文章目录

Eigen Quick Reference
- Eigen array, matrix and vector types
- Which should I choose: matrix or array
- Initialization
- - Repeating
  - Miss those elements out
- Slicing
- Indexing and accessing elements
- Maximum and minimum
- Assignment
- Concatenation
- Reshape and flatten matrices
- Coefficient-wise math functions
- - Basic operatoins
  - Exponential and logarithm
  - Round off
  - Power functions
  - Trigonometric functions
Eigen exercise

Eigen Quick Reference

一个 Eigen 的参考手册，包含 Python(Numpy) 与 Eigen 的对应用法，以及一些练习题。持续更新中。

Eigen array, matrix and vector types

在 Eigen 中，Matrix 用来表示数学意义上的矩阵和向量，Array 用来表示 1D 和 2D 的数组，你可以这样定义它们:

typedef Matrix<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyMatrixType;
typedef Array<Scalar, RowsAtCompileTime, ColsAtCompileTime, Options> MyArrayType;

Scalar 表示系数的类型(例如float, double, boll, int 等等)。
RowsAtCompileTime 和 ColsAtCompileTime 表示矩阵行数和列数（必须在编译时期给定），或者使用 Dynamic 表示其行数或者列数在运行时期给定。
Options 可以是 ColMajor 或者 RowMajor，它们表示存储数据的顺序，默认是 ColMajor。（这里有存储顺序的更多介绍）

你可以随意的组合上面的参数来创建 Matrix，例如

Matrix<double, 6, Dynamic>                  // 动态列数（堆分配）
Matrix<double, Dynamic, 2>                  // 动态行数（堆分配）
Matrix<double, Dynamic, Dynamic, RowMajor>  // 动态大小，RowMajor（堆大小）
Matrix<double, 13, 3>                       // 固定大小（通常是栈大小）

你还可以使用更加简易形式来表示一些常用的 Matrix 或者 Array，例如

Matrices	Arrays
`Matrix<float, Dynamic, Dynamic> <=> MatrixXf`	`Array<float,Dynamic,Dynamic> <=> ArrayXXf`
`Matrix<double, Dynamic, 1> <=> VectorXd`	`Array<double,Dynamic,1> <=> ArrayXd`
`Matrix<int, 1, Dynamic> <=> RowVectorXi`	`Array<int,1,Dynamic> <=> RowArrayXi`
`Matrix<float, 3, 3> <=> Matrix3f`	`Array<float,3,3> <=> Array33f`
`Matrix<float, 4, 1> <=> Vector4f`	`Array<float,4,1> <=> Array4f`

Which should I choose: matrix or array

如果你也有类似的困惑，很大几率是因为你比较熟悉 Python(NumPy) 或者 MATLAB。

在 MATLAB 中所有变量都是多维数组，矩阵是指通常用来进行线性代数运算的二维数组（它还是个数组），MATALB 具有两种运算：数组运算和矩阵运算。所以 MATLAB 并不区分数组类型或者矩阵类型，它只区分数组操作或者矩阵操作。

在 Python（NumPy）中，你可以用 np.matrix 创建矩阵，np.array 创建数组，但是官方推荐用 np.array，原因是：

Use arrays.

They are the standard vector/matrix/tensor type of numpy. Many numpy functions return arrays, not matrices.

There is a clear distinction between element-wise operations and linear algebra operations.

You can have standard vectors or row/column vectors if you like.

Until Python 3.5 the only disadvantage of using the array type was that you had to use dot instead of * to multiply (reduce) two tensors (scalar product, matrix vector multiplication etc.). Since Python 3.5 you can use the matrix multiplication @ operator.
Given the above, we intend to deprecate matrix eventually.

