# 1. Reductions

## 1.1 求和

`sum()`返回matrix或array中所有元素的和。

## 1.2 范数计算

L2范数 `squareNorm()`，等价于计算vector的自身点积，也等价于vector中所有元素的平方和。

`norm()`返回`squareNorm()`的开方根。

``````#include <Eigen/Dense>
#include <iostream>
using namespace std;
using namespace Eigen;
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;
}
``````

``````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。

``````#include <Eigen/Dense>
#include <iostream>
using namespace Eigen;
using namespace std;
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
``````

## 1.2 布尔归约

all() 如果Matrix或Array中所有元素的对应操作返回Ture则返回True
any() 如果Matrix或Array中所有元素中至少一个元素的对应操作返回Ture则返回True
count() 返回Matrix或Array中所有元素对应操作返回True的数量

``````#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;
}
``````

``````(a > 0).all()   = 1
(a > 0).any()   = 1
(a > 0).count() = 4

(a > 2).all()   = 0
(a > 2).any()   = 1
(a > 2).count() = 2
``````

# 2. Visitors

``````#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
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

``````

# 3. Partial reductions-部分归约

Eigen中支持对Matrx或Array的行或列进行归约操作，返回该行或该列中的相应值。部分归约可以使用`colwise()``rowwise()`函数。其中，`colwise()`返回一个行向量，`rowwise()`返回一个列向量。

``````#include <iostream>
#include <Eigen/Dense>
using namespace std;
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
``````

## 3.1 结合部分归约和其他操作

``````#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
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;
}
``````

``````Maximum sum at position 2
The corresponding vector is:
6
7
And its sum is is: 13
``````

``````#include <iostream>
#include <Eigen/Dense>
using namespace std;
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;
}
``````

``````Broadcasting result:
1 2 6 9
4 2 8 3
``````

## 4.1 结合广播和其他操作

``````#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;
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;
}
``````

``````Nearest neighbour is column 0:
1
3
``````

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