1.What Is the Symbolic Math Toolbox?
The Symbolic Math Toolbox is a functionality in Matlab for operations on symbolic objects. It introduces a special data type – symbolic objects;
This data type includes symbolic numbers, symbolic variables, symbolic expressions, and symbolic functions, as well as symbolic matrices and symbolic arrays composed of the above variables, which are also referred to as symbolic types;
The Symbolic Math Toolbox provides functionalities for solving, plotting, and manipulating symbolic math equations, which are referred to as symbolic computations.
2.Why Use Symbolic Computation?
Firstly, the conventional numeric data types (floating-point) in Matlab typically use decimal approximations for storage and computation. For example, 10/3=3.3333, due to hardware storage limitations, the result 3.3333 is not the true result of 3 and 1/3, but in general applications, this is not a significant issue;
When precision in calculations is particularly important, this issue becomes significant;
Thus, symbolic computation was born. It stores variables in a symbolic manner. For example, the above 10/3 uses symbolic computation to express the result as 10/3, which is indeed in fractional form;
3.Creating Symbolic Objects (Variables)
Symbolic variables can be directly used after declaration (creation) without output results, but they are stored in the workspace;
To create a symbolic variable, use the functions sym and syms
Example:
sym(‘x’) % Syntax 1
syms x % Syntax 2
syms x y z % Create multiple symbolic variables
sym(‘x’,[1 5]) % Create symbolic variables x1/x2/x3/x4/x5
4.Precision Differences Between Symbolic and Numeric Types
Example:
1/6 % Double-precision floating-point 1/6, expressed in Matlab as 0.1667
x=sym(1/6) % Create a symbolic variable, expressed in fractional form as 1/6
sin(pi) % Double-precision floating-point pi value calculation sin value, with rounding error 1.2246e-16
sin(sym(pi)) % Compute the sin value of the symbolic object pi, fully expressed as 0
5.Symbolic Expressions
Example:
syms x y z
f=2*x+1 % Symbolic expression
g=x*y+z
6.Symbolic Functions
The usage of symbolic functions is the same as that of conventional functions, with the difference being the data types handled. When performing operations with symbolic objects, one must first declare the symbolic objects and can express results in a formal manner.
Example:
syms x
f(x)=2*x+1 % Symbolic function
f(1) % Call the symbolic function, resulting in a symbolic object3
syms f(x,y) % Directly create (declare) a function without a function expression
f(x,y) % Call
f(1,2) % Result of the operationf(1,2)
7.Symbolic Matrices
The usage of symbolic matrices is the same as that of conventional matrices, with the difference being the data types handled. When performing operations with symbolic objects, one must first declare the symbolic objects and can express results in a formal manner.
Example:
syms a b c d % Declare symbolic variables
A=[a b;c d] % Symbolic matrix
B=[a 1;c 3]
sum(A) % Symbolic matrix operation, same as conventional matrix algorithms
sum(B)
sym(‘x’,[2 2]) % Create a 2 by 2 matrix, automatically adding subscripts
sym(‘x%d%d’,[2 2]) % Return result[x11, x12; x21, x22]
sym(‘x%d2018%d’,[2 2]) %2 Add numbers in between two subscripts
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