由于 NumPy 的便利性，我们通常用 np.array 也能够完成线性代数相关的任务，进一步导致 np.matrix 存在感很弱。

但在 Eigen 中 matrix 与 array 是有明确区别的，总的来说，Eigen 中的 matrix 与线性代数息息相关，它设计的初衷就是为了解决线性代数问题，例如解线性方程组、求矩阵特征值、矩阵的秩、QR分解等等。而 array 则负责系数运算，例如所有系数加上一个常数或者两个 array 系数相乘。

因此，如果你需要线性代数的操作时，请使用 matrix；如果你需要系数操作时，请使用 array。但有时候事情不会那么简单，你可能需要同时使用 matrix 和 array，这种情况下，你需要对 matrix 和 array 进行相互转换。
一个 matrix 通过 .array() 来得到一个 array 表达式；相似的，一个 array 通过 .matrix() 来得到一个 matrix 表达式。.array() 和 .matrix() 不会有任何运行时开销，它们是在编译期完成的。

Matrix 和 Array 之间可以相互转换：

Array44f a1, a1;
Matrix4f m1, m2;
m1 = a1 * a2;               // 系数相乘，隐式地从 Array 转到 Matrix
a1 = m1 * m2;               // 矩阵相乘，隐式地从 Matrix 转到 Array
a2 = a1 + m1.array();        // Array 和 Matrix 混合使用是不被允许的，需要显式地转换
m2 = a1.matrix() + m1;
ArrayWrapper<Matrix4f> m1a(m1); // m1a 是 m1 的别名，它们共享同样的数据
MatrixWrapper<Array44f> a1m(a1);// a1m 是 a1 的别名，它们共享同样的数据

Initialization

int rows = 5;
int cols = 3;
int size = 3;
// fixed-size array
{Array33f  a1 = Array33f::Zero();                    // np.zeros((3,3))Array33f  a2 = Array33f::Random();                  // np.random.rand(3,3)Array33f  a3 = Array33f::Ones();                    // np.ones((3,3))Array3f   a4 = Array3f::LinSpaced(3, 0, 2);         // np.linspace(0,2,3)Array33f  a5 = Array33f::Constant(1.0);             // np.full((3,3), 1.0)
}// one-dimensional dynamic-size
{ArrayXf a1 = ArrayXf::Zero(cols);                   // np.zeros(cols)ArrayXf a2 = ArrayXf::Random(cols);                 // np.random.rand(cols)ArrayXf a3 = ArrayXf::Ones(cols);                   // np.ones(cols)ArrayXf a4 = ArrayXf::LinSpaced(size, 0, 2);        // np.linspace(0,2,size)ArrayXf a5 = ArrayXf::Constant(cols, 1.0);          // np.full(cols, 1.0)
}// two-dimensional dynamic-size
{ArrayXXf a1  = ArrayXXf::Zero(rows,cols);           // np.zeros((rows, cols)))ArrayXXf a2  = ArrayXXf::Random(rows,cols);         // np.random.rand(rows, cols)ArrayXXf a3  = ArrayXXf::Ones(rows, cols);          // np.ones((rows, cols))ArrayXXf a4  = ArrayXXf::Constant(rows, cols, 1.0); // np.full((rows, cols), 1.0)MatrixXf m1  = MatrixXf::Identity(rows, cols);      // np.eye(rows, cols)MatrixXf m2  = Vector3f{1,2,3}.asDiagonal();        // np.diag((1,2,3))
}

Repeating

ArrayXXf a(2,2);
a << 1,2,3,4;a.replicate(2, 1); // np.tile(a, (2,1)), repeat array by row
a.replicate(1, 2); // np.tile(a, (1,2)), repeat array by column
a.replicate(2, 3); // np.tile(a, (2,3))

Miss those elements out

ArrayXf a = ArrayXf::Random(10);// NumPy
a.head(2);                         // a[:2], the first two element
a.tail(a.size() - 1);              // a[1:], miss the first element
a.segment(1, 2);                   // a[1:3], middle element
a.tail(1);                         // a[-1], last element
a.tail(2);                         // a[-2:], last two elementArrayXXf A = ArrayXXf::Random(10,10);
A.leftCols(2);                     // a[:, :2], the first two columns
A.topRows(2);                      // a[:2, :], the first two rowsA.bottomRows(A.rows() - 1);        // a[1:, :], miss the first row
A.rightCols(A.cols() - 1);         // a[:, 1:], miss the first columnA.middleCols(1, 2);                // a[:, 1:3], middle columns
A.middleRows(1, 2);                // a[1:3, :], middle rowsA.rightCols(1);                    // a[:, -1], last column
A.bottomRows(1);                   // a[-1, :], last rowA.rightCols(2);                    // a[:, -2:], last two column
A.bottomRows(2);                   // a[-2:, :], last two row

