MATLAB

MATLAB Symbolic Math

This chapter introduces Symbolic Math in MATLAB, a technique for performing mathematical computations symbolically rather than numerically. The Symbolic Math Toolbox in MATLAB provides tools for algebraic computation, symbolic differentiation, integration, and equation solving, making it ideal for precise analytical tasks in engineering, mathematics, and science.

Chapter Goal

The goal of this chapter is to provide a comprehensive understanding of MATLAB’s symbolic math capabilities, focusing on:

  1. Learning how to define symbolic variables and expressions.
  2. Exploring operations such as simplification, differentiation, and integration.
  3. Understanding how to solve equations symbolically.
  4. Implementing best practices for combining symbolic and numeric computations.

Key Characteristics for MATLAB Symbolic Math

  1. Precise Computation: Perform calculations exactly, without numerical approximation errors.
  2. Flexible Expression Handling: Manipulate mathematical expressions symbolically.
  3. Wide Range of Functions: Includes tools for calculus, algebra, transforms, and equation solving.
  4. Integration with Numeric MATLAB: Convert symbolic results to numeric values when needed.
  5. Visualization Support: Graph symbolic functions and equations directly.

Basic Rules for MATLAB Symbolic Math

  1. Define Symbolic Variables: Use syms to create symbolic variables.
  2. Perform Symbolic Operations: Use functions like simplify, diff, int, and solve.
  3. Use Assumptions: Apply constraints to symbolic variables using assume.
  4. Combine Symbolic and Numeric: Convert symbolic expressions to numeric with double or vpa.
  5. Validate Results: Compare symbolic results with numerical methods for accuracy.

Best Practices

  1. Simplify Results: Use simplify or expand to make symbolic results more interpretable.
  2. Leverage Plotting: Visualize symbolic functions with fplot or ezplot for better insights.
  3. Optimize Computations: Use assumptions to reduce complexity and improve solver performance.
  4. Document Workflows: Clearly label symbolic computations for readability and reproducibility.
  5. Check Compatibility: Ensure symbolic expressions are compatible with subsequent numeric operations.

This chapter equips you with the knowledge to perform precise analytical computations using MATLAB Symbolic Math Toolbox, enhancing your ability to solve complex mathematical problems. Subsequent sections will delve into syntax, examples, and real-world applications.

Syntax Table

Serial No Component Syntax Example Description
1 Define Symbolic Variable syms x Creates a symbolic variable x.
2 Simplify Expression simplify(x^2 – 2*x + 1) Simplifies the symbolic expression to its simplest form.
3 Solve Equation solve(x^2 – 4 == 0, x) Solves the equation x^2 – 4 = 0 for x.
4 Differentiate Symbolically diff(x^3, x) Computes the symbolic derivative of x^3 with respect to x.
5 Integrate Symbolically int(x^2, x) Computes the symbolic integral of x^2 with respect to x.
6 Convert to Numeric double(sin(pi/2)) Converts the symbolic result sin(pi/2) to a numeric value.
7 Visualize Symbolic Function fplot(x^2, [0, 5]) Plots the symbolic function x^2 over the interval [0, 5].

Syntax Explanation

1. Define Symbolic Variable

What is Defining a Symbolic Variable?

Defining symbolic variables allows you to perform algebraic and calculus operations symbolically.

Syntax:

syms x;

Detailed Explanation:

  • Use the syms command to declare variables as symbolic.
  • Symbolic variables represent unknowns or parameters in mathematical expressions.

Example:

syms x;

expr = x^2 + 2*x + 1;

disp(expr);

Output:

x^2 + 2*x + 1

2. Simplify Expression

What is Simplifying an Expression?

Simplifying reduces a symbolic expression to its simplest form.

Syntax:

simplify(expression);

Detailed Explanation:

  • The simplify function reduces redundancy in expressions.
  • Useful for making results more interpretable.

Example:

expr = (x + 1)^2 – x^2 – 2*x – 1;

simplifiedExpr = simplify(expr);

disp(simplifiedExpr);

Output:

0

3. Solve Equation

What is Solving an Equation?

