Contents: Numerical analysis toolbox and iterative solution toolbox for linear/nonlinear equations

Update1: Programs about some common interpolation and fitting

• Lagrange interpolation: 2-points, 3-points, n-points; Runge's phenomenon —— lagra_2points.m, largra_3points.m, lagra_npoints.m and Runge.m
• Newton interpolation: 2-points, 3-points, n-points —— New_2points.m, New_3points.m and New_npoints.m
• Piecewise-linear Lagrangian interpolation —— fenduan_linear.m
• Unary nonlinear fitting —— nonlinear_fitting.m
• More detailed interpretation about Interpolation: https://www.jianshu.com/p/add2e938271c
• More detailed interpretation about Fitting: https://www.jianshu.com/p/41caace02f39

Update2: Programs about Numerical intergration

• Lagrangian quadrature(basic): Trapezoidal formula, Simpson formula, Newton-Coates formula —— trapezoid.m, simpson.m and newton_cotes.m
• Lagrangian quadrature(intermediate): Composite trapezoid formula, Composite Simpson formula —— f_trapezoid.m and f_simpson.m
• Lagrangian quadrature(advanced): Composite trapezoidal encryption formula, Romberg quadrature formula —— jm_f_trapezoid.m and romberg_js.m
• Gauss quadrature: Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite —— gauss_legendre2.m, gauss_laguerre.m, gauss_hermite.m
• You can see more detailed interpretation from my blog: https://www.jianshu.com/p/c6fdfe11e6bc

Update3：Programs about iterative solution for linear equations

• Original iteration methods: Jacobi iteration, Gauss-Seidel iteration, super-relaxation iteration —— jacobian_iteration.m, seidel_iteration.m, relaxation_iteration.m
• Universal iteration method after preprocessing(Recommand!):pre-Gauss-Seidel iteration, pre-super-relaxation iteration —— pre_seidel.m, pre_relaxation.m
• Matrix diagonal maximization preprocessing: This is not a panacea, but it can improve the convergence probability of iterative methods! —— diagonal_maximization.m
• Reference: Jacobi和Gauss-Seidel迭代法的预处理
• You can see more detailed interpretation from my blog: https://www.jianshu.com/p/e14d9e910984

Update4: Using the least square approximate solution to solve overdetermined incompatible linear equations

• The least square approximation —— overdetermined_linear.m
• You can see more detailed interpretation from my blog: https://www.jianshu.com/p/3d8abb3ba7d3

Update5: Using generalized plus inverse matrix to solve linear equations

• The full-rank decomposition of any matrix —— FRD.M
• The linear equation has no solution(Condition1): using the generalized plus inverse to find *all least squares solutions and unique minimal-norm least squares solution —— wujie.m
• The linear equation has infinite solutions(Condition2): using the generalized plus inverse to find general solutions and unique minimal-norm solution —— wuqiongjie.m
• The linear equation has only one solution(Condition3): using the generalized plus inverse to find out the unique solution —— weiyijie.m
• You can see more detailed interpretation from my blog: https://www.jianshu.com/p/8777e5d11a03

Update6: The solution methods of the nonlinear equations. The programs are in the Nonlinear_Equations folder.

• Original Newton method + pre-Gauss-Seidel iteration —— niudun.m and pre_seidel.m
• Modified Newton method —— xzniudun.m
• Quasi-Newton Method(single rank inverse Broyden Ⅰ) —— nbroyden1.m
• Quasi-Newton Method(single rank inverse Broyden Ⅱ) —— nbroyden2.m
• Quasi-Newton Method(rank-two BFS) —— BFS.m
• You can see more detailed interpretation from my blog: https://www.jianshu.com/p/4e2d6a45aa67

内容：数值分析相关程序包括插值、拟合、数值积分；方程求解包括线性方程组迭代求解、非线性方程(组)求解、常微分方程数值解

更新1：多项式插值相关程序；最小二乘一元非线性拟合程序。

• 拉格朗日插值：2点、3点、n点拉格朗日插值、龙格现象(文件名：lagra_2points.m、largra_3points.m、lagra_npoints.m、Runge.m)；
• 牛顿插值：2点、3点、n点牛顿插值(文件名：New_2points.m、New_3points.m、New_npoints.m)；
• 分段线性拉格朗日插值(文件名：fenduan_linear.m)；
• 一元非线性拟合(文件名：nonlinear_fitting.m)。
• 插值说明参考这里；拟合说明参考这里

更新2：数值积分相关程序。

• 拉格朗日型积分(基础款)：梯形公式、辛普森公式、牛顿-科茨公式(文件名：trapezoid.m、simpson.m、newton_cotes.m)；
• 拉格朗日型积分(进阶款)：复化梯形公式、复化辛普森公式(文件名：f_trapezoid.m、f_simpson.m)；
• 拉格朗日型积分(高级款)：复化梯形加密公式、龙贝格公式(文件名：jm_f_trapezoid.m、romberg_js.m)；
• 高斯型积分公式：高斯-勒让德、高斯-拉盖尔、高斯-埃尔米特；包括插值节点和系数的求取，以及实例(文件名：gauss_legendre2.m、gauss_laguerre.m、gauss_hermite.m)。
• 相关说明参考这里

更新3：线性方程组迭代求解相关程序。

• 未预处理原始迭代方法：雅克比迭代、高斯-赛德尔迭代、(超)松弛迭代(文件名：jacobian_iteration.m、seidel_iteration.m、relaxation_iteration.m)；
• 预处理后万能迭代方法(推荐√)：预处理后万能高斯-赛德尔迭代、预处理后万能(超)松弛迭代(文件名：pre_seidel.m、pre_relaxation.m)。
• 对角最大化预处理：非万能，但还是可以提高迭代收敛的几率，值得参考(文件名: diagonal_maximization.m)
• 预处理参考文献：《Jacobi和Gauss-Seidel迭代法的预处理》
• 相关说明参考这里

更新4：超定不相容线性方程组最小二乘近似解

• 最小二乘法近似求解(文件名：overdetermined_linear.m)；
• 相关说明参考这里

更新5：广义加号逆矩阵求解线性方程组

• 任意矩阵的满秩分解(文件名：FRD.m)；
• 线性方程组无解：广义加号逆求全部最小二乘解和唯一极小范数最小二乘解(文件名：wujie.m)；
• 线性方程组无穷解：广义加号逆求通解和唯一极小范数解(文件名：wuqiongjie.m)；
• 线性方程组唯一解：广义加号逆求唯一解(文件名：weiyijie.m)；
• 相关说明参考这里

更新6：非线性方法组的求解方法，详见文件夹Nonlinear_Equations

• 原始牛顿法(文件名：niudun.m)，辅助求解线性方法组的万能高斯-赛德尔迭代函数(pre_seidel.m)；
• 修正牛顿法(文件名：xzniudun.m)；
• 拟牛顿法_逆Broyden秩1法(nbroyden1.m)；
• 拟牛顿法_逆Broyden秩1第二方法(nbroyden2.m)；
• 拟牛顿法_BFS秩2法(BFS.m)；
• 相关说明参考这里