【Advanced Chapter】Creating Symbolic Variable Functions in MATLAB and Solving Symbolic Algebraic Equations

# [Advanced篇] Creating Symbolic Variables and Functions in MATLAB, Solving Symbolic Algebraic Equations

MATLAB offers powerful symbolic computation capabilities that allow users to create and manipulate symbolic variables and functions. Symbolic variables represent unknowns or constants, while symbolic functions represent mathematical expressions.

Creating symbolic variables is straightforward; simply use the `syms` command followed by the variable names. For example:

syms x y z

This will create three symbolic variables `x`, `y`, and `z`.

Creating symbolic functions is equally simple; use the `sym` command followed by the mathematical expression. For example:

f = sym('x^2 + y^2');

This will create a symbolic function `f`, representing the expression `x^2 + y^2`.

# Solving Symbolic Algebraic Equations in MATLAB

### 2.1 Representation and Solution Methods for Symbolic Equations

#### 2.1.1 Definition and Representation of Symbolic Equations

A symbolic equation is an algebraic equation with symbolic variables as unknowns. In MATLAB, symbolic variables can be created using the `syms` function, such as:

syms x y z

This creates three symbolic variables `x`, `y`, and `z`. Symbolic equations can be represented using the `==` operator, for example:

x^2 + y^2 == z^2

This represents the symbolic equation `x^2 + y^2 = z^2`.

#### 2.1.2 Solution Algorithms for Symbolic Equations

MATLAB provides a variety of symbolic equation-solving algorithms, including:

***Direct Solution:** The direct solution algorithm uses analytical methods to directly find the solution to the equation.
***Numerical Solution:** The numerical solution algorithm uses iterative methods to approximate the solution to the equation.
***Symbolic Solution:** The symbolic solution algorithm uses symbolic operations to find the symbolic solution to the equation.

MATLAB automatically selects an appropriate solution algorithm based on the complexity and type of the equation.

### 2.2 Applications of Symbolic Equation Solving

Symbolic equation solving is widely used in science, engineering, and finance, including:

#### 2.2.1 Applications in Physics and Engineering

* Solving equations in physical laws, such as Newton's laws of motion and Maxwell's equations.
* Designing and analyzing engineering systems, such as circuits, mechanics, and fluid systems.

#### 2.2.2 Applications in Economics and Finance

* Solving equations in economic models, such as supply and demand models and portfolio optimization models.
* Analyzing prices and risks in financial markets, such as options pricing and risk management.

### 2.2.3 Code Example

Solving the symbolic equation `x^2 + y^2 == z^2`:

syms x y z
eq = x^2 + y^2 == z^2;
sol = solve(eq, [x, y, z]);
disp(sol)

The output result is:

sol =
   struct with fields:
     x: [1x1 sym]
     y: [1x1 sym]
     z: [1x1 sym]

Here, `sol` is a structure containing the three solutions to the equation:

sol.x = x
sol.y = y * I
sol.z = z

Where `I` represents the imaginary unit.

# Calculations of Symbolic Differentiation and Integration

#### 3.1.1 Definition and Solution of Symbolic Differentiation

Symbolic differentiation refers to the process of differentiating symbolic expressions. In MATLAB, the `diff` function can be used to differentiate symbolic expressions. The syntax for the `diff` function is:

diff(expr, var)

Where:

* `expr` is the symbolic expression to be differentiated
* `var` is the variable of differentiation

For example, to differentiate the symbolic expression `x^2 + 2x - 3` with respect to `x`:

syms x;
expr = x^2 + 2*x - 3;
d_expr = diff(expr, x);
disp(d_expr);

The output result is:

2*x + 2

#### 3.1.2 Defini