Slicing

ArrayXf a = ArrayXf::LinSpaced(20, 0, 19);
Map<ArrayXf, 0, InnerStride<2>> v1(a.data(), a.size()/2);        // a[::2]
Map<ArrayXf, 0, InnerStride<2>> v2(a.middleRows(1, 6).data(), 3);// a[1:7:2]ArrayXXf A = ArrayXXf::Random(8,10);// Column-major storage
Map<ArrayXXf, 0, OuterStride<>> A1(A.data(), A.rows(), (A.cols() + 1)/2, OuterStride<>(A.outerStride()*2));               // A[:,::2]typedef Array<float, Dynamic, Dynamic, RowMajor> RowMajorArrayXXf;
RowMajorArrayXXf A2(A);
Map<RowMajorArrayXXf, 0, OuterStride<>> A3(A2.data(), (A2.rows() + 1)/2, A2.cols(), OuterStride<>(A2.outerStride()*2));   // A[::2, :]

Indexing and accessing elements

更多关于索引和元素访问的细节请参看 block operations

ArrayXXf a = ArrayXXf::Random(3,4);// NumPy
a(1,2);                        // a[1,2], element 1,2
a.row(0);                      // a[0,],  first row
a.col(0);                      // a[:,0], first column
a.middleRows(1, 2);            // a[1:3,], middle rowa.middleRows(1, a.rows()-1);   // a[1:,], all, except first row
a.bottomRows(a.rows()-1);a.bottomRows(2);               // a[-2:,], last tow rows

Maximum and minimum

ArrayXXf a = ArrayXXf::Random(3,3);
ArrayXXf b = ArrayXXf::Random(3,3);a.maxCoeff();             // np.max(a), max in array
a.colwise().maxCoeff();   // np.amax(a, axis=0), max in each column
a.rowwise().maxCoeff();   // np.amax(a, axis=1), max in each rowa.max(b);                 // np.maximum(a,b), pairwise max

Assignment

ArrayXXf a = ArrayXXf::Random(3,3);// NumPy
a.col(0) = 99;                   // a[:,0] = 99;
a.col(0) = Array3f{99,98,97};    // a[:,0] = np.array([99, 98, 97])
(a > 90).select(a, 90);          // (a>90).choose(a, 90)

Concatenation

// Concatenation(vectors)
Array3f a{1,1,1};
Array3f b{2,2,2};ArrayXf c( a.size() + b.size() );
c << a, b;                              // np.concatenate((a,b))// Concatenation(matrix)
int rows = 3;
int cols = 3;
ArrayXXf A = ArrayXXf::Constant(rows, cols, 1.0f);
ArrayXXf B = ArrayXXf::Constant(rows, cols, 2.0f);ArrayXXf BindRows(A.rows() + B.rows(), cols);
BindRows << A,B;                        // np.concatenate((A,B), axis=0)  // or np.vstack((a,b))ArrayXXf BindCols(rows, A.cols() + B.cols());
BindCols << A,B;                        // np.concatenate((A,B), axis=1) // or np.hstack((a,b))ArrayXf C(A.size() + B.size());         // np.concatenate((a,b), axis=None)
C << Map<ArrayXf>(A.data(), A.size()),  // concatenate matrices into a 1d arrayMap<ArrayXf>(B.data(), B.size());

Reshape and flatten matrices

ArrayXXf a = ArrayXXf::Random(3,4);           // Column-major storage// NumPy
Map<ArrayXXf> b(a.data(), 2, 6);              // np.reshape(a, (2,6))
Map<ArrayXf> v(a.data(), a.size());           // a.flatten("F"),  flatten in column-major

Coefficient-wise math functions

下面列举一些常用的 coefficient-wise 数学运算函数，更多详细的内容请参看 Catalog of coefficient-wise math functions