Solving finds the values of variables that satisfy an equation.

Syntax:

solve(equation, variable);

Detailed Explanation:

  • The solve function computes solutions to algebraic equations.
  • Specify the equation and the variable to solve for.

Example:

solution = solve(x^2 – 4 == 0, x);

disp(solution);

Output:

 2

 -2

4. Differentiate Symbolically

What is Symbolic Differentiation?

Differentiation computes the derivative of a symbolic expression.

Syntax:

diff(expression, variable);

Detailed Explanation:

  • Use the diff function to find the derivative of a function with respect to a variable.

Example:

derivative = diff(x^3, x);

disp(derivative);

Output:

3*x^2

5. Integrate Symbolically

What is Symbolic Integration?

Integration computes the indefinite or definite integral of a symbolic expression.

Syntax:

int(expression, variable);

Detailed Explanation:

  • Use the int function for symbolic integration.
  • Specify bounds for definite integrals.

Example:

integral = int(x^2, x);

disp(integral);

Output:

x^3/3

6. Convert to Numeric

What is Converting to Numeric?

Converting symbolic results to numeric values allows for numerical computations.

Syntax:

double(expression);

Detailed Explanation:

  • Use the double function to evaluate symbolic expressions as double-precision numbers.

Example:

numericResult = double(sin(pi/2));

disp(numericResult);

Output:

1

7. Visualize Symbolic Function

What is Visualizing a Symbolic Function?

Visualizing plots the graph of a symbolic expression over a specified interval.

Syntax:

fplot(expression, [xmin, xmax]);

Detailed Explanation:

  • Use fplot for high-quality plots of symbolic functions.
  • Specify the range [xmin, xmax] for the plot.

Example:

fplot(x^2, [0, 5]);

title(‘Plot of x^2’);

xlabel(‘x’);

ylabel(‘y’);

Output:

A plot of the function from to .

Notes

  • Symbolic computations are exact but can be computationally intensive for large expressions.
  • Use symbolic math to derive formulas or validate numeric results.

Warnings

  • Avoid using symbolic math for large-scale numerical problems as it may cause performance issues.
  • Always simplify results before interpreting complex symbolic expressions.

Real-Life Project: Solving and Visualizing a Spring-Mass System

Project Name: Analyzing the Motion of a Spring-Mass System

Project Goal:

To use MATLAB’s symbolic math capabilities to model and solve the differential equation of a spring-mass system and visualize its motion.

Steps in the Project:

  1. Define the Differential Equation:
    • Write the equation of motion for a spring-mass system.
  2. Solve the Equation Symbolically:
    • Use MATLAB to find the solution for displacement as a function of time.
  3. Visualize the Solution:
    • Plot the motion of the spring-mass system over time.

Code for This Project:

% Step 1: Define symbolic variables

syms t x(t) m k;

 

% Define constants

m = 1; % mass in kg

k = 10; % spring constant in N/m

 

% Define the differential equation

ode = diff(x, t, 2) + (k/m)*x == 0;

 

% Step 2: Solve the differential equation

sol = dsolve(ode, x(0) == 1, diff(x, t)(0) == 0);

 

% Simplify the solution

displacement = simplify(sol);

 

% Step 3: Visualize the solution

fplot(displacement, [0, 10]);

title(‘Spring-Mass System Motion’);

xlabel(‘Time (t)’);

ylabel(‘Displacement (x)’);

grid on;

Save and Run:

  1. Save the script as spring_mass_system.m.
  2. Run the script in MATLAB.

Expected Output:

  1. Console Output:
    • A symbolic expression for the displacement as a function of time.
  2. Plot:
    • A graph showing the oscillatory motion of the spring-mass system.

Learning Outcomes:

  • Understand how to model physical systems using differential equations.
  • Learn to solve second-order ODEs symbolically in MATLAB.
  • Gain insights into visualizing dynamic system behaviors.