main()
{
MatrixXf m(2,2);
m << 1,-2,
-3,4;
cout << "1-norm(m) = " << m.cwiseAbs().colwise().sum().maxCoeff()
<< " == " << m.colwise().lpNorm<1>().maxCoeff() << endl;
cout << "infty-norm(m) = " << m.cwiseAbs().rowwise().sum().maxCoeff()
<< " == " << m.rowwise().lpNorm<1>().maxCoeff() << endl;
}
1-norm(m) = 6 == 6
infty-norm(m) = 7 == 7
布尔归约
all()=true matrix/array中的所有算术是true any()=true matrix/array中至少有一个元素是true count() 返回为true元素的数目
#include <Eigen/Dense>
#include <iostream>
using namespace std;
using namespace Eigen;
int main()
{
ArrayXXf a(2,2);
a << 1,2,
3,4;
cout << "(a > 0).all() = " << (a > 0).all() << endl;
cout << "(a > 0).any() = " << (a > 0).any() << endl;
cout << "(a > 0).count() = " << (a > 0).count() << endl;
cout << endl;
cout << "(a > 2).all() = " << (a > 2).all() << endl;
cout << "(a > 2).any() = " << (a > 2).any() << endl;
cout << "(a > 2).count() = " << (a > 2).count() << endl;
}
输出
(a > 0).all() = 1
(a > 0).any() = 1
(a > 0).count() = 4
(a > 2).all() = 0
(a > 2).any() = 1
(a > 2).count() = 2
迭代器(遍历)
当我们想获取某元素在Matrix或Array中的位置的时候，迭代器是必须的。常用的有：minCoeff和maxCoeff。
int main()
{
Eigen::MatrixXf m(2,2);
m << 1, 2,
3, 4;
//get location of maximum
MatrixXf::Index maxRow, maxCol;
float max = m.maxCoeff(&maxRow, &maxCol);
//get location of minimum
MatrixXf::Index minRow, minCol;
float min = m.minCoeff(&minRow, &minCol);
cout << "Max: " << max << ", at: " <<
maxRow << "," << maxCol << endl;
cout << "Min: " << min << ", at: " <<
minRow << "," << minCol << endl;
}
Max: 4, at: 1,1
Min: 1, at: 0,0
部分归约
Eigen中支持对Matrx或Array的行/行进行归约操作。部分归约可以使用colwise()/rowwise()函数。
int main()
{
Eigen::MatrixXf mat(2,4);
mat << 1, 2, 6, 9,
3, 1, 7, 2;
std::cout << "Column's maximum: " << std::endl
<< mat.colwise().maxCoeff() << std::endl;
}
Column's maximum:
3 2 7 9
类似，针对行也可以，只是返回的是列向量而已。
int main()
{
Eigen::MatrixXf mat(2,4);
mat << 1, 2, 6, 9,
3, 1, 7, 2;
std::cout << "Row's maximum: " << std::endl
<< mat.rowwise().maxCoeff() << std::endl;
}
Row's maximum:
9
7
结合部分归约和其他操作
例子：寻找和最大的列向量。
int main()
{
MatrixXf mat(2,4);
mat << 1, 2, 6, 9,
3, 1, 7, 2;
MatrixXf::Index maxIndex;
float maxNorm = mat.colwise().sum().maxCoeff(&maxIndex);
std::cout << "Maximum sum at position " << maxIndex << std::endl;
std::cout << "The corresponding vector is: " << std::endl;
std::cout << mat.col( maxIndex ) << std::endl;
std::cout << "And its sum is is: " << maxNorm << std::endl;
}
输出
Maximum sum at position 2
The corresponding vector is:
6
7
And its sum is is: 13
广播
广播是针对vector的，将vector沿行/列重复构建一个matrix，便于后期运算。
int main()
{
Eigen::MatrixXf mat(2,4);
Eigen::VectorXf v(2);
mat << 1, 2, 6, 9,
3, 1, 7, 2;
v << 0,
1;
//add v to each column of m
mat.colwise() += v;
std::cout << "Broadcasting result: " << std::endl;
std::cout << mat << std::endl;
}
输出
Broadcasting result:
1 2 6 9
4 2 8 3
注意：对Array类型，*=，/=和/这些操作可以进行行/列级的操作，但不使用与Matrix，因为会与矩阵乘混淆。
结合广播和其他操作
示例：计算矩阵中哪列与目标向量距离最近。
int main()
{
Eigen::MatrixXf m(2,4);
Eigen::VectorXf v(2);
m << 1, 23, 6, 9,
3, 11, 7, 2;
v << 2,
3;
MatrixXf::Index index;
// find nearest neighbour
(m.colwise() - v).colwise().squaredNorm().minCoeff(&index);
cout << "Nearest neighbour is column " << index << ":" << endl;
cout << m.col(index) << endl;
}
输出
Nearest neighbour is column 0:
1
3