Basic operatoins

Python	Eigen	Description
abs(a)	a.abs(); Eigen::abs(a); m.cwiseAbs()	absolute value (∣ai∣\vert a_i\vert∣ai∣)
reciprocal(a)	a.inverse(); Eigen::inverse(a); m.cwiseInverse()	inverse vaule(1/ai1/a_i1/ai)
a.conj()	a.conjugate(); Eigen::conj(a); m.conjugate();	complex conjugate(aiˉ\bar{a_i}aiˉ)

Exponential and logarithm

Python	Eigen	Description
math.log(a)	a.log(); log(a)	logarithm, base eee(natural)
math.log10(a)	a.log10(); log10(a)	logarithm, base 10
math.exp(a)	a.exp(); exp(a)	exponential function
math.log1p(a)	a.log1p(); log1p(a)	natural (base eee) logarithm of 1 plus the given number ( ln⁡(1+ai)\ln({1+a_i})ln(1+ai))

Round off

Python	Eigen	Description
around(a) or math.round(a)	a.round()	Round
ceil(a)	a.ceil()	Round up
floor(a)	a.floor()	Round down

Power functions

Python	Eigen	Description
power(a,b)	a.pow(b)	power, aibia_i ^ {b_i}aibi
math.sqrt(a)	a.sqrt(); Eigen::sqrt(a); m.cwiseSqrt()	square root ((ai)\sqrt(a_i)(ai))
square(a)	a.square(); Eigen::square(a)	square power(ai2a_i^2ai2)
power(a,3)	a.cube(); Eigen::cube(a)	cubic power(ai3a_i^3ai3)
abs(a).square()	a.abs2(); Eigen::abs2(a);m.cwiseAbs2()	squared absolute value(∣ai∣2\vert a_i \vert^2∣ai∣2)

Trigonometric functions

Python	Eigen	Description
sin(a)	a.sin(); Eigen::sin(a)	computes sine
cos(a)	a.cos(); Eigen::cos(a)	computes cosine
tan(a)	a.tan(); Eigen::tan(a)	computes tangent
arcsin(a)	a.asin(); Eigen::asin(a)	computes arc sine(sin⁡−1ai\sin^{-1} a_isin−1ai)
arccos(a)	a.acos(); Eigen::acos(a)	computes arc cosine(cos⁡−1ai\cos^{-1} a_icos−1ai)
arctan(a)	a.atan(); Eigen::atan(a)	computes arc tangent(tan⁡−1ai\tan^{-1} a_itan−1ai)

