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Chapter I

Introduction

Background

In the field of engineering electricity it is necessary to know the properties of

the materials to be used for a certain project of development or for a study to be

conducted. Failure to know the properties of the material to be made of, may lead

to the disappointment of the project. Thus, resistivity is one of the physical

properties of a material as of density, specific heat and many others. Resistivity is

primarily defined as the strength or capacity of the material to oppose the flow of

an electric current.Any object made of the same material has the common

resistivity. When talking about resistivity it doesn't matter how big or what shape

the sample is, it is the object’s resistance which depends on its length, cross

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sectional area, etc. The relationship between the resistivity and resistance is shown

by a formula,

where R is the resistance , r is the resistivity , L is the length of the material, and A

is the cross sectional area. Thus, resistivity related to resistance may be obtained

through mathematical process. Hence the SI unit for resistivity is ohm meter (Ω.m).

Another way of obtaining the resistivity (ρ) of an object is based on its

electron content. It is defined that the resistivity(ρ) of a certain material is inversely

proportional to the electron charge (q), the electron density (n), and the electron

mobility (µ) measured in cm2.

Electron density is themeasure of the probability of an electron being present

at a specific location and the electron mobility is a quantity relating the drift

velocity of electrons to the applied electric field across a material. Electron density

(n) is given by the formula,

where N is determined as the doping density and ni as the intrinsic carrier density

measured in cm-3. And also the electron mobility (μ) is described by the final

R=ρ LA (1)

ρ= 1qn μ (2)

n=12

¿ ) (2)

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temperature and reference temperature T and T o respectively, and a reference

electron mobility ( μo). And it is given by a formula,

Engineering Problem Posed

The van der Waal equation of state for a vapor is

(P+ av2 ) (v−b )=RT

where P is the pressure (Pa = N/m2), v is the specific volume (m3/kg), the

temperature (K), R is the gas constant (J/kg-K), and a and b are empirical constants.

Consider water vapor, for which R=461.495J/kg-K, a=1703.28Pa-(m3/kg)3, and

b=0.00169099(m3/kg). Calculate the specific volume v for P = 10,000kPa and T

=800K.

Mathematical Analysis

Different mathematical methods should be used to obtain the value in the

given problem. Thus, it needs to transform and express functions or working

equations to the form the problem requires.

Engineering Problem Expressed Mathematically

Determine the specific volume (v) given

R=461.495 J /kg−K ,a=1703.28 Pa−(m3 /kg)3 , b=0.00169099(m 3/kg) ,P=10,000kPa ,

and T=800K .

μ=μo(TTO

)−2.42

(3)

(3)

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The given equation can be rearranged by algebraic manipulation:

(P+ av2 ) (v−b )=RT

Pv−Pb+ av−abv2 =RT

Pv3−Pbv2+av−ab=RT v2

Pv3− (Pb+RT ) v2+av−ab=0

f ( v )=Pv3− (Pb+RT ) v2+av−ab

Using the derived function, the value of (v) can be obtained with the aid of the

different mathematical numerical methods of solving problems.

Chapter II

Methods in Obtaining Roots of the Equation

Referring to Appendix A are the figures of the flowchart of the programs

being constructedfor each methods of obtaining root of an equation, it follows a

subroutine procedure. The program contains a main class obtaining the necessary

methods calling for inputs, and subclasses (Bisection, FalsePosition, FixedPoint

Iteration, NewtonRaphson, Secant, Brent) which leads to the computation of the

root using any method.

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Bracketing Method

It comprises different methods which the roots may be found within the two

initial guesses which are typically changes the signs. The methods present here

give strategies which reduces the width of the bracket until the root will be found.

Bisection Method

It is called the binary chopping or the Bolzano’s method. A Bracketing

method which finds root of a given continuous function over an interval x l andxu

such that f(x l) and f(xu) will have an opposite signs that gives f(x l) f(xu) < 0. The

method divides the interval in two by computing the midpoint xr= (x l+xu)/2 of the

interval. Either f(x l) and f(xr) or f(xr) and f(xu) will have opposite signs and it

brackets a root, we must select a subinterval within the interval and apply the same

bisection step. There will be a 50% of chance of getting a function equals to zero. If

f(x l) f(xr) < 0, then the method sets equal xu toxr, and if f(xu) f(xr)< 0, then the

method sets x lequal to xr. For both cases, the new f(x l) and f(xu) will have opposite

signs, so that the method is applicable to this smaller interval.

