chapter 2 mmcs1

8/3/2019 Chapter 2 Mmcs1

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Chapter II

Methods in Obtaining Roots of the Equation

Appendix A contains the figures of the flowchart of the programs being

constructed for each methods of obtaining root of an equation. In ths program, it

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

necessary methods calling for inputs, and subclasses (Bairstow, Bisection, Brent,

FalsePosition, FixedPoint, Muller, NewtonRaphson, and Secant), which lead to the

computation of the root using any method.

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

Bisection is a numerical method for estimating the roots of a polynomial f(x) . It

is one of the simplest and most reliable but it is not the fastest method. It is called the

binary chopping or the Bolzanos method. It is a bracketing method which finds root

of a given continuous function over an interval and such that f ( ) and f ( ) will

have an opposite signs that gives f ( ) f ( ) < 0. The method divides the interval in two

by computing the midpoint = ( + ) of the interval. Either f ( ) and f ( ) or f ( )

and f ( ) 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 ( ) f ( ) < 0, then the method sets equal to ,


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and if f ( ) f ( ) < 0, then the method sets equal to . For both cases, the new f ( )

and f ( ) will have opposite signs, so that the method is applicable to this smaller

interval.

The continuous function on the given interval [ , ] and f ( ) f ( ) < 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 ( ) and f ( ) 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;

False Position Method

The false-position method is a modification on the bisection method: if it is

known that the root lies on [ a , b ], then it is reasonable that we can approximate the

function on the interval by interpolating the points ( a , f( a )) and ( b , f( b )). 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 and such that the functions

f ( ) and f ( 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:

(2.2)

(2.1)


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The root is from the graphical representation of joining the function f ( ) and

f ( 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 ( ) and f ( with the same sign as f (

so that the root is always at the interval of the two point and .

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

and same as the algorithm, but the equation for finding 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 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.


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Simple Fixed Point Method

Fixed Point Method 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 can be used to estimate as and can be expressed as =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 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,

N ewton 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

(2.3)


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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

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 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

(2.4)

(2.5)

(2.6)


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Bairstows Method

Bairstow Method is an iterative method used to find both the real and complex

roots of a polynomial. It is based on the idea of synthetic division of the given

polynomial by a quadratic function and can be used to find all the roots of a

polynomial. 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

Newtons method to adjust the coefficients u and v in the quadratic x 2 + u x + 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

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

the Newtons 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.

In a given polynomial equation, a n x n + a n-1 x n-1 +...+ a o =0, the root of the equation

can be solved using the equation given below:

where r and s are guesses.

Mller'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 theres no information about the derivatives

(2.6)


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that exists. Its the generalizes the secant method but it uses three points of quadratic

interpolation noted by as x k , x k -1 and x k - 2 .The The parabola going through the three

points (xk, f(x k )), (x k -1 , f(x k -1)) and (x k - 2 , f(x k - 2 )) when

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

divided differences:

where:

Brents Method

Brent's method is a complicated but popular root-finding algorithm combining

the bisection method, the secant method, and inverse quadratic interpolation. It has

the reliability of bisection but it can be as quick as some of the less reliable methods.

The idea is to use the secant method or inverse quadratic interpolation if possible,

because they converge faster, but to fall back to the more robust bisection method if

necessary.a method that combines that bisection method, the secant method. The idea

is to use the secant method because they converge faster, but to fall back to the more

robust bisection method if necessary.

Given a specific numerical tolerane , must hold and the

results is used in the iteration and if previous step is perform interpolation then the

inequality gives . Also, if the previous step used the bisection

(2.8)

(2.9)

(2.7 )


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method, the inequality must hold, otherwise the bisectionmethod 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 N 2 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 2 mmcs1

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