Multiplicity Matlab Solve Only Giving Solution Once

Solving Equations in MATLAB: Why You Only Get One Solution and How to Find More

MATLAB's solve function, while powerful for symbolic computation, often presents a challenge when dealing with equations that have multiple solutions. This article explains why you might only get one solution from solve and provides techniques to uncover all possible solutions to your equations. Understanding this behavior is crucial for accurate mathematical modeling and problem-solving in various engineering and scientific disciplines.

Many equations, especially those involving polynomials or trigonometric functions, possess multiple roots or solutions. However, MATLAB's solve function, by default, returns only one solution, often the simplest or principal solution. This isn't a bug; it's a consequence of the algorithm's design and the inherent complexity of finding all solutions symbolically. The function prioritizes finding a solution rather than exhaustively finding all solutions.

Why solve Might Only Give One Solution

The solve function uses symbolic manipulation techniques to find solutions. These techniques are not always exhaustive, especially for complex equations. Here are the key reasons why you might only see one solution:

• Implicit Assumptions: solve often makes implicit assumptions about the solution domain (e.g., real numbers vs. complex numbers). If you're looking for complex roots and the function only returns real roots, it's because it hasn't explored the complex plane.

• Algorithmic Limitations: The algorithms used by solve are designed for efficiency. Finding all solutions can be computationally expensive, especially for high-degree polynomials or transcendental equations. The algorithm may terminate after finding a single solution.

• Branch Cuts: When dealing with multi-valued functions like square roots or logarithms, the choice of branch cut influences the solution obtained. solve may choose a specific branch cut, leading to only one solution being presented.

Techniques to Find All Solutions

To overcome the limitations of solve and obtain all possible solutions, employ these strategies:

• Specify Solution Domains: For example, if you're searching for complex solutions, explicitly state that in your solve command. Using solve(equation, variable, 'Real', true) explicitly limits solutions to the real domain. Leaving it out or setting it to false allows for complex solutions.

• Utilize Numerical Methods: For equations that are difficult to solve symbolically, numerical methods such as fzero or fsolve are powerful alternatives. These functions find numerical approximations to the roots and can be more effective in locating multiple solutions. You'll often need to provide initial guesses close to each solution to ensure they are found.

• Graphical Analysis: Plotting the function can provide valuable insights into the number and approximate locations of solutions. This visual inspection can guide the use of numerical methods.

• Equation Manipulation: Sometimes, strategically manipulating the original equation can help uncover more solutions. This might involve factoring, using trigonometric identities, or making substitutions.

• Symbolic Toolbox Capabilities: Explore advanced features of MATLAB's Symbolic Math Toolbox, such as examining the output of solve more carefully (looking for implicit assumptions), or using more sophisticated symbolic manipulation functions.

Example: Finding All Roots of a Polynomial

Let's consider a simple example: finding the roots of the quadratic equation x² - 5x + 6 = 0.

A naive approach using solve:

syms x;
sol = solve(x^2 - 5*x + 6 == 0, x);
disp(sol);

This might only return one solution (e.g., x = 2). However, we know there are two solutions (x=2 and x=3).

Using factorization or the quadratic formula directly will reveal both solutions explicitly. Numerical methods are less necessary for this simple case but would be important for polynomials with no easy factorization.

In conclusion, while MATLAB's solve function is a convenient tool, it's essential to understand its limitations. By combining symbolic methods with numerical techniques and a good understanding of the underlying mathematical principles, you can effectively find all solutions to your equations, leading to more complete and accurate results. Remember to always check your solutions and consider the context of your problem to ensure their validity.