Eigen exercise

/** Eigen exercise** This is quick reference of Eigen.** We use Eigen to implement 100 Numpy(some of them) (https://github.com/rougier/numpy-100)** It is highly recommended to read Eigen document before starting if you never read it**/#include <Eigen/Core>
#include <iostream>
#include <vector>
#include <set>
#include <algorithm>
#include <map>
#include <unsupported/Eigen/CXX11/Tensor>
#include <complex>
#include <cmath>
using namespace std;
using namespace Eigen;void exercise_2()
{// 2. Print the eigen versioncout << "Eigen version " << EIGEN_MAJOR_VERSION << "."<< EIGEN_MINOR_VERSION << endl;
}void exercise_3()
{// 3 Create a null vector of size 10 (★☆☆)VectorXf Z = VectorXf::Zero(10);cout << Z << endl;
}void exercise_4()
{// 4. How to find the memory size of any array (★☆☆)MatrixXf Z = MatrixXf::Zero(10, 10);cout << Z.size() * sizeof(MatrixXf::Scalar) << endl;
}void exercise_6()
{// 6. Create a null vector of size 10 but the fifth value which is 1 (★☆☆)VectorXf Z = VectorXf::Zero(10);Z(4) = 1;cout << Z << endl;
}void exercise_7()
{// 7. Create a vector with values ranging from 10 to 49 (★☆☆)const int start = 10;const int end = 49;const int size = end - start + 1;VectorXf Z = VectorXf::LinSpaced(size, start, end);cout << Z << endl;
}void exercise_8()
{// 8. Reverse a vector (first element becomes last) (★☆☆)VectorXf Z = VectorXf::LinSpaced(10, 1, 10);Z = Z.reverse().eval();cout << Z << endl;// Z = Z.reverse() is aliasing// you can use .eval() or inplace to solve this://           Z = Z.reverse().eval()//           Z.reverseInPlace()
}void exercise_9()
{//9. Create a 3x3 matrix with values ranging from 0 to 8 (★☆☆)// Eigen does not expose convenient methods to take slices or to reshape a matrix yet.// Nonetheless, such features can easily be emulated using the Map class.VectorXf Z1 = VectorXf::LinSpaced(9, 1, 9);Map<MatrixXf> Z2(Z1.data(), 3, 3);cout << Z2 << endl;
}void exercise_10()
{// 10. Find indices of non-zero elements from [1,2,0,0,4,0] (★☆☆)VectorXf Z(6);Z << 1,2,0,0,4,0;std::vector<Index> nz;for(Index i = 0; i < Z.size(); ++i){if(Z(i)){nz.push_back(i);}}Map<Matrix<Index , 1, Dynamic>> nzz(nz.data(), nz.size());cout << nzz;
}void exercise_11()
{// 11. Create a 3x3 identity matrix (★☆☆)MatrixXf Z = MatrixXf::Identity(3,3);cout << Z << endl;
}void exercise_12()
{// 12. Create a 3x3x3 array with random values (★☆☆)// NOTE: Tensor is unsupported module in 3.3.7Tensor<float,3> T(3,3,3);T.setRandom();cout << T << endl;// 12.1 Create a 3x3 array with random values (★☆☆)MatrixXf Z = MatrixXf::Random(3,3);cout << Z << endl;
}void exercise_13()
{// 13. Create a 10x10 array with random values and find the minimum and maximum values (★☆☆)MatrixXf Z = MatrixXf::Random(10, 10);cout << Z.maxCoeff() << "," << Z.minCoeff() << endl;
}void exercise_14()
{// 14. Create a random vector of size 30 and find the mean value (★☆☆)VectorXf Z = VectorXf::Random(30);cout << Z.mean() << endl;
}void exercise_15()
{// 15. Create a 2d array with 1 on the border and 0 inside (★☆☆)MatrixXf Z = MatrixXf::Zero(5, 5);VectorXf padding = VectorXf::Constant(5, -1);Z.topRows<1>() = padding;Z.bottomRows<1>() = padding;Z.leftCols<1>() = padding;Z.rightCols<1>() = padding;cout << Z << endl;
}void exercise_16()
{// 16. How to add a border (filled with 0's) around an existing array? (★☆☆)MatrixXf Z = MatrixXf::Ones(5, 5);Z.conservativeResize(6,6);VectorXf padding = VectorXf::Zero(6);Z.topRows<1>() = padding;Z.bottomRows<1>() = padding;Z.leftCols<1>() = padding;Z.rightCols<1>() = padding;cout << Z << endl;
