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