Polynomials: Roots, Evaluation, and Interpolation in MATLAB

Polynomials are fundamental building blocks in mathematics and engineering, used to model a vast array of phenomena. In MATLAB, you can efficiently work with polynomials for tasks like finding roots, evaluating polynomial values, and performing interpolation.

Representing Polynomials in MATLAB

In MATLAB, a polynomial is represented by a row vector of its coefficients, ordered from the highest power of the variable to the lowest. For example, the polynomial $p(x) = 3x^3 - 2x^2 + 5$ is represented as

code

[3 -2 0 5]

How would you represent the polynomial

p(x) = 5x^4 + 2x - 7

in MATLAB?

As the row vector [5 0 0 2 -7].

Evaluating Polynomials

The

code

polyval

function is used to evaluate a polynomial at specific points. It takes the coefficient vector and the point(s) at which to evaluate as input.

The polyval(p, x) function evaluates the polynomial whose coefficients are in vector p at the value x. If x is a vector or matrix, polyval returns a vector or matrix of the same size containing the evaluated polynomial values. This is highly efficient for evaluating a polynomial at multiple points simultaneously, leveraging MATLAB's vectorized computation capabilities.

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For instance, to evaluate $p(x) = 3x^3 - 2x^2 + 5$ at $x=2$ , you would use

code

polyval([3 -2 0 5], 2)

What is the output of polyval([1 -3 2], 3)?

The polynomial is $x^2 - 3x + 2$ . Evaluating at $x=3$ gives $3^2 - 3(3) + 2 = 9 - 9 + 2 = 2$ .

Finding Polynomial Roots

The

code

roots

function finds the roots of a polynomial. It takes the coefficient vector as input and returns a column vector containing the roots.

For example, to find the roots of $x^2 - 3x + 2 = 0$ , you would use

code

roots([1 -3 2])

. The output would be

code

[2; 1]

The roots function uses numerical methods to find the roots, which may include complex numbers.

What command would you use in MATLAB to find the roots of

x^3 - 6x^2 + 11x - 6

roots([1 -6 11 -6]).

Polynomial Interpolation

Polynomial interpolation involves finding a polynomial that passes through a given set of data points. MATLAB provides functions for this purpose, such as

code

polyfit

and

code

polyval

The

code

polyfit(x, y, n)

function fits a polynomial of degree

code

to the data points

code

(x, y)

. It returns the coefficients of the fitted polynomial.

Once you have the coefficients from

code

polyfit

, you can use

code

polyval

to evaluate the interpolated polynomial at new points.

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What is the primary function used to find the coefficients of a polynomial that fits a set of data points in MATLAB?

polyfit.

Practical Applications

Understanding and manipulating polynomials is crucial for various engineering and scientific tasks, including curve fitting, signal processing, control systems design, and solving differential equations numerically.

Learning Resources

MATLAB Documentation: Polynomials(documentation)

Official MATLAB documentation covering polynomial representation, evaluation, roots, and fitting.

MATLAB Tutorial: Working with Polynomials(tutorial)

A guided tutorial from MathWorks on how to use MATLAB's polynomial functions.

Finding Roots of Polynomials in MATLAB(video)

A video demonstration of how to use the `roots` function in MATLAB.

Polynomial Interpolation in MATLAB using polyfit(video)

Explains how to use `polyfit` for polynomial interpolation with examples.

MATLAB `polyval` Function Explained(blog)

A clear explanation of the `polyval` function and its usage.

MATLAB `polyfit` Function Explained(blog)

Details on how to use the `polyfit` function for curve fitting.

Numerical Methods for Engineers - Polynomials(documentation)

Covers polynomial evaluation and root finding from a numerical methods perspective.

Introduction to Polynomial Interpolation(wikipedia)

An overview of polynomial interpolation concepts, useful for understanding the underlying theory.

Solving Equations and Data Fitting(documentation)

Discusses polynomial interpolation and fitting methods in the context of numerical analysis.

MATLAB Central: File Exchange - Polynomial Tools(documentation)

A repository of user-contributed MATLAB code, including various tools for polynomial manipulation.

Polynomials: Roots, Evaluation, Interpolation

Polynomials: Roots, Evaluation, and Interpolation in MATLAB

Representing Polynomials in MATLAB

Evaluating Polynomials

Finding Polynomial Roots

Polynomial Interpolation

Practical Applications

Learning Resources