American University of Armenia
Intensive Computer Program
Matlab
LESSON 6
LESSON 7
TOPIC 9. Polinomials.
A polynomial of order n is an algebraic expression of type p(x) = c0 + c1x + c2x2 + … +
cnx . In order to specify the polynomial, it is necessary and sufficient to specify the set of its
coefficients. In Matlab it is natural to keep them in a vector, and namely, in a row vector with
ascending or descending order of coefficients. There are predefined functions working with
arrays as polynomials.
Polynomial predefined function summary:
n
Function
Description
conv
deconv
poly
polyder
polyval
polyvalm
roots
Multiply polynomials.
Divide polynomials.
Polynomial with specified roots.
Polynomial derivative.
Polynomial evaluation.
Matrix polynomial evaluation.
Find polynomial roots.
Our immediate tasks, however, will be to implement our own versions of these functions
and add some new.
In this section polynomials will be represented by the arrays of their coefficients, where
the first element is the constant, the second one is the coefficient of the linear member and the
last element is the coefficient of the highest order. For example, p(x) = x2 – 4 is to be
represented as [-4 0 1], and p(x) = 7 x5 as [0 0 0 0 0 7]. It is obvious that the length of the
array of coefficients exceeds by 1 the order of the polynomial.
Task 1. Write a function that, given a polynomial, returns the latter’s value at the
specified point.
1. Start a new M-function as follows:
function result = polyval(coeffs, x)
2. In order to compute the value of the polynomial, x0 must be raised into powers from 0 to
the order of the polynomial. So, the range of the required exponents will be:
powers = x .^ (0 : length(coeffs) - 1);
3. The last touch of the brush:
result = powers * coeffs(:);
ATTENTION: length (X) returns the length of vector X.
American University of Armenia
Intensive Computer Program
Matlab
LESSON 6
After raising x0 into powers on element- by-element basis, the scalar product of the produced
row with the transposed coeffs is computed. This statement will correctly run only if the
argument coeffs is a row-vector, which is usually the case. An alternative and more general
statement, working for both row-vector and column vector types of coeffs, sums up elementby-element products:
result = sum( powers .* coeffs);
4. Save the function in C:\Matlab\Pol folder under polyval.m name. In order to create a new
folder, press “Create New Folder” button in the “Save as” window, type in the name, press
“Enter” and click on the “Open” button.
Task 2. Write a function that returns the sum of two specified polynomials.
1. Start a new M-function:
function result = polysum (this, that)
The polynomials to be summed up are formally called this and that respectively.
2. Coefficients in the resulting polynomial are sums of the respective pairs, if the arguments
are of equal lengths. Otherwise, extra zeros must be attached at the end of both vectors to get
the same length:
same_length_sum = [this, zeros(1, length(that))] + [that, zeros(1, length(this))];
3. In the obtained vector same_length_sum extract the indexes of non zero elements:
non_zeros = find(same_length_sum);
ATTENTION: I = find(vector) returns the indices of the vector that are non-zero.
I = find(A > 100) returns the indices of the elements of A that are greater than 100.
[I , J] = find (X) returns the row and column indices of the nonzero entries in the matrix X.
4. Find the maximum index of the last non zero element:
last_non_zero = max(non_zeros);
5. And finally, vector same_length_sum is assigned to the result with missing zeros:
result = same_length_sum(1 : last_non_zero);
American University of Armenia
Intensive Computer Program
Matlab
LESSON 6
Task 3. Write a function that computes the first derivative of a polinomial.
1. Start a new M-function:
function result = polyder (coeffs)
2. The derivative of the constant is 0, therefore the first element of the vector coeffs is cut out
by assigning it empty array:
coeffs(1) = [];
3. Numbers from the range 1 : length(coeffs) now are multiplied element – by – element by
the coeffs:
result = (1 : length(coeffs)) .* coeffs;
Task 4. Create a function that computes the nth derivative of the polinomial.
1. Start a new M-function:
function result = polydern (coeffs, n)
2. If the amount of arguments the function was called with is equal to 1 or order of the
derivative is equal to 1 we just compute the first derivative calling polyder function.
if nargin == 1 | n == 1
polyder(coeffs);
3. Otherwise, polydern function is called recursively to compute the (n - 1)th derivative of the
polynomial.
else
result = polydern(polydern(coeffs), n - 1);
end
Task 5. Write a function that computes the integral of the polinomial.
1. Start a new M-function:
function result = polygral (coeffs, x0, x)
2. If the amount of arguments the function was called with is equal to 1, then an indefinite
integral is computed. Recall that the integral from a polynomial is another polynomial of
higher by 1 degree with coefficients equal to the old coefficients divided by the powers: c0x +
(c2 / 2) x2 + … + (cn / n)xn.
if nargin == 1
result = [0 coeffs ./ (1 : length(coeffs))];
American University of Armenia
Intensive Computer Program
Matlab
LESSON 6
Adding 0 to the vector increases its length and, hence the polynomial degree, by 1.
Meantime, the rest of the coefficients are divided element-by-element by the range from 1 to
the old length.
3. Otherwise, it is assumed that the definite integral is computed. The definite integral is the
difference between the indefinite integral taken at specific points x and x0.
else
result = polyval(polygral(coeffs), x) - polyval(polygral(coeffs), x0);
end