c++ mathmatics

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本文介绍并对比了C++中的三个科学计算库：SP++, Eigen和GSL。SP++专注于信号处理，Eigen则是一个高效的线性代数库，而GSL提供了全面的科学计算功能。文章详细介绍了Eigen的使用方法，包括矩阵定义、基础操作、特殊矩阵生成等，并给出了GSL的安装步骤和示例代码。

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1) C++信号处理库 SP++

2)线性算术的C++模板库 Eigen

3)C 科学计算库 GSL

https://my.oschina.net/zmjerry/blog/13049,sp++ 用啥调啥，模板设计如kalman filter

https://my.oschina.net/zmjerry/blog/8517

Kalman滤波算法的C++实现

/*
 * Copyright (c) 2008-2011 Zhang Ming (M. Zhang), zmjerry@163.com
 *
 * This program is free software; you can redistribute it and/or modify it
 * under the terms of the GNU General Public License as published by the
 * Free Software Foundation, either version 2 or any later version.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 *
 * 1. Redistributions of source code must retain the above copyright notice,
 *    this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * This program is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
 * more details. A copy of the GNU General Public License is available at:
 * http://www.fsf.org/licensing/licenses
 */


/*****************************************************************************
 *                                   kalman.h
 *
 * Kalman Filter.
 *
 * The Kalman filter is an efficient recursive filter that estimates the
 * internal state of a linear dynamic system from a series of noisy
 * measurements. In most applications, the internal state is much larger
 * (more degrees of freedom) than the few "observable" parameters which are
 * measured. However, by combining a series of measurements, the Kalman
 * filter can estimate the entire internal state.
 *
 * A wide variety of Kalman filters have now been developed, from Kalman's
 * original formulation, now called the simple Kalman filter, the Kalman-Bucy
 * filter, Schmidt's extended filter, the information filter, and a variety
 * of square-root filters that were developed by Bierman, Thornton and so on.
 *
 * Zhang Ming, 2010-10, Xi'an Jiaotong University.
 *****************************************************************************/


#ifndef KALMAN_H
#define KALMAN_H


#include <vector.h>
#include <matrix.h>
#include <inverse.h>


namespace splab
{

    template<typename Type>
    Vector<Type> kalman( const Matrix<Type>&, const Matrix<Type>&,
                         const Matrix<Type>&, const Matrix<Type>&,
                         const Vector<Type>&, const Vector<Type>&,
                         const Vector<Type>& );


    #include <kalman-impl.h>

}
// namespace splab


#endif
// KALMAN_H

实现文件：

/*
 * Copyright (c) 2008-2011 Zhang Ming (M. Zhang), zmjerry@163.com
 *
 * This program is free software; you can redistribute it and/or modify it
 * under the terms of the GNU General Public License as published by the
 * Free Software Foundation, either version 2 or any later version.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 *
 * 1. Redistributions of source code must retain the above copyright notice,
 *    this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * This program is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
 * more details. A copy of the GNU General Public License is available at:
 * http://www.fsf.org/licensing/licenses
 */


/*****************************************************************************
 *                              kalman-impl.h
 *
 * Implementation for Kalman Filter.
 *
 * Zhang Ming, 2010-10, Xi'an Jiaotong University.
 *****************************************************************************/


/**
 * The simple Kalman filter for one step.
 * A ---> system matrix defining linear dynamic system
 * C ---> measurement matrix defining relationship between system's state
 *        and measurements
 * Q ---> covariance matrix of process noise in system state dynamics
 * R ---> covariance matrix of measurements uncertainty
 * y ---> measurements at time t
 * xPrev ---> previous estimated state vector of the linear dynamic system
 * initDiagV ---> diagonal vector for initializing the covariance matrix of
 *                state estimation uncertainty
 */
template <typename Type>
Vector<Type> kalman( const Matrix<Type> &A, const Matrix<Type> &C,
                     const Matrix<Type> &Q, const Matrix<Type> &R,
                     const Vector<Type> &xPrev, const Vector<Type> &y,
                     const Vector<Type> &initDiagV )
{
    int N = xPrev.size();