}void exercise_18()
{// 18. Create a 5x5 matrix with values 1,2,3,4 just below the diagonal (★☆☆)// Is difficult to implement in Eigen, but create a diagonal is easyVectorXf V(4);V << 1, 2, 3, 4;MatrixXf Z = V.asDiagonal();cout << Z << endl;
}void exercise_21()
{// 21. Create a checkerboard 8x8 matrix using the tile function (★☆☆)MatrixXf Z(2,2);Z << 0,1,1,0;cout <<  Z.replicate(4, 4) << endl;
}void exercise_22()
{// 22. Normalize a 5x5 random matrix (★☆☆)MatrixXf Z = MatrixXf::Random(5,5);float mean = Z.mean();float std = std::sqrt( (Z.array() - mean).square().sum() / (Z.size() - 1) );Z = (Z.array() - mean) / (std);cout << Z << endl;
}void exercise_24()
{// 24. Multiply a 5x3 matrix by a 3x2 matrix (real matrix product) (★☆☆)MatrixXf A = MatrixXf::Ones(5,3);MatrixXf B = MatrixXf::Ones(3,2);MatrixXf Z = A * B;cout << Z << endl;
}
void exercise_25()
{// 25. Given a 1D array, negate all elements which are between 3 and 8, in place. (★☆☆)VectorXf Z = VectorXf::LinSpaced(11, 0, 10);Matrix<float, Dynamic, 1> B = (3 < Z.array() && Z.array() <= 8).cast<float>() * -1.0;Matrix<float, Dynamic, 1> C = B.array() + 1.0;cout << Z.array() * B.array() + Z.array() * C.array() << endl;
}void exercise_30()
{// 30. How to find common values between two arrays? (★☆☆)std::mt19937 gen(0);std::uniform_int_distribution<int> dis(0, 10);// generate random int numbers in range [0, 10]auto func = [&](int x){return dis(gen);};VectorXi A = VectorXd::Zero(10).unaryExpr(func);VectorXi B = VectorXd::Zero(10).unaryExpr(func);std::set<int> commom_values_set;auto find_common_values = [&](int x){if( (B.array() == x).any() ){commom_values_set.insert(x);}return x;};A = A.unaryExpr(find_common_values);for(const auto& v : commom_values_set){cout << v << " ";}
}void exercise_39()
{// 39. Create a vector of size 10 with values ranging from 0 to 1, both excluded (★★☆)¶VectorXf Z = VectorXf::LinSpaced(12, 0, 1);Z = Z.segment(1, 10);cout << Z << endl;
}void exercise_40()
{// 40. Create a random vector of size 10 and sort it (★★☆)VectorXf Z = VectorXf::Random(10);sort(Z.data(), Z.data()+Z.size(), [](float x, float y){return x < y;});cout << Z << endl;
}void exercise_40_1()
{// 40_1. Create a random matrix of size 10x10 and sort it row by row (★★☆)MatrixXf Z = MatrixXf::Random(10, 10);auto sort_func = [](float x, float y){return x<y;};std::vector<MatrixXf::Scalar> data(Z.cols());for(int i = 0; i < Z.rows(); ++i){// copy row to data arrayfor(int j = 0; j < Z.cols(); ++j){data[j] = Z(i, j);}// sort data arraysort(data.begin(), data.end(), sort_func);// copy back to rowfor(int j = 0; j < Z.cols(); ++j){Z(i, j) = data[j];}}cout << Z << endl;
}void exercise_40_2()
{// 40_2. Create a random matrix of size 10x10 and sort it col by col (★★☆)MatrixXf Z = MatrixXf::Random(10, 10);auto sort_func = [](float x, float y){return x<y;};std::vector<MatrixXf::Scalar> data(Z.rows());for(int i = 0; i < Z.cols(); ++i){// copy row to data arrayfor(int j = 0; j < Z.rows(); ++j){data[j] = Z(j, i);}// sort data arraysort(data.begin(), data.end(), sort_func);// copy back to rowfor(int j = 0; j < Z.rows(); ++j){Z(j, i) = data[j];}}cout << Z << endl;}void exercise_42()
{// 42. Consider two random array A and B, check if they are equal (★★☆)MatrixXf A = MatrixXf::Random(5,5);MatrixXf B = MatrixXf::Random(5,5);bool equal = (A.array() == B.array()).all();cout << equal << endl;
}void exercise_44()