The continuous function on the given interval [x l,xu ] and f(x l) f(xu) < 0 states

that the bisection converges to a root of the function and the true error is halved in

each step and the method converges linearly if f(x l) and f(xu) will have different

signs. This method gives only a range where the root exists and not the estimation

where is the roots location. The smallest bracket is where the root can be found. Its

true error of n steps can be solved by the equation;

ε t=x l+ xu

2 (2.1) M

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False Position Method/Regula Falsi Method

It is also called the linear interpolation method. An alternative method based

on the graphical method. The false position method starts with a two points x l and

xu such that the functions f(x l) and f((xu ) will have an opposite signs then one of the

end-points will converges and the other will remain fixed for all the iterations

function f a root. It is given by the formula,

The root xr is from the graphical representation of joining the function f(x l)

and f((xu ) by a straight line and which the point that intersects the line and the axis

is the improve root. The value of the root replaces f(x l) and f((xu ) with the same sign

as f( xr ¿¿ so that the root is always at the interval of the two point x l and xu.

The termination of the computation will be the same as the bisection method

and same as the algorithm, but the equation for finding xr is used. The error of the

regula falsi is more efficient for root finding than the bisection since one of the

points will stay throughout the computation and the others converges quickly and

makes the approximate error conservative.

Modified False Position Method

xr=xu−f ( xu ) (xu−x l)f (xl )−f (xu)

(2.2)

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It is the remedy of being one-sided of the false position method. It divides the

function value that was stuck. The algorithm implements the strategies on how the

counters are used to determine the root when the one is bound stays fixed for the

two iterations and through this, the function value is bound halved.

It is more than the bisection and the false position method for setting the

stopping criterion as 1.01% since it gives only 12 iterations compare with the 14

and 25 of the bisection and false position method.

Open Method

It composed of different methods that are based on the formulas that

requires only a single starting value of x or two starting values that do not

necessarily bracket the root. It may diverge or converges as the computation

progresses.

Simple Fixed Point Method

It is also called the One-point iteration or the successive substitution method.

It rearranges the function f(x)=0 to x=g(x) It can be obtained by adding both

sides a x of the equation or by simply doing algebraic manipulation. The guess roots

x i can be used to estimate as x i+1 and can be expressed as x i+1=g(x).

The convergence or the divergence of this method can be depicted graphically

through its behavior and structure or it can also be predicted by separating the it

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into two components parts and the x values obtained by the intersections are the

roots of the function f(x)=0. The two-curve method also shows the convergence and

the divergence of the simple fixed-point method. To find for the approximate error

of this method can be solve using this formula,

Newton Raphson Method

The widely used for finding the root for approximations to the zeroes of a real

valued function. It converges quickly for the iterations which are near on the desired

root. It also detects and overcomes the convergences failure.

This method starts with an initial guess which is close to the true root, the

given function is approximated by its tangent line then computes the x-intercept of

this tangent line. This x-intercept will be the approximation to the function's root

than the original guess, and the method can be repeated. The formula for this

method is given by

The termination of the Newton- Raphson method is the same as for

computing the other methods. The convergence depends on the accuracy of the

initial guess root and the nature of the problem.

Secant Method

ε a=( x i+1−xix i+1

)100 % (2.3)

x i+1=x i+f (x i)f ' (x i)

(2.4)

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It is an open method which assumes a function that can be approximately

linear in the region of interest. The formula for the needs two initial estimates of x

but the f(x) is not required to change the signs between the two estimates and is

given by this equation,

The two values can sometimes lie on the same root and sometimes this can

cause the divergence. The convergence of this method is that the root is within the

bracketing which is the reason that it was compared with the false position method.