    // covariance matrix of state estimation uncertainty
    static Matrix<Type> V = diag(initDiagV);

    // previoused state vector
    Vector<Type> xPred = A * xPrev;

    // inovation
    Vector<Type> alpha = y - C * xPred;

    Matrix<Type> CTran = trT( C );
    Matrix<Type> VPred = A*V*trT(A) + Q;

    // Kalman gain matrix
    Matrix<Type> KGain = VPred*CTran * inv(C*VPred*CTran+R);

    V = ( eye(N,Type(1.0)) - KGain*C ) * VPred;

    // return the estimation of the state vector
    return xPred + KGain * alpha;
}

测试代码：

/*****************************************************************************
 *                                   kalman.h
 *
 * Kalman filter testing.
 *
 * Zhang Ming, 2010-10, Xi'an Jiaotong University.
 *****************************************************************************/


#define BOUNDS_CHECK

#include <iostream>
#include <kalman.h>


using namespace std;
using namespace splab;


typedef double  Type;
const   int     N = 2;
const   int     M = 2;
const   int     T = 20;


int main()
{
    Matrix<Type> A(N,N), C(M,N), Q(N,N), R(M,M);
    A = eye( N, Type(1.0) );    C = eye( N, Type(1.0) );
    Q = eye( N, Type(1.0) );    R = eye( N, Type(2.0) );

    Vector<Type> x(N,Type(1.0)), y(M), ytInit(M);
    ytInit[0] = Type(0.5);  ytInit[1] = Type(2.0);
    Matrix<Type> yt(M,T);
    for( int t=0; t<T; ++t )
        yt.setColumn( ytInit, t );

    Vector<Type> intV( N, Type(10.0) );
    for( int t=0; t<T; ++t )
    {
        y = yt.getColumn(t);
        x = kalman( A, C, Q, R, x, y, intV );
        cout << "Estimation of xt at the " << t << "th iteratin:   " << x << endl;
    }

    cout << "The theoretical xt should converge to:   " << ytInit << endl;

    return 0;
}

运行结果：

2) Eigen

https://www.cnblogs.com/python27/p/EigenQuickRef.html

C++矩阵库 Eigen 快速入门

最近需要用 C++ 做一些数值计算，之前一直采用Matlab 混合编程的方式处理矩阵运算，非常麻烦，直到发现了 Eigen 库，简直相见恨晚，好用哭了。 Eigen 是一个基于C++模板的线性代数库，直接将库下载后放在项目目录下，然后包含头文件就能使用，非常方便。此外，Eigen的接口清晰，稳定高效。唯一的问题是之前一直用 Matlab，对 Eigen 的 API 接口不太熟悉，如果能有 Eigen 和 Matlab 对应的说明想必是极好的，终于功夫不负有心人，让我找到了，原文在这里，不过排版有些混乱，我将其重新整理了一下，方便日后查询。

Eigen 矩阵定义

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#include <Eigen/Dense>

Matrix<double, 3, 3> A;               // Fixed rows and cols. Same as Matrix3d.
Matrix<double, 3, Dynamic> B;         // Fixed rows, dynamic cols.
Matrix<double, Dynamic, Dynamic> C;   // Full dynamic. Same as MatrixXd.
Matrix<double, 3, 3, RowMajor> E;     // Row major; default is column-major.
Matrix3f P, Q, R;                     // 3x3 float matrix.
Vector3f x, y, z;                     // 3x1 float matrix.
RowVector3f a, b, c;                  // 1x3 float matrix.
VectorXd v;                           // Dynamic column vector of doubles
// Eigen          // Matlab           // comments
x.size()          // length(x)        // vector size
C.rows()          // size(C,1)        // number of rows
C.cols()          // size(C,2)        // number of columns
x(i)              // x(i+1)           // Matlab is 1-based
C(i,j)            // C(i+1,j+1)       //

