(F) [Matlab] numerical calculus and solving equations

【Matlab】

A numerical differentiation and numerical integration

Aa numerical differentiation

After calculating the difference can be calculated $f（x）$ difference quotient at some point, is calculated $f'(x)$approximation.
example:

Ab numerical integration

Examples:




Examples:



Examples:

example:

Solving linear equations B

Ba direct Act

example:

L and U are triangular matrix.


example:

Bb iterative method

jacobi.m

function [y,n]=jacobi(A,b,x0,ep)
	D=diag(diag(A)); % 对角阵
	L=-tril(A,1);
	U=-triu(A,1);
	B=D\(L+U);
	f=D\b;
	y=B*x0+f;
	n=1;
	while norm(y-x0)>=ep
		x0=y;
		y=B*x0+f;
		n=n+1;	
	end

gauseidel.m

function [y,n]=jacobi(A,b,x0,ep)
 	D=diag(diag(A)); % 对角阵
	L=-tril(A,-1);
 	U=-triu(A,1);
 	B=(D-L)\U;
 	f=(D-L)\b;
 	y=B*x0+f;
 	n=1;
 	while norm(y-x0)>=ep
	  	x0=y;
 	 	y=B*x0+f;
  		n=n+1; 
 	end

Example:

Sometimes Gauss - Seidel iteration for solving linear equations method may not converge.

C and the nonlinear equation calculation function extremum

Numerical solving nonlinear equation Ca

example:

example:

Calculating Cb function extremum

Matlab only consider the problem of computing the minimum value, if required $f(x)$ the maximum difference can be $-f(x)$ minimum.

fminbnd: a univariate function
fminsearch: simplex method. Multivariate
fminunc: Quasi-Newton method. Multivariate
example:

AX <= b (linear inequality constraints)
AeqX = BEQ (linear equality constraints)
G (X) <= 0 (linear inequality constraints)
the Ceq (X-) = 0 (linear equality constraints)
Lbnd <= X-<= Ubub (variable constraints)
examples:

D numerical solution of ordinary differential equations

The general concept of the numerical solution of ordinary differential equations Da

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Numerical solution of ordinary differential equations function Db

Examples:

Examples:

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Dc stiff problems

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Source:
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