Modified Secant Position Method

This method uses an alternative approach which involves the fractional

perturbation of the independent variable to estimate the f’(x) instead of using the

two arbitrary values. The formula for the iteration is given by

Bairstow’s Method

It is a method that finds complex roots of a polynomial of a quadratic formula

and can be used for solving the root all kinds of a polynomial. It uses the Newton’s

method to adjust the coefficients u and v in the quadratic x2 + ux + v until its roots

are also roots of the polynomial being solved. The root can be found be found by

dividing the polynomial by the quadratic to eliminate the roots and then it can be

repeated until the polynomial becomes quadratic or linear and all roots will be

x i+1=x1−f (x i ) f ( xi−1−x i)f (x i−1−x i )−f (xi)

(2.5)

x i+1=x i−δ x i f (x i )

f (x i+δ x i )−f (x i )(2.6)

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determined. The values of u and v can be found by picking the starting and

repeating the Newton’s method in two dimensions until it converges, for the

quadratic equations of multiplicity higher than one it converges to that factor is a

linear and quadratic factors that have a small value which has real roots will tend to

diverge to infinity. To find for the zero of polynomial can be implemented with a

programming language.

Müller's method

A root finding method that solves for the root of the form f(x) = 0 of the single

variable x and a scalar function whenever there’s no information about the

derivatives that exists. It’s the generalizes the secant method but it uses three

points of quadratic interpolation noted by as xk, xk-1 and xk-2.The The parabola

going through the three points (xk, f(xk)), (xk-1, f(xk-1)) and (xk-2, f(xk-2)) when

It can be written in the Newton form, where f[xk, xk-1] and f[xk, xk-1, xk-2] denote

divided differences ;

where;

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Brent’s Method

It is a method that combines that bisection method, the secant method. The

idea is to use thesecant method because they converge faster, but to fall back to

the more robust bisection method if necessary.

Given a specific numerical tolerane δ, | δ | < | bk − bk − 1 |must hold and the

results is used in the iteration and if previous step is performinterpolation then the

inequality gives | δ | < | bk − 1 − bk − 2 |. Also, if the previous step used the bisection

method, the inequality must hold, otherwise the bisection

method is performed and the result used for the next iteration. If the previous step

performed interpolation, then the inequality is used

instead. Most of the N2 iterations, where N denotes the number of iterations for the

bisection method, if the function f is well-behaved, and this method will usually

proceed by either inverse quadratic or linear interpolation, in which case it will

converge linearly.

Chapter III

Source Code

The following are the listing for the source code of the different methods of

obtaining the root of the function.

Listing 1. Class Main

/* Main Class - contains the main method which executes the Java application - calls the needed methods*///import Java extension packagesimport javax.swing.JOptionPane;

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/* Main Class - contains the main method which executes the Java application - calls the needed methods*///import Java extension packagesimport javax.swing.JOptionPane;

break;// done processing case case 1: c.FalsePosition(); break;// done processing case case 2: d.FixedPointIteration(); break;// done processing case case 3: e.NewtonsRaphson(); break;// done processing case case 4: f.Secant(); break;// done processing case case 5: g.Brent();

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Listing 1 contains the source code of the Main class, the class which contains the main method, a method that executes the application.

Listing 2.Class MRZ

break;// done processing case case 1: c.FalsePosition(); break;// done processing case case 2: d.FixedPointIteration(); break;// done processing case case 3: e.NewtonsRaphson(); break;// done processing case case 4: f.Secant(); break;// done processing case case 5: g.Brent();

/*MRZ class - declares method of the function- declares methods to prompt user to input values needed in the methods - declares the method in giving the output*///Declaring methods//import Java extension packagesimport javax.swing.*;//import Java core packagesimport java.text.DecimalFormat;public class MRZ

//Create JTextArea to display output JTextArea Mrz = new JTextArea (25,75); //Puts a scrollbar in the JTextArea JScrollPane mRz = new JScrollPane(Mrz); DecimalFormat df = new DecimalFormat("0.00000000");

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/*MRZ class - declares method of the function- declares methods to prompt user to input values needed in the methods - declares the method in giving the output*///Declaring methods//import Java extension packagesimport javax.swing.*;//import Java core packagesimport java.text.DecimalFormat;public class MRZ

//Create JTextArea to display output JTextArea Mrz = new JTextArea (25,75); //Puts a scrollbar in the JTextArea JScrollPane mRz = new JScrollPane(Mrz); DecimalFormat df = new DecimalFormat("0.00000000");

//end method input

//Function of the given problempublic double func(double N)return ((Math.pow(j, -1)))-(N/2)- (Math.sqrt(Math.pow(N,2)+ 4*Math.pow(ni,2)))/2;//returns the function //g(x) of the function of the given problempublic double gfunc(double N)return (Math.pow(j,-1))-(N/2)- (Math.sqrt(Math.pow(N,2)+4*Math.pow(ni,2)))/2+N; //Derivative of the function of the given problempublic double dfunc(double N)return -(0.5)-((N*(1/(Math.sqrt(Math.pow(N,2)+

