Root Finding

Section 9.4 Root Finding

The roots or zeros of a function, including polynomials, are the values of the argument at which the function evaluates to zero.

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The Fundamental Theorem of Algebra states that an nth degree polynomial has n roots:

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These roots are not necessarily distinct (some roots occur with higher multiplicity)

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Not all roots are necessarily real (they can also be complex)

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The root finding problem is one of the oldest in numerical approximation dating back to 1700 BC, yet it is still a topic of research.

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There are many techniques and algorithms used to find (approximate) roots or zeros of functions: bisection, Newton’s Method, secant method, Müller’s Method.

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In MATLAB, we can easily find the roots of a polynomial using the roots function:

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r = roots(p)

Here, \(p\) is a row vector of polynomial coefficients. The result \(r\) is a column vector of the roots of the polynomial.

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Example:

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$ p = [1 -12.1 40.59 -17.015 -71.95  35.88];

$ r = roots(p)
r =
     6.5000      % n roots of an nth order polynomial
     4.0000
     2.3000
    -1.2000
     0.5000

Activity 9.4.

Find the zeros (roots) of the polynomials

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\begin{equation*} f(x) = x^3 - x^2 - 2x \end{equation*}

\begin{equation*} g(x) = x^4 - x^3 - 2x^2 \end{equation*}

Please put your MATLAB code into the textbook.

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Subsection 9.4.1 Degenerate Roots

Consider the polynomial \(p(x) = x^2\text{.}\) This polynomial only has one x-value, at which it takes the value 0, namely x = 0. Since the graph of p does not cross the x-axis at x=0, this root is actually of multiplicity greater than 1 and is called a degenerate root. MATLAB figures this out and lists such roots multiple times in the vector of roots (according to multiplicity). Take a look:

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$ p = [1, 0, 0]
p =
    1  0  0

$ r = roots(p)
r =
    0
    0

Activity 9.5.

Find a polynomial \(h\) that has zeros at 0, 1 and 2.

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Please express the polynomial as a MATLAB polynomial and put your vector into the textbox.

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How did you accomplish this? Unless you have magic powers, most likely you multiplied together x, (x-1) and (x-2).

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Note that the roots of a polynomial by themselves actually define the polynomial, up to multiplication by a scalar.

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If the roots of a polynomial are known, MATLAB can actually construct the polynomial, that is, the coefficients of the polynomial, assuming that the leading coefficient equal 1. Here is an example: Suppose we wanted to find a polynomial with roots 6.5, 4, 2.3, 0.5 and -1.2. We know that the polynomial is given by

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\begin{equation*} f(x) = (x - 6.5)(x - 4)(x - 2.3)(x + 1.2)(x - 0.5) \end{equation*}

But in order to find the polynomial coefficients we’d actually have to multiply through. MATLAB does this for us:

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$ r = [6.5; 4; 2.3; 0.5; -1.2]
r =
     6.5
     4.0
     2.3
     0.5
    -1.2

Note that this is a column vector of the roots. The command poly finds the polynomial for us:

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$ p = poly(r)
p =
    1.00  -12.10  40.59  -17.02  -71.95  35.88

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