Numerical Analysis

Determination of roots of polynomials and transcendental equations by Newton Raphson, Secant and Bairstow's method.

Motivation:
Let us look at a set of problems of scientific and engineering interest to get a feel of what is root finding and why to find roots. Later we learn how to find them.

Problem 1:
Suppose you are asked to cut a rectangular sheet with one of its sides 1.25 mts longer than the other and the area being 0.875 mts$^{2}$ from a thin iron sheet of 5 mts$^{2}$ area. What will be length of the 'smallest side'.

Say, length of the smallest side = $x$
Length of the other side $=x+1.25$
Area of rectangle $=x(x+1.25)$
                         $=0.875$   mt$^{2}$
$\therefore \quad\quad x(x+1.25)=0.875$
i.e.          $x^{2}+1.25x-0.875=0$
i.e. say

$$f(x)=x^2+1.25x-0.875=0$$ (1)

So you need to solve a quadratic equation to find the required quantity. i.e. You have to find the roots of a quadratic equation.

You know that the roots of a quadratic equation $f(x)=ax^{2}+bx+c=0$ (2)
are given by

$$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\qquad(a\neq0)$$ (3)

Problem 2:
Concepts of thermodynamics are used extensively in their works by say aerospace, mechanical and chemical engineers. Here, the zero-pressure specific heat of dry air say $C_{p}$ KJ/(KgK)
is related to temperature 'T' by.'

$$C_{p}=0.99403+1.671\times10^{-4} T+9.7215\times10^{-8} T^{2} -9.5838\times10^{-11} T^{3}+1.9520\times10^{-14} T^{4}$$

Now, determine the temperature that corresponds to a specific heat of 1.2 KJ/(KgK). So, here you have to solve or find the roots of

$$f(T)=0.99403+1.671\times10^{-4}T+9.7215\times10^{-8}T^{2} -9.5838\times10^{-11}T^{3}+1.9520\times10^{-14}T^{4}$$

                 =1.2      i.e. find roots of

$$f(T)=1.9520\times10^{-14}T^{4}-9.5838\times10^{11}T^{3}+9.7215\times10^{-8}T^{2}+1.671\times10^{-4}T-0.20597=0$$ (4)

Problem 3:
The concentration of pollutant bacteria 'C' in a lake decreases as per the model:

$$C=70e^{-1.5t}\quad+\quad25e^{-0.075t}$$

Where 't' is the time variable. Determine the time required for the bacteria concentration to be reduced to 9.
Here, you have to find the roots of

$$f(t)=70e^{-1.5t}+25e^{-0.075t}-9=0$$ (5)

Problem 4:
The volume of liquid in a hollow horizontal cylinder of radius r and length L is related to the depth of the liquid h by

$$V=[r^{2}\quad \cos^{-1}\left(\frac{r-h}{r}\right)-(r-h)\sqrt{2rh-h^{2}}]L$$

Determine h given $r=2m,\quad l=5m^{3},\quad V=8m^{3}$
Here you have to find the roots of

$$f(h)=5\times[2^{2}Cos^{-1}\left(\frac{2-h}{2}\right)-(2-h)\sqrt{2\times2\times h-h^{2}}]-8=0$$ (6)

So you have seen that finding roots of $f(x)=0$ is very important in finding solution to several scientific and engineering problems.

The equation $f(x)=0$ may be a polynomial equation or a transcendental equation.

Polynomial Equations:
Polynomial equations in one independent variable 'x' are a simple class of algebraic equations that are represented as follows:

$$a_{n}x^{n}+a_{n-1}x^{n-1}+....+a_{1}x+a_{0}=0$$

The $n^{th}$ degree polynomial has $\tilde{n}$ roots. They roots may be real or complex.
Examples:
$f(x)=x^{2}+1.25x-0.875=0$
$f(x)=1.9520\times10^{-4}x^{4}-9.5838\times10^{-11}x^{3}+9.7215\times10^{-8}x^{2}+1.671\times10^{-4}x-0.20597=0$
$f(x)=x^{5}-3.5x^{4}+2.75x^{3}+2.125x^{2}-3.875x+1.25=0$

Transcendental Equation:
The equations include trigonometric or exponential or logarithmic functions.
Examples:
$f(x)=70e^{-1.5t}+25e^{-0.075t}-9=0$
$f(x)=5x[4\cos^{-1}\left(\frac{2-x}{2}\right)-(2-x)\sqrt{4h-h^{2}}]-8=0$
$f(x)=\log x^{2}-1=0$
You may note that the example are nonlinear functions.

Method of solution:
Some of the ways of finding the roots or solution of $f(x)=0$ are:

• Direct analytical methods
• graphical approach
• iterative methods etc.

Direct analytical methods: One may we able to find a mathematical expression for the solution
(root) of $f(x)=0$.
For example, for quadratic equations (2), we have solutions given by (3). However a large number of equations cannot be solved by direct analytical methods.

Graphical Method: This approach involves plotting the given function and determining the points where it crosses the x-axis. These points, extracted approximately from the plot, represent approximate values of the roots of the function.
Example:
Find the positive roots of $f(x)=3x-2\sin x-1=0$
Rewrite      $3x-2\sin x-1=0$ as
$\quad\qquad 3x-1=2 \sin x$
Now consider $y=3x-1$ and $y=2 \sin x$ and plot them

The x-co-ordinate of the point of intersection of $y=3x-1$, and $y=2 \sin x$ gives the required positive root of the given function. Clearly thus approach is cumbersome and time consuming.

Iterative Methods: Starting with a initial guess solution these methods generate a sequence of estimates to the solution which is expected converge to the true solution. They are grouped into two categories :

• Bracketing methods
• open methods

Bracketing Methods: These methods exploit the fact that a function typically changes sign in the vicinity of a root. They start with two initial guesses that bracket the root and then systematically reduce the width of the bracket until the solution to a desired accuracy is reached. The popular bracketing methods are: (a) Bisection Method, (b) False Position (or) Regula Falsi method, (c) Improved or modified Regula Falsi Method.
Open methods: These methods are based on formulas that require only a single starting (or guess) values of solution or two starting values that do not necessarily bracket the root. They may sometimes diverge or move away from true root as the computation progresses. However when the open methods converge they do so much more quickly than the Bracketing methods. Some of the popular open methods are: (a) Secant method, (b) Newton-Raphson method, (c) Bairstow's method (d) Muller's method etc.