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The Bisection Method For Root Finding Within Matlab 2020

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The bisection method in math is the key finding method that continually intersect the interval and then selects a sub interval where a root must lie in order to perform the more original process. This is a very simple and powerful method, but it is also relatively slow. Because of this, it is often used to roughly sum up a solution that is used as a starting point for a more rapid conversion method. The method is also known as intra-ablation method, binary search method or dichotomy method.

ACTUAL APPLICATION OF THIS METHOD:

The bisection method is an iterative algorithm used to find the roots of continuous functions. One of the main advantages of the method is that it is guaranteed to convert when the primary break is chosen correctly and is relatively easy to implement. The main difficulty, however, is that the expression is slower than other expressions.

You usually choose the method for complex situations that cause problems for other methods. For example, if your choices are bisection and Newton / Raphson, then the fission derivative is effective for some iterations equal to zero, since this condition fails Newton’s method to develop hybrid algorithms that are not uncommon for some iterations for Basic and other iterations. Using for approach.

Explain this method:

Bisection method is based on theorem, which states that if a function f(x) is continuous between f(a) and f(b) are of opposites signs. Then there exist at least one root between (a) and (b). For example, let f(a) be negative and f(b) be positive.

bisection method
bisection method curve

Then the root lies between (a) and (b) let its approximate value be given, x = (a+b)/2. If f(x) = 0, that is x is root of the equation f(x) = 0. The root lies either between x and b, or between x and depend on f(x) is negative or positive.

Let’s solve a common problem for a clearer understanding:

Example:

Find a real root of the equation, X^3-2x-5=0 by using bisection method.

Formula, a+b/2

Solution: 

Let, f(x) =

a = 3, y = 16 (positive)

b = 2, y= -1 (negative)  

Here, the root lies between 2 and 3;

X1 = (3+2)/2 = 2.5,                                 y = 5.6250

Since f(x1) = f (2.5) = 5.6250, the root lies between 2 and 2.5;

X2 = (2.5+2)/2=2.25,                              y = 1.8906

Since f(x2) = f (2.25) = 1.8906, the root lies between 2 and 2.25;    

X3 = (2.25+2)/2 =2.125,                         y = 0.3457

Since f(x3) = f (2.125) = 0.3457, the root lies between 2 and 2.125;

X4 = (2.125+2)/2 = 2.0625,                     y = -0.3513

Since f(x3) = f (2.0625) = -0.3513, the root lies between 2.0625 and 2.125;      

X5 = (2.125+2.0625)/2 = 2.09375,          y = -0.0089

Since f(x3) = f (2.09375) = -0.0089, the root lies between 2.09375 and 2.125;

X6 = (2.125+2.09375)/2 = 2.10938,              y = 0.1668

Since f(x3) = f (2.10938) = 0.1668, the root lies between 2.09375 and 2.10938;

X7 = (2.10938+2.09375)/2 = 2.10156,          y = 0.07856

Since f(x3) = f (2.10156) = 0.07856, the root lies between 2.09375 and 2.10156;

X8 = (2.10156+2.09375)/2 = 2.09766,          y = 0.03471

Since f(x3) = f (2.09766) = 0.03471, the root lies between 2.09375 and 2.09766;

X9 = (2.09766+2.09375)/2 = 2.09570,          y = 0.01266 

Since f(x9) = f (2.09570) = 0.01266, the root lies between 2.09375 and 2.09570;

X10 = (2.09570+2.09375)/2 = 2.09473,         y = 0.00195

The table is shown as:

B, y(+ve) A, y(-ve) X= (a+b)/2 Y=f(x)
3 2 2.5 5.6250
2.5 2 2.25 1.8906
2.25 2 2.125 0.3457
2.125 2 2.0625 -0.3513
2.125 2.0625 2.09375 -0.0089
2.125 2.09375 2.10938 0.1668
2.10938 2.09375 2.10156 0.07856
2.10156 2.09375 2.09766 0.03471
2.09766 2.09375 2.09570 0.01266
2.09570 2.09375 2.09473 0.00195

We will see how to write a matlab program for false positioning.

MATLAB PROGRAM:

Write a MATLAB program to solve the equation described in problem X^3-2x-5=0 by using bisection method 9 iteration.

Code:

bisection method
bisection method in matlab code

Output:

Here the values of p, q, m and y are shown:

Bisection method2
bisection method in matlab output

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