Symbolic operations

Symbolic operation is also called computer algebra , Generally speaking, it is to use computers to deduce mathematical formulas , Such as factoring the expression 、 Simplification 、 differential 、 integral 、 Solve algebraic equations 、 Solve ordinary differential equations, etc . Compared with numerical operation , Symbolic computation has the following characteristics ：（1） The operation is carried out by reasoning , Therefore, it is not affected by truncation error and cumulative error ;（2） The speed of symbol calculation is relatively slow .

1.1 Creation of symbol objects

stay MATLAB in , Provides two functions for creating symbol objects , namely sym and syms function .

1.sym
sym Function is used to establish a single symbolic quantity , The call format of the function is ：

var=sym('var') % Create a symbolic variable var.
symexpr=sym(h) % Create a symbolic expression through the function handle of an anonymous function .
A=sym('a', [m,n]) % Create a m×n  Symbolic variable of matrix .

2.syms
sym Functions can only define one symbolic variable at a time , Inconvenient to use . MATLAB Another function is provided syms, Multiple symbolic variables can be defined at a time .

clc, clear
syms a % Define symbolic variables a
syms x y integer % Define integer symbolic variables x  and y
syms s(t) f(x,y) % Define two symbolic functions 

1.2 Symbolic solutions of algebraic equations

stay MATLAB in , Provides solve Function to solve symbolic algebraic equations or systems of equations , The call format is as follows ：
$S=solve(eqn,var)$% Find the expression eqn Algebraic equation of , The solution variable is var.
$S=solve(eqn,var,Name,Value)$% Find the expression eqn Algebraic equation of , The solution variable is var, It specifies
One or more attribute value names and their corresponding attribute values .
$[y1,\dots ,yN]$=solve(eqns,vars) % Find the expression eqns Algebraic equations of , The solution variable group is vars.

Numerical solutions of algebraic equations

Numerical solutions of linear equations

System of linear equations $Ax=b$ The solution command of is $x=pinv(A)\ast b$, When A When the rank is full , You can use commands KaTeX parse error: Undefined control sequence: \b at position 4: x=A\̲b̲.
（1） Mathematically unique solution ,MATLAB The unique solution is also given ;
（2） No solution in Mathematics ,MATLAB Give the least square solution ;
（3） Mathematically, there are infinite solutions ,MATLAB Give the minimum norm solution .

Numerical solutions of nonlinear equations

1. Find all the root commands of polynomials roots
   roots The call format of is
   r=roots§ % Find all roots of a polynomial , here p It's a vector , Is a polynomial from a high power coefficient to a constant
   The vector of terms , Return value r Is all roots of a polynomial .
2. Find the zero point of nonlinear function or the solution of nonlinear equation
   fzero function
   fsolve function