Let's consider the following equation:

*eˣ = 2x² + 1*We may write the above expression as,

*f(x) = eˣ - 2x² - 1*To find the value of

*or roots of the equation, we may apply the Secant method. In this method, the derivative may be approximated by a backward finite divided difference approach. So, the function may be expressed as,***x**This method is convenient when evaluating derivative for some function is difficult in Newton-Raphson method. Now, the above equation may be substituted in Newton-Raphson’s formula to get the following algorithm.

In contrast to Newton-Raphson method, two initial assumptions are made for the two unknowns’

**and***xi-1***.***xi*For the proper initial guess for x, graphical method is applied to visualize the characteristics of the function. The MATLAB commands generate the plot of the function

**. This function is plotted with respect to the values of***f(x)***from***x**-10*to*+10*. The following figure shows the function plot. From the plot, the function changes sign when**is between***x**2*to*3*. So, the root lies in this interval. And, in the program developed for Secant method, two initial guesses for**and***xi-1**are***xi***2*and*3*respectively.However, more close observation reveals two roots, which are shown in the following figure. In this time, the range of

*is decreased significantly for precise investigation between***x***-2*to*+2*as this interval seems unclear in the previous figure for the presence of any roots.So, the above figure shows the existence of other two roots, which lie in between

*0.1*to*1*and at the origin. The origin is obvious for this case, and there is no need for any further estimation by numerical methods. But, for the interval*0.1*to*1*, the accurate and precise approximation is necessary. For this, the two initial guesses are set to*0.1*and*1*for*and***xi-1***accordingly.***xi**The following Secant formula is implemented to approximate the two roots lie in the intervals

*2*to*3*and*0.1*to*1*. The results are shown after the program.**Secant Formula Implementation by a MATLAB Program**

**% Definition of a function "secant" to solve equation by Secant Method**

**function [x,ea] = secant(X,X0,etol)**

**format long;**

**% Input: X,X0, etol=Tolerance definition for error**

**% Output: x, ea=Calculated error in loop**

**% Iterative Calculations**

**while (1)**

**% Implementation of Secant Algorithm**

**solution = X-((exp(X)-(2*X^2)-1)*(X0-X))/((exp(X0)-(2*X0^2)-1)-(exp(X)-(2*X^2)-1));**

**solutionprevious=X;**

**X0=X;**

**X=solution;**

**if (solution-solutionprevious)~=0 % Approximate percent relative error**

**ea=abs((solution - solutionprevious)/solution)*100;**

**end**

**if ea<=etol,**

**break,**

**end**

**x = solution;**

**disp(x);**

**end**

**% Display of output parameters**

**disp(x);**

**disp(ea);**

**%disp(solution-(0.1*abs(ea)*solution));**

**end**

**Outputs:**

**>> secant (3, 2, 1e-5)**

**% Iterations**

**2.597424571529533**

**2.796706966294471**

**2.858723109017894**

**2.841817367110900**

**2.842657964680533**

**2.842673428005184**

**2.842673428005184**

**>> secant(1, 0.1, 1e-5)**

**% Iterations**

**0.308929151717719**

**0.570046187158243**

**1.157486960694727**

**0.682985699401547**

**0.723807551583747**

**0.742065590882227**

**0.740827136083267**

**0.740850398734311**

**0.740850398734311**

So, the solutions for the equation are

*2.842673428005184*,*0.740850398734311*and*0*.#SecantMethod #RootsofEquations #Matlab #NumericalMethod #Blog #Blogger