4*Math.pow(ni,2)))))/2);//return the derivative of the function //Calculates the Approximation Error

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//end method input

//Function of the given problempublic double func(double N)return ((Math.pow(j, -1)))-(N/2)- (Math.sqrt(Math.pow(N,2)+ 4*Math.pow(ni,2)))/2;//returns the function //g(x) of the function of the given problempublic double gfunc(double N)return (Math.pow(j,-1))-(N/2)- (Math.sqrt(Math.pow(N,2)+4*Math.pow(ni,2)))/2+N; //Derivative of the function of the given problempublic double dfunc(double N)return -(0.5)-((N*(1/(Math.sqrt(Math.pow(N,2)+

4*Math.pow(ni,2)))))/2);//return the derivative of the function //Calculates the Approximation Error

NU=JOptionPane.showInputDialog("Input upper limit: ");Nl=Double.parseDouble(NL);Nu=Double.parseDouble(NU);//check if the given interval contains one root if (mrz.func(Nl)*mrz.func(Nu)>0) b=0;

JOptionPane.showMessageDialog(null,"The root is not " +"located in the given interval or there are more " +"than one root","Notification",JOptionPane.WARNING_MESSAGE);

else b=1; //end while structure // determine the method to use based on user's choice switch(s)

case 2://Fixed-Point Iteration Method

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NU=JOptionPane.showInputDialog("Input upper limit: ");Nl=Double.parseDouble(NL);Nu=Double.parseDouble(NU);//check if the given interval contains one root if (mrz.func(Nl)*mrz.func(Nu)>0) b=0;

JOptionPane.showMessageDialog(null,"The root is not " +"located in the given interval or there are more " +"than one root","Notification",JOptionPane.WARNING_MESSAGE);

else b=1; //end while structure // determine the method to use based on user's choice switch(s)

case 2://Fixed-Point Iteration Method

NO=JOptionPane.showInputDialog("Enter initial estimate of x: ");

N0=Double.parseDouble(NO); if (N0==0) b=0;

JOptionPane.showMessageDialog(null,"INVALID! Enter another" +" value for x(non-zero).","Notice",

JOptionPane.WARNING_MESSAGE); else b=1; //end while structure

SIGMA=JOptionPane.showInputDialog("Input pertubation fraction: ");

sigma=Double.parseDouble(SIGMA); break;// done processing case case 5://Brent's Method

EPS=JOptionPane.showInputDialog(null,"Enter the

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Listing 2 contains the source code of the MRZ class. This class is the Mother

class with six subclasses ModBisection, ModFalsePosition, FixedPointIteration,

NewtonRaphson, ModSecant and Brent. This class contains the conditions of

obtaining correct initial inputs for each of the subclasses.

Listing 3. Listing for Bisection Method

NO=JOptionPane.showInputDialog("Enter initial estimate of x: ");

N0=Double.parseDouble(NO); if (N0==0) b=0;

JOptionPane.showMessageDialog(null,"INVALID! Enter another" +" value for x(non-zero).","Notice",

JOptionPane.WARNING_MESSAGE); else b=1; //end while structure

SIGMA=JOptionPane.showInputDialog("Input pertubation fraction: ");

sigma=Double.parseDouble(SIGMA); break;// done processing case case 5://Brent's Method

EPS=JOptionPane.showInputDialog(null,"Enter the

/*ModBisection Class - class containing the method of the "Modified Bisection method".*/public class ModBisection extends MRZ public void Bisection() //initialize String

wee=" Bisection\n----------\nNo of iter.\tXl\t\tXu\t\tXr\t\tEa\n";

fl = mrz.func(Nl); for(int iter = 0; iter<imax;iter++) Nrold=Nr; Nr=(Nl+Nu)/2; fr = mrz.func(Nr);

if(Nr != 0) ea = mrz.ea(); else ea=es; if (fl*fr<0)

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Next is Listing 3 is the source code for the class ModBisection. This class only consists of the process of the method. The inputs are located in the mother class MRZ.