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Eigen 基础使用

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// Basic usage
// Eigen        // Matlab           // comments
x.size()        // length(x)        // vector size
C.rows()        // size(C,1)        // number of rows
C.cols()        // size(C,2)        // number of columns
x(i)            // x(i+1)           // Matlab is 1-based
C(i, j)         // C(i+1,j+1)       //

A.resize(4, 4);   // Runtime error if assertions are on.
B.resize(4, 9);   // Runtime error if assertions are on.
A.resize(3, 3);   // Ok; size didn't change.
B.resize(3, 9);   // Ok; only dynamic cols changed.
                  
A << 1, 2, 3,     // Initialize A. The elements can also be
     4, 5, 6,     // matrices, which are stacked along cols
     7, 8, 9;     // and then the rows are stacked.
B << A, A, A;     // B is three horizontally stacked A's.
A.fill(10);       // Fill A with all 10's.

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Eigen 特殊矩阵生成

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// Eigen                            // Matlab
MatrixXd::Identity(rows,cols)       // eye(rows,cols)
C.setIdentity(rows,cols)            // C = eye(rows,cols)
MatrixXd::Zero(rows,cols)           // zeros(rows,cols)
C.setZero(rows,cols)                // C = ones(rows,cols)
MatrixXd::Ones(rows,cols)           // ones(rows,cols)
C.setOnes(rows,cols)                // C = ones(rows,cols)
MatrixXd::Random(rows,cols)         // rand(rows,cols)*2-1        // MatrixXd::Random returns uniform random numbers in (-1, 1).
C.setRandom(rows,cols)              // C = rand(rows,cols)*2-1
VectorXd::LinSpaced(size,low,high)  // linspace(low,high,size)'
v.setLinSpaced(size,low,high)       // v = linspace(low,high,size)'

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Eigen 矩阵分块

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// Matrix slicing and blocks. All expressions listed here are read/write.
// Templated size versions are faster. Note that Matlab is 1-based (a size N
// vector is x(1)...x(N)).
// Eigen                           // Matlab
x.head(n)                          // x(1:n)
x.head<n>()                        // x(1:n)
x.tail(n)                          // x(end - n + 1: end)
x.tail<n>()                        // x(end - n + 1: end)
x.segment(i, n)                    // x(i+1 : i+n)
x.segment<n>(i)                    // x(i+1 : i+n)
P.block(i, j, rows, cols)          // P(i+1 : i+rows, j+1 : j+cols)
P.block<rows, cols>(i, j)          // P(i+1 : i+rows, j+1 : j+cols)
P.row(i)                           // P(i+1, :)
P.col(j)                           // P(:, j+1)
P.leftCols<cols>()                 // P(:, 1:cols)
P.leftCols(cols)                   // P(:, 1:cols)
P.middleCols<cols>(j)              // P(:, j+1:j+cols)
P.middleCols(j, cols)              // P(:, j+1:j+cols)
P.rightCols<cols>()                // P(:, end-cols+1:end)
P.rightCols(cols)                  // P(:, end-cols+1:end)
P.topRows<rows>()                  // P(1:rows, :)
P.topRows(rows)                    // P(1:rows, :)
P.middleRows<rows>(i)              // P(i+1:i+rows, :)
P.middleRows(i, rows)              // P(i+1:i+rows, :)
P.bottomRows<rows>()               // P(end-rows+1:end, :)
P.bottomRows(rows)                 // P(end-rows+1:end, :)
P.topLeftCorner(rows, cols)        // P(1:rows, 1:cols)
P.topRightCorner(rows, cols)       // P(1:rows, end-cols+1:end)
P.bottomLeftCorner(rows, cols)     // P(end-rows+1:end, 1:cols)
P.bottomRightCorner(rows, cols)    // P(end-rows+1:end, end-cols+1:end)
P.topLeftCorner<rows,cols>()       // P(1:rows, 1:cols)
P.topRightCorner<rows,cols>()      // P(1:rows, end-cols+1:end)
P.bottomLeftCorner<rows,cols>()    // P(end-rows+1:end, 1:cols)
P.bottomRightCorner<rows,cols>()   // P(end-rows+1:end, end-cols+1:end)

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Eigen 矩阵元素交换

// Of particular note is Eigen's swap function which is highly optimized.
// Eigen                           // Matlab
R.row(i) = P.col(j);               // R(i, :) = P(:, i)
R.col(j1).swap(mat1.col(j2));      // R(:, [j1 j2]) = R(:, [j2, j1])

Eigen 矩阵转置

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// Views, transpose, etc; all read-write except for .adjoint().
// Eigen                           // Matlab
R.adjoint()                        // R'
R.transpose()                      // R.' or conj(R')
R.diagonal()                       // diag(R)
x.asDiagonal()                     // diag(x)
R.transpose().colwise().reverse(); // rot90(R)
R.conjugate()                      // conj(R)

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Eigen 矩阵乘积

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// All the same as Matlab, but matlab doesn't have *= style operators.
// Matrix-vector.  Matrix-matrix.   Matrix-scalar.
y  = M*x;          R  = P*Q;        R  = P*s;
a  = b*M;          R  = P - Q;      R  = s*P;
a *= M;            R  = P + Q;      R  = P/s;
                   R *= Q;          R  = s*P;
                   R += Q;          R *= s;
                   R -= Q;          R /= s;

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Eigen 矩阵单个元素操作

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// Vectorized operations on each element independently
// Eigen                  // Matlab
R = P.cwiseProduct(Q);    // R = P .* Q
R = P.array() * s.array();// R = P .* s
R = P.cwiseQuotient(Q);   // R = P ./ Q
R = P.array() / Q.array();// R = P ./ Q
R = P.array() + s.array();// R = P + s
R = P.array() - s.array();// R = P - s
R.array() += s;           // R = R + s
R.array() -= s;           // R = R - s
R.array() < Q.array();    // R < Q
R.array() <= Q.array();   // R <= Q
R.cwiseInverse();         // 1 ./ P
R.array().inverse();      // 1 ./ P
R.array().sin()           // sin(P)
R.array().cos()           // cos(P)
R.array().pow(s)          // P .^ s
R.array().square()        // P .^ 2
R.array().cube()          // P .^ 3
R.cwiseSqrt()             // sqrt(P)
R.array().sqrt()          // sqrt(P)
R.array().exp()           // exp(P)
R.array().log()           // log(P)
R.cwiseMax(P)             // max(R, P)
R.array().max(P.array())  // max(R, P)
R.cwiseMin(P)             // min(R, P)
R.array().min(P.array())  // min(R, P)
R.cwiseAbs()              // abs(P)
R.array().abs()           // abs(P)
R.cwiseAbs2()             // abs(P.^2)
R.array().abs2()          // abs(P.^2)
(R.array() < s).select(P,Q);  // (R < s ? P : Q)

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Eigen 矩阵化简

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// Reductions.
int r, c;
// Eigen                  // Matlab
R.minCoeff()              // min(R(:))
R.maxCoeff()              // max(R(:))
s = R.minCoeff(&r, &c)    // [s, i] = min(R(:)); [r, c] = ind2sub(size(R), i);
s = R.maxCoeff(&r, &c)    // [s, i] = max(R(:)); [r, c] = ind2sub(size(R), i);
R.sum()                   // sum(R(:))
R.colwise().sum()         // sum(R)
R.rowwise().sum()         // sum(R, 2) or sum(R')'
R.prod()                  // prod(R(:))
R.colwise().prod()        // prod(R)
R.rowwise().prod()        // prod(R, 2) or prod(R')'
R.trace()                 // trace(R)
R.all()                   // all(R(:))
R.colwise().all()         // all(R)
R.rowwise().all()         // all(R, 2)
R.any()                   // any(R(:))
R.colwise().any()         // any(R)
R.rowwise().any()         // any(R, 2)

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Eigen 矩阵点乘

// Dot products, norms, etc.
// Eigen                  // Matlab
x.norm()                  // norm(x).    Note that norm(R) doesn't work in Eigen.
x.squaredNorm()           // dot(x, x)   Note the equivalence is not true for complex
x.dot(y)                  // dot(x, y)
x.cross(y)                // cross(x, y) Requires #include <Eigen/Geometry>

Eigen 矩阵类型转换

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//// Type conversion
// Eigen                           // Matlab
A.cast<double>();                  // double(A)
A.cast<float>();                   // single(A)
A.cast<int>();                     // int32(A)
A.real();                          // real(A)
A.imag();                          // imag(A)
// if the original type equals destination type, no work is done

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Eigen 求解线性方程组 Ax = b

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// Solve Ax = b. Result stored in x. Matlab: x = A \ b.
x = A.ldlt().solve(b));  // A sym. p.s.d.    #include <Eigen/Cholesky>
x = A.llt() .solve(b));  // A sym. p.d.      #include <Eigen/Cholesky>
x = A.lu()  .solve(b));  // Stable and fast. #include <Eigen/LU>
x = A.qr()  .solve(b));  // No pivoting.     #include <Eigen/QR>
x = A.svd() .solve(b));  // Stable, slowest. #include <Eigen/SVD>
// .ldlt() -> .matrixL() and .matrixD()
// .llt()  -> .matrixL()
// .lu()   -> .matrixL() and .matrixU()
// .qr()   -> .matrixQ() and .matrixR()
// .svd()  -> .matrixU(), .singularValues(), and .matrixV()

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Eigen 矩阵特征值

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// Eigenvalue problems
// Eigen                          // Matlab
A.eigenvalues();                  // eig(A);
EigenSolver<Matrix3d> eig(A);     // [vec val] = eig(A)
eig.eigenvalues();                // diag(val)
eig.eigenvectors();               // vec
// For self-adjoint matrices use SelfAdjointEigenSolver<>

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参考文献

【1】http://eigen.tuxfamily.org/dox/AsciiQuickReference.txt

【2】http://blog.youkuaiyun.com/augusdi/article/details/12907341

https://blog.youkuaiyun.com/liu_tian_wei/article/details/83615441

Linux系统
1. 在gnu的ftp站点http://ftp.gnu.org/gnu/gsl/ 上，下载最新的gsl-2.x.tar.gz

2. 解压下载好的gsl-2.x.tar.gz 压缩包，$tar -zxvf gsl-2.x.tar.gz. 直接解压在了Downloads文件夹中。

3. $ cd gsl-2.x，进入到gsl文件夹中，运行$./configure --prefix=/usr ，该命令指定之后安装的include, lib, bin等文件夹都安装在usr目录下。

4. 相继运行，$make，$make check(可不使用）, $make install, gsl库便安装在了/usr 目录下。

test.cpp

#include<iostream>
#include<gsl/gsl_fit.h>
using namespace std;

int main(){
   double c0 = 0, c1 = 0, cov00 = 0, cov01 = 0, cov11 = 0, sumsq = 0;
   double x[]={1,2,3,4,5};
   double y[]={3,4,5,6,7};
   gsl_fit_linear(x, 1, y, 1, 5, &c0, &c1, &cov00, &cov01, &cov11, &sumsq);
   cout<<c0<<" "<<c1<<endl;
}
g++:

g++ -o Test test.cpp -std=c++11 -lgsl -lgslcblas
执行

./Test
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原文链接：https://blog.youkuaiyun.com/liu_tian_wei/article/details/83615441