{// 44. Consider a random 10x2 matrix representing cartesian coordinates, convert them to polar coordinates (★★☆)MatrixXf Z = MatrixXf::Random(10, 2);VectorXf X = Z.col(0);VectorXf Y = Z.col(1);VectorXf R = (X.array().square() + Y.array().square()).sqrt();VectorXf T = (Y.array()/X.array()).atan();cout << "R:\n" << R << endl;cout << "T:\n" << T << endl;
}void exercise_45()
{// 45. Create random vector of size 10 and replace the maximum value by 0 (★★☆)VectorXf Z = VectorXf::Random(10);VectorXf::Index max_index;Z.maxCoeff(&max_index);Z(max_index) = 0.0;cout << Z << endl;
}void exercise_47()
{// 47. Given two arrays, X and Y, construct the Cauchy matrix C (Cij =1/(xi - yj))VectorXf X = VectorXf::LinSpaced(8, 0, 7);VectorXf Y = X.array() + 0.5;MatrixXf C(X.size(), Y.size());for(int i = 0; i < C.cols(); ++i){C.col(i) = 1.0/(X.array() - Y(i));}cout << C << endl;
}void exercise_50()
{// 50. How to find the closest value (to a given scalar) in a vector? (★★☆)VectorXf Z = VectorXf::LinSpaced(100, 0, 99);float v = 10;VectorXf::Index index;(Z.array() - v).abs().minCoeff(&index);cout << index << endl;
}void exercise_52()
{//    52. Consider a random vector with shape (10,2) representing coordinates, find point by point distances (★★☆)MatrixXf Z = MatrixXf::Random(10, 2);Matrix<float, Dynamic, 1> X = Z.col(0);Matrix<float, Dynamic, 1> Y = Z.col(1);MatrixXf XX = X.rowwise().replicate(10);MatrixXf YY = Y.rowwise().replicate(10);MatrixXf D = (XX - XX.transpose()).array().square() + (YY - YY.transpose()).array().square();cout << D.cwiseSqrt() << endl; // D.cwiseSqrt() = D.array().sqrt()
}void exercise_56()
{// 56. Generate a generic 2D Gaussian-like array (★★☆)
//    Matrix<float,Dynamic,1> X =VectorXf V = VectorXf::LinSpaced(10, -1, 1);MatrixXf Y = V.rowwise().replicate(10);MatrixXf X = Y.transpose();MatrixXf G = (X.array().square() + Y.array().square()).cwiseSqrt();float sigma = 1.0;float mu = 0.0;MatrixXf result = ( -(G.array() - mu).square() / (2.0 * sigma*sigma) ).exp();cout << result << endl;
}void exercise_58()
{// 58. Subtract the mean of each row of a matrix (★★☆)¶MatrixXf Z = MatrixXf::Random(5, 10);VectorXf mean = Z.rowwise().mean();cout << Z.colwise() - mean << endl;
}void exercise_59()
{// 59. How to sort an array by the nth column? (★★☆)MatrixXf Z = MatrixXf::Random(3,3);cout << Z << endl << endl;int n = 1; // first columnauto compare_nth = [&n](const VectorXf& lhs, const VectorXf& rhs){return lhs(n) < rhs(n);};std::vector<VectorXf> vec;for(int i = 0; i < Z.rows(); ++i){vec.emplace_back(Z.row(i));}std::sort(vec.begin(), vec.end(), compare_nth);for(int i = 0; i < Z.rows(); ++i){Z.row(i) = vec[i];}cout << Z << endl;
}void exercise_60()
{// 60. How to tell if a given 2D array has null columns? (★★☆)MatrixXf Z = MatrixXf::Random(5,5);Z(0,0) = sqrt(-1); // genenrate a nancout << Eigen::isnan(Z.array()).any() << endl;cout << Eigen::isnan(Z.array()).select(0, Z) << endl;
}void exercise_61()
{// 61. Find the nearest value from a given value in an array (★★☆)VectorXf Z = VectorXf::Random(10);float z = 0.5;VectorXf::Index min_index;(Z.array() - z).abs().minCoeff(&min_index);cout << Z(min_index) << endl;
}void exercise_64()
{// 64. Consider a given vector, how to add 1 to each element indexed by a second vector (be careful with repeated indices)? (★★★)VectorXf Z = VectorXf::Ones(10);VectorXi I(20);I << 7, 4, 2, 6, 4, 8, 6, 3, 0, 8, 1, 0, 5, 1, 9, 7, 5, 1, 8, 7;std::map<int, int> bins;for(int i = 0; i < I.size(); ++i){const int key = I(i);if(bins.find(key) != bins.end()){++bins[key];} else{bins[key] = 1;}}for(int i = 0; i < Z.size(); ++i){Z(i) += bins[i];}cout << Z << endl;
}

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Eigen 简明教程之如何从Python转到Eigen