Listing 4. Listing for False Position

/*ModBisection Class - class containing the method of the "Modified Bisection method".*/public class ModBisection extends MRZ public void Bisection() //initialize String

wee=" Bisection\n----------\nNo of iter.\tXl\t\tXu\t\tXr\t\tEa\n";

fl = mrz.func(Nl); for(int iter = 0; iter<imax;iter++) Nrold=Nr; Nr=(Nl+Nu)/2; fr = mrz.func(Nr);

if(Nr != 0) ea = mrz.ea(); else ea=es; if (fl*fr<0)

package casestudy;/*ModFalsePosition Class - class containing the method of the "False Position method".*/public class ModFalsePosition extends MRZ public void FalsePosition() double il=0,iu=0; //initialize String

wee="False Position\n----------\nNo of iter.\tXl\t\tXu\t\tXr\t\tEa\n";

fl = mrz.func(Nl); fu = mrz.func(Nu); for(int iter=0; iter<imax;iter++) Nrold=Nr; Nr=Nu-fu*(Nl-Nu)/(fl-fu);

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Listing 4 contains the source code of the class ModFalsePosition. This

class contains only the method’s process, input proper appears in the

Mother class MRZ

Listing 5. Listing for Fixed Point Iteration

package casestudy;/*ModFalsePosition Class - class containing the method of the "False Position method".*/public class ModFalsePosition extends MRZ public void FalsePosition() double il=0,iu=0; //initialize String

wee="False Position\n----------\nNo of iter.\tXl\t\tXu\t\tXr\t\tEa\n";

fl = mrz.func(Nl); fu = mrz.func(Nu); for(int iter=0; iter<imax;iter++) Nrold=Nr; Nr=Nu-fu*(Nl-Nu)/(fl-fu);

/*FixedPointIteration Class - class containing the method of the "Fixed-Point Iteration method".*/public class FixedPointIteration extends MRZ

public void FixedPointIteration() //initialize string

wee = "Fixed-Point Iteration\n----------\nNo of iter.\tXo\t\tXr\t\tEa\n";

Nr=N0; for( int iter = 0; iter<imax;iter++)

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Listing 5 shows the source code of the class FixedPointIteration. Just like the

previous listings, this class contains only the process of the method, the inputs are

located in the Mother class MRZ.

Listing 6. Listing for Newton-Raphson

/*FixedPointIteration Class - class containing the method of the "Fixed-Point Iteration method".*/public class FixedPointIteration extends MRZ

public void FixedPointIteration() //initialize string

wee = "Fixed-Point Iteration\n----------\nNo of iter.\tXo\t\tXr\t\tEa\n";

Nr=N0; for( int iter = 0; iter<imax;iter++)

/*NewtonsRaphson Class - class containing the method of the "Newton's Raphson method".*/public class NewtonsRaphson extends MRZ public void NewtonsRaphson()

//initialize Stringwee = "Newton's Raphson Method\n----------\nNo of iter.\tXo\t\tXr\t\tEa\n";

Nr=N0;

for( int iter = 0; iter<imax;iter++) Nrold=Nr; Nr = Nrold-(mrz.func(Nrold))/(mrz.dfunc(Nrold)); if(Nr!=0) ea=mrz.ea(); else ea=es; if (iter==0)wee += (iter+1)+"\t"+df.format(Nrold)+"\t"+ df.format(Nr)+"\t-----\n"; else

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Listing 6 contains the source code of the Newton-Raphson method. This

contains the process of the said method. The inputs are located in the Mother class

MRZ.

Listing 7. Listing for Secant Method

/*NewtonsRaphson Class - class containing the method of the "Newton's Raphson method".*/public class NewtonsRaphson extends MRZ public void NewtonsRaphson()

//initialize Stringwee = "Newton's Raphson Method\n----------\nNo of iter.\tXo\t\tXr\t\tEa\n";

Nr=N0;

for( int iter = 0; iter<imax;iter++) Nrold=Nr; Nr = Nrold-(mrz.func(Nrold))/(mrz.dfunc(Nrold)); if(Nr!=0) ea=mrz.ea(); else ea=es; if (iter==0)wee += (iter+1)+"\t"+df.format(Nrold)+"\t"+ df.format(Nr)+"\t-----\n"; else

/*ModSecant Class - class containing the method of the "Modified Secant method".*/public class ModSecant extends MRZ

public void Secant() //initialize String

wee ="Secant Method\n----------\nNo of iter.\tXo\t\tXr\t\tEa\t\n";

double fa,fb; Nr=N0;for( int iter = 0; iter<imax;iter++)

Nrold=Nr; fa = mrz.func(Nrold); fb = mrz.func(Nrold+(sigma*Nrold)); Nr = Nrold -(sigma*Nrold*fa)/(fb-fa); if(Nr!=0)ea=mrz.ea(); else ea=es; if(iter==0)

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Listing 7 contains the source code of the class ModSecant. Just like the

previous subclasses, this only contains the process proper of the method. The

inputs are located in the Mother class MRZ. The next is Listing 8, which contains the

source code of the class Brent, like previous subclasses, this contains the process of

the method. The input proper is located in the Mother class MRZ.

Listing 8. Brent’s Method

/*ModSecant Class - class containing the method of the "Modified Secant method".*/public class ModSecant extends MRZ

public void Secant() //initialize String

wee ="Secant Method\n----------\nNo of iter.\tXo\t\tXr\t\tEa\t\n";

double fa,fb; Nr=N0;for( int iter = 0; iter<imax;iter++)

Nrold=Nr; fa = mrz.func(Nrold); fb = mrz.func(Nrold+(sigma*Nrold)); Nr = Nrold -(sigma*Nrold*fa)/(fb-fa); if(Nr!=0)ea=mrz.ea(); else ea=es; if(iter==0)

/*Brent Class - class containing the method of the "Brent's method".*/public class Brent extends MRZ

public void Brent () //Initialize String wee = "Brent's Method\n----------\nNo of iter.\tXr\t\tEa\n";double fc,c, c0, c1, c2,temp, mtflag, d;int mflag,iter = 1; c = Nl;d = c; fl = mrz.func(Nl);//fa=flfu = mrz.func(Nu);//fb=fufc = mrz.func(c); if ( Math.abs(fl) < Math.abs(fu)) temp = Nl; Nl = Nu; Nu = temp;temp = fl;

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/*Brent Class - class containing the method of the "Brent's method".*/public class Brent extends MRZ

public void Brent () //Initialize String wee = "Brent's Method\n----------\nNo of iter.\tXr\t\tEa\n";double fc,c, c0, c1, c2,temp, mtflag, d;int mflag,iter = 1; c = Nl;d = c; fl = mrz.func(Nl);//fa=flfu = mrz.func(Nu);//fb=fufc = mrz.func(c); if ( Math.abs(fl) < Math.abs(fu)) temp = Nl; Nl = Nu; Nu = temp;temp = fl;

else mflag = 0; fr = mrz.func(Nr); d = c; c = Nu; fc = fu;if(Nr !=0)ea=mrz.ea(); else ea=es; if ( (fl*fr)< 0) Nu = Nr; else Nl = Nr; if ( Math.abs(fl) < Math.abs(fu)) temp = Nl; Nl = Nu;Nu = temp; temp = fl; fl = fu; fu = temp; if (iter==1)

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Chapter IV

Result

A. Graphical Result

Chapter IV

Results and Discussion

A. Graphical Representation

else mflag = 0; fr = mrz.func(Nr); d = c; c = Nu; fc = fu;if(Nr !=0)ea=mrz.ea(); else ea=es; if ( (fl*fr)< 0) Nu = Nr; else Nl = Nr; if ( Math.abs(fl) < Math.abs(fu)) temp = Nl; Nl = Nu;Nu = temp; temp = fl; fl = fu; fu = temp; if (iter==1)

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Figure 1 shows the graphical representation between the approximate errors(

ϵ a) and the number of iterations. The results were obtained through the use of the

different methods stated.

B. Screenshot of outputs

The following figures are the actual results or screenshots obtained using the

source codes accessible on Chapter 3.M

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Figure 3. Screenshot of the Result for False Position Method

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Figure 5. Screenshot of the Result for Newton-Raphson Method

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Figure 7. Screenshot of the result for Brent’s Method

C. Table of Results

The following are the tabular presentation of the results obtained after

employing the different methods for solving the function eq. 5.

Table 1. Results for Bisection Method

Iterations Nr εa

1 15000000000 -----2 12500000000 203 11250000000 11.11111114 11875000000 5.263157895 11562500000 2.70270276 11718750000 1.33333333

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Table 1 shows the obtained results for solving the roots of eqn. 5 using the

Bisection Method. As shown, the root was obtained after 11 iterations and is equal

to Nr = 11752929688, where the condition εa< εs was satisfied after 11 iterations

such that εa = 0.0415454, which is less than the inputted stopping εs =0.05. The

limits used in solving the problem are 1x1010 and 2x1010, the lower and upper limits

respectively.

Table 2 shows the obtained results for solving the roots of the function eqn. 5

using the False Position Method. As seen above, the root was obtained after

4iterations, which was evaluated from the lower and upper limits, 1x1010 and 2x1010

respectively, where the value of the approximation error is εa = 0.04073353, which

satisfies the condition εa< εs, where εs = 0.05.

Table 1. Results for Bisection Method

Iterations Nr εa

1 15000000000 -----2 12500000000 203 11250000000 11.11111114 11875000000 5.263157895 11562500000 2.70270276 11718750000 1.33333333

Table 2. Results for False Position Method

Iterations Nr εa

1 11655417267 -----2 11751253575 0.815541153 11761402241 0.086287884 11756613357 0.04073353

Table 3. Results for Fixed-Point Iteration Method

Iterations Nr εa

1 11456538023 -----2 11709015931 2.156269233 11749157662 0.341656244 11755447971 0.05350973

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Table 3 shows the obtained results for solving the roots of the function in

eqn. 5 using the Fixed-Point Iteration Method. From the table, a root of Nr =

11756431421 is obtained after 5 iterations, where the condition εa< εs is satisfied,

where εa = 0.00836521 and εs = 0.05. The initial guess used is 1x1010.

Table 4 shows the obtained results for solving the function in eqn.5 using the

Newton-Raphson Method. From the table, a root of Nr = 11756613591 is obtained

after 3 iterations with a relative approximate error of εa = 0.00008794, which

satisfies the condition εa< εs, where εs = 0.05. The initial guess used is 1x1010.

Table 3. Results for Fixed-Point Iteration Method

Iterations Nr εa

1 11456538023 -----2 11709015931 2.156269233 11749157662 0.341656244 11755447971 0.05350973

Table 4. Results for Newton-Raphson Method

Iterations Nr εa

1 11790305201 -----2 11756623930 0.286487623 11756613591 0.00008794

Table 5. Results forSecant Method

Iterations Nr εa

1 11780101085 -----2 11756741026 0.198695023 11756614257 0.00107828

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Table 5 shows the results after computing for the root of the function in eqn.

5 using Modified Secant Method where the root was found out to be equal to

Nr=11756614257. This root was obtained in the 3rd iteration where the relative

approximate error is equal εa=0.00107828 which is less than the required stopping

condition εs=0.5. The perturbation factor used is 0.05, and the initial estimate used

is 1x1010.

The table above shows the results obtained after computing for the root of

the function in eqn. 5 by employing Brent’s Method. As shown above, the root, Nr =

11748046875, was obtained in the 10th iteration where the computed approximate

error Es=0.08312552 satisfies the terminating condition εa<εs where εs = 0.05.

In evaluating the root of the function using theformula for Brent’s Method,

the lower and upper limit used were 1x1010 and 2x1010 respectively, and the

numerical tolerance was chosen to be equal to 0.000001.

Table 6. Results forBrent’s Method

Iterations Nr εa

1 15000000000 -----2 12500000000 203 11250000000 11.111111114 11875000000 5.2631578955 11562500000 2.7027027036 11718750000 1.3333333337 11796875000 0.6622516568 11757812500 0.3322259149 11738281250 0.166389351

10 11748046875 0.08312552

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Chapter V

Conclusion

The Newton-Raphson Method gives the least approximate true error over

other methods in finding the root of equation (5). Equation (5) is much more

approximately equal to zero when the root obtained using the Fixed Point Iteration

methodis substituted. In addition, these methods give a lesser relative

approximation error, though εa and εt for this technique has a considerable

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With our given function, the Newton Raphson is the most efficient technique

in finding the root of the function in equation (5). This is true since only a single

value is used to predict succeeding values for the root estimate since it uses the

least number of iterations to obtain the root. A better accuracy is also attained with

this method since, it ensures a better convergence other than the methods, and

since equation (5) is a linear function the Muller and Bairstow’s Method cannot be

applied because primarily these methods can only be applied for polynomial

functions.

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