polynomial_operations.pdf

Polynomials:

Representation, Evaluation, Operations

MATH2070: Numerical Methods in Scientific Computing I

Location: http://people.sc.fsu.edu/∼jburkardt/classes/math2070 2019/polynomial operations/polynomial operations.pdf

Evaluating three representations of the same polynomial.

Polynomial Computations

• What choices are there for representing a polynomial?

• What is an efficient evaluation procedure?

• How do we add, multiply, divide, differentiate, and integrate?

the coefficient c

i

matches the corresponding power x

i

We assume that c

MATLAB does not allow the use of 0 as an array index. The simplest way to deal with this issue is to use

a “1-based” power form, in which c

multiplies x

Thus, the degree-3 polynomial

could be stored as a MATLAB array of 4 entries:

If we want to print out a polynomial, we could just print the vector of coefficients c. It is much more

intelligible to try to reproduce the full mathematical form, if possible. Here is a very basic example that

suggest how we might do this:

Listing 1: Basic poly print.m code.

While we could work harder to make nicer output, here’s what happens if we call the function to print out

3x

5

− 2x

3

+ x

2

− 5x − 17:

Listing 3: Using the fancier r8poly print().

which prints the polynomial in the more traditional form in which the highest power is shown first. It also

omits powers of x with zero coefficient.

Listing 4: Polynomial addition.

Why couldn’t we simply say the following?

Listing 5: Explain why this is a bad idea.

Mathematically, if p

can be computed by

So computationally, multiplying two polynomials means computing every possible product c1(i) ∗ c2(j) and

adding that to c3(i + j). But we have to remember to increment our coefficient indices by 1. The degree of

the product will be the sum of the degrees of the factors: