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M.Sc. (Mathematics)IV - Semester

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NUMERICAL ANALYSIS

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Authors:Dr. N. Dutta, Professor of Mathematics, Head - Department of Basic Sciences & Humanities, Heritage Institute of Technology,KolkataUnits (2, 4, 6-8, 10-13)Vikas® Publishing House: Units (1, 3, 5, 9, 14)

SYLLABI-BOOK MAPPING TABLENumerical Analysis

BLOCK - I : POLYNOMIAL EQUATIONS AND EIGEN VALUEPROBLEMS

UNIT - 1Transcendental and Polynomial Equations: Rate of Convergenceof Iterative Methods.UNIT - 2Methods for Finding Complex Roots - Polynomial Equations.UNIT - 3Birge - Vieta Method, Bairstow’s Method, Graeffe’s Root SquaringMethod.UNIT - 4System of Linear Algebraic Equations and Eigen Value Problems:Error Analysis of Direct and Iteration Methods.

BLOCK - II : EIGEN VECTORS, INTERPOLATION,APPROXIMATION, DIFFERENTIATIONAND INTEGRATION

UNIT - 5Finding Eigen Values and Eigen Vectors - Jacobi and PowerMethods.UNIT - 6Interpolation and Approximation: Hermite Interpolations -Piecewise and Spline Interpolation - Bivariate Interpolation.UNIT - 7Approximation - Least Square Approximation and BestApproximations.UNIT - 8Differentiation and Integration: Numerical Differentiation -Optimum Choice of Step - Length - Extrapolation Methods.

BLOCK - III : PDE, ODE AND EULER METHODSUNIT - 9Partial Differentiation - Methods Based on UndeterminedCoefficient - Gauss Methods.UNIT - 10Ordinary Differential Equations: Local Truncation Error - Problems.UNIT - 11Euler, Backward Euler, Midpoint, -Problems.

BLOCK - IV: TAYLOR’S METHOD, R.K METHODAND STABILITY ANALYSIS

UNIT - 12Taylor’s Method -Related Problems.UNIT - 13Second Order Runge Kutta Method - Stability Analysis.UNIT - 14Stability Analysis.

Syllabi Mapping in Book

Unit 1: Transcendental andPolynomial Equations

(Pages 3-29);Unit 2: Methods for Finding Complex

Roots and Polynomial Equations(Pages 30-53);

Unit 3: Birge – Vieta, Bairstow’sand Graeffe’s RootSquaring Methods

(Pages 54-65);Unit 4: Solution of Simultaneous

Linear Equation(Pages 66-85);

Unit 5: Eigen Values andEigen Vectors

(Pages 86-106);Unit 6: Interpolation and

Approximation(Pages 107-146);

Unit 7: Approximation(Pages 147-171);

Unit 8: Numerical Integration andNumerical Differentiation

(Pages 172-220)

Unit 9: Partial Differential Equations(Pages 221-283);

Unit 10: Ordinary Differential Equations(Pages 284-299);

Unit 11: Euler’s Method(Pages 300-307)

Unit 12: Taylor’s Method(Pages 308-312);

Unit 13: Runge Kutta Method(Pages 313-321);

Unit 14: Stability Analysis(Pages 322-328)

BLOCK I: POLYNOMIAL EQUATIONS AND EIGEN VALUE PROBLEMS

UNIT 1 TRANSCENDENTAL AND POLYNOMIAL EQUATIONS 1-291.0 Introduction1.1 Objectives1.2 Transcendental and Polynomial Equations1.3 Answers to Check Your Progress Questions1.4 Summary1.5 Key Words1.6 Self Assessment Questions and Exercises1.7 Further Readings

UNIT 2 METHODS FOR FINDING COMPLEX ROOTSAND POLYNOMIAL EQUATIONS 30-53

2.0 Introduction2.1 Objectives2.2 Methods for Finding Complex Roots2.3 Polynomial Equations2.4 Answers to Check Your Progress Questions2.5 Summary2.6 Key Words2.7 Self Assessment Questions and Exercises2.8 Further Readings

UNIT 3 BIRGE – VIETA, BAIRSTOW’S AND GRAEFFE’SROOT SQUARING METHODS 54-65

3.0 Introduction3.1 Objectives3.2 Birge – Vieta Method3.3 Bairstow’s Method3.4 Graeffe’s Root Squaring Method3.5 Answers to Check Your Progress Questions3.6 Summary3.7 Key Words3.8 Self-Assessment Questions and Exercises3.9 Further Readings

UNIT 4 SOLUTION OF SIMULTANEOUS LINEAR EQUATION 66-854.0 Introduction4.1 Objectives4.2 System of Linear Equations

4.2.1 Classical Methods4.2.2 Elimination Methods4.2.3 Iterative Methods4.2.4 Computation of the Inverse of a Matrix by using Gaussian Elimination Method

CONTENTS

4.3 Answers to Check Your Progress Questions4.4 Summary4.5 Key Words4.6 Self Assessment Questions and Exercises4.7 Further Readings

BLOCK II: EIGEN VECTORS, INTERPOLATION, APPROXIMATION,DIFFERENTIATION AND INTEGRATION

UNIT 5 EIGEN VALUES AND EIGEN VECTORS 86-1065.0 Introduction5.1 Objectives5.2 Finding Eigen Values and Eigen Vectors5.3 Jacobi and Power Methods5.4 Answers to Check Your Progress Questions5.5 Summary5.6 Key Words5.7 Self Assessment Questions and Exercises5.8 Further Readings

UNIT 6 INTERPOLATION AND APPROXIMATION 107-1466.0 Introduction6.1 Objectives6.2 Interpolation and Approximation6.3 Answers to Check Your Progress Questions6.4 Summary6.5 Key Words6.6 Self Assessment Questions and Exercises6.7 Further Readings

UNIT 7 APPROXIMATION 147-1717.0 Introduction7.1 Objectives7.2 Approximation7.3 Least Square Approximation7.4 Answers to Check Your Progress Questions7.5 Summary7.6 Key Words7.7 Self Assessment Questions and Exercises7.8 Further Readings

UNIT 8 NUMERICAL INTEGRATION AND NUMERICALDIFFERENTIATION 172-220

8.0 Introduction8.1 Objectives8.2 Numerical Integration8.3 Numerical Differentiation

8.4 Optimum Choice of Step Length8.5 Extrapolation Method8.6 Answers to Check Your Progress Questions8.7 Summary8.8 Key Words8.9 Self Assessment Questions and Exercises

8.10 Further ReadingsBLOCK III: PDE, ODE AND EULER METHODS

UNIT 9 PARTIAL DIFFERENTIAL EQUATIONS 221-2839.0 Introduction9.1 Objectives9.2 Partial Differential Equation of the First Order Lagrange’s Solution9.3 Solution of Some Special Types of Equations9.4 Charpit’s General Method of Solution and Its Special Cases9.5 Partial Differential Equations of Second and Higher Orders

9.5.1 Classification of Linear Partial Differential Equations of Second Order9.6 Homogeneous and Non-Homogeneous Equations with

Constant Coefficients9.7 Partial Differential Equations Reducible to Equations with

Constant Coefficients9.8 Answers to Check Your Progress Questions9.9 Summary

9.10 Key Words9.11 Self Assessment Questions and Exercises9.12 Further Readings

UNIT 10 ORDINARY DIFFERENTIAL EQUATIONS 284-29910.0 Introduction10.1 Objectives10.2 Ordinary Differential Equations10.3 Answers to Check Your Progress Questions10.4 Summary10.5 Key Words10.6 Self Assessment Questions and Exercises10.7 Further Readings

UNIT 11 EULER’S METHOD 300-30711.0 Introduction11.1 Objectives11.2 Euler Method11.3 Answers to Check Your Progress Questions11.4 Summary11.5 Key Words11.6 Self Assessment Questions and Exercises11.7 Further Readings

BLOCK IV: TAYLOR’S METHOD, R.K METHOD AND STABILITY ANALYSIS

UNIT 12 TAYLOR’S METHOD 308-31212.0 Introduction12.1 Objectives12.2 Taylor’s Method12.3 Answers to Check Your Progress Questions12.4 Summary12.5 Key Words12.6 Self Assessment Questions and Exercises12.7 Further Readings

UNIT 13 RUNGE KUTTA METHOD 313-32113.0 Introduction13.1 Objectives13.2 Runge Kutta Method13.3 Answers to Check Your Progress Questions13.4 Summary13.5 Key Words13.6 Self Assessment Questions and Exercises13.7 Further Readings

UNIT 14 STABILITY ANALYSIS 322-32814.0 Introduction14.1 Objectives14.2 Stability Analysis14.3 Answers to Check Your Progress Questions14.4 Summary14.5 Key Words14.6 Self Assessment Questions and Exercises14.7 Further Readings

INTRODUCTION

Numerical analysis is the study of algorithms to find solutions for problems ofcontinuous mathematics. It helps in obtaining approximate solutions while maintainingreasonable bounds on errors. Although numerical analysis has applications in allfields of engineering and the physical sciences, yet in the 21st century life sciencesand both the arts have adopted elements of scientific computations. Ordinarydifferential equations are used for calculating the movement of heavenly bodies,i.e., planets, stars and galaxies. Besides, it evaluates optimization occurring inportfolio management and also computes stochastic differential equations to solveproblems related to medicine and biology. Airlines use sophisticated optimizationalgorithms to finalize ticket prices, airplane and crew assignments and fuel needs.Insurance companies too use numerical programs for actuarial analysis. The basicaim of numerical analysis is to design and analyse techniques to compute approximateand accurate solutions to unique problems.

In numerical analysis, two methods are involved, namely direct and iterativemethods. Direct methods compute the solution to a problem in a finite number ofsteps whereas iterative methods start from an initial guess to form successiveapproximations that converge to the exact solution only in the limit. Iterative methodsare more common than direct methods in numerical analysis. The study of errors isan important part of numerical analysis. There are different methods to detect andfix errors that occur in the solution of any problem. Round-off errors occur becauseit is not possible to represent all real numbers exactly on a machine with finitememory. Truncation errors are assigned when an iterative method is terminated ora mathematical procedure is approximated and the approximate solution differsfrom the exact solution.

This book, Numerical Analysis, is divided into four blocks that are furtherdivided into fourteen units which will help you understand how to solvetranscendental and polynomial equations, rate of convergence of iterative methods,methods for finding complex roots – polynomial equations, Birge-Vieta method,Bairstow’s method, Graeffe’s root squaring method, system of linear algebraicequations and eigenvalue problems, error analysis of direct and iteration methods,finding eigenvalues and eigenvectors – Jacobi and power methods, interpolationand approximation, Hermite interpolations, piecewise and spline interpolation,approximation, least square approximation and best approximations, differentiationand integration, numerical differentiation, partial differentiation, ordinary differentialequations, Euler, backward Euler, Taylor’s method, second order Runge Kuttamethods, and stability analysis.

The book follows the Self-Instruction Mode or the SIM format whereineach unit begins with an ‘Introduction’ to the topic followed by an outline of the‘Objectives’. The content is presented in a simple, organized and comprehensiveform interspersed with ‘Check Your Progress’ questions and answers for betterunderstanding of the topics covered. A list of ‘Key Words’ along with a ‘Summary’and a set of ‘Self Assessment Questions and Exercises’ is provided at the end ofthe each unit for effective recapitulation. Logically arranged topics, relevant solvedexamples and illustrations have been included for better understanding of the topics.

NOTES

Self-InstructionalMaterial

Transcendental andPolynomial Equations

NOTES

Self-InstructionalMaterial 1

BLOCK - IPOLYNOMIAL EQUATIONS AND

EIGEN VALUE PROBLEMS

UNIT 1 TRANSCENDENTAL ANDPOLYNOMIAL EQUATIONS

Structure1.0 Introduction1.1 Objectives1.2 Transcendental and Polynomial Equations1.3 Answers to Check Your Progress Questions1.4 Summary1.5 Key Words1.6 Self Assessment Questions and Exercises1.7 Further Readings

1.0 INTRODUCTION

In mathematics, a polynomial is an expression consisting of variables (alsocalled indeterminate) and coefficients, that involves only the operationsof addition, subtraction, multiplication, and non-negative integer exponents ofvariables. An example of a polynomial of a single indeterminate, x, is x2 – 4x + 7.An example in three variables is x3 + 2xyz2 – yz + 1.

Polynomials appear in many areas of mathematics and science. For example,they are used to form polynomial equations, which encode a wide range of problems,from elementary word problems to complicated scientific problems; they are usedto define polynomial functions, which appear in settings ranging frombasic chemistry and physics to economics and social science; they are usedin calculus and numerical analysis to approximate other functions. In advancedmathematics, polynomials are used to construct polynomial rings and algebraicvarieties, central concepts in algebra and algebraic geometry.

In this unit, you will study about transcendental and polynomial equations,and rate of convergence of iterative methods.

1.1 OBJECTIVES

After going through this unit, you will be able to:Understand linear integral equations and some basic identities


NOTES

Self-Instructional2 Material

Reduce initial value problems to Volterra integral equationsKnow the methods of successive approximation and successive substitutionto solve Volterra equations of second kind, iterated kernels and Neumannseries for Volterra equationsExpress resolvent kernel as a series in Know Laplace transform method for a difference kernelFind the solution of a Volterra equation of the first kindReduce boundary value problems to Fredholm integral equationsKnow the method of successive approximation and successive substitutionto solve Fredholm equations of the second kindKnow iterated kernels and Neumann series for Fredholm equationsExpress resolvent kernel as a sum of series and Fredholm resolvent kernelas a ratio of two seriesKnow Fredholm equations with separable kernels, approximation of a kernelby a separable kernel and Fredholm alternative

1.2 TRANSCENDENTAL AND POLYNOMIALEQUATIONS

In mathematics, an integral equation is an equation in which an unknown functionappears under an integral sign.

An integral equation in u(x) is given by,

…(1.1)

where K(x, t) is called the kernel of the integral Equation (1.1) and (x)and β(x) are the limits of integration. It can be easily observed that the unknownfunction u(x) appears under the integral sign. It is to be noted here that both thekernel K(x, t) and the function f(x) in Equation (1.1) are given functions; and isa constant parameter. We have to determine the unknown function u(x) that willsatisfy Equation (1.1).

An integral equation can be classified as a linear or nonlinear integral equation.The most frequently used integral equations fall under two major classes, namelyVolterra and Fredholm integral equations. In this unit we will distinguish followingintegral equations:

Volterra integral equationsFredholm integral equations


NOTES


Volterra Integral Equations

The most standard form of Volterra linear integral equations is of the form

where the limits of integration are function of x and the unknown functionu(x) appears linearly under the integral sign.

If the function (x) = 1, then equation becomes

and this equation is known as the Volterra integral equation of the secondkind; whereas if (x) = 0, then the equation becomes

which is known as the Volterra equation of the first kind.

Fredholm Integral Equations

The most standard form of the Fredholm linear integral equations is givenby,

1.2)

where the limits of integration a and b are constants and the unknown functionu(x) appears linearly under the integral sign. If the function (x) = 1, then Equation(1.2) becomes,

and this equation is called Fredholm integral equation of second kind; whereasif (x) = 0, then Equation (1.2) gives,

which is called Fredholm integral equation of the first kind.

Initial Value Problems Reduced to Volterra Integral Equations

Consider the integral equation,


NOTES


The Laplace transform of f(t) is defined as

L{f(t)} =

Using this definition the above integral equation can be transformed to

In a similar manner if y( ) = then

This is inverted by convolution theorem to give

If

Then . Using the convolution theorem, we getthe Laplace inverse as

Thus the n-fold integrals can be expressed as a single integral as,


NOTES


Method of Successive Approximation to Solve Volterra IntegralEquations of Second Kind

Volterra integral equation of the second kind is of the form,

where K(x, t) is the kernel of the integral equation, f (x) a continuous functionof x and a parameter. Here, f (x) and K(x, t) are the given functions but u(x) isan unknown function that needs to be determined. The limits of integral for theVolterra integral equations are functions of x.

In this method of approximation, we replace the unknown function u(x)under the integral sign of the Volterra equation by any selective real valued continuousfunction u0(x), called the zeroth approximation. This substitution will give the firstapproximation u1(x) by

It is obvious that u1(x) is continuous if f (x), K(x, t) and u0(x) are continuous.The second approximation u2(x) can be obtained similarly by replacing u0(x) inthe above equation by u1(x) obtained above. And we find,

Proceeding in a similar way, we obtain an infinite sequence of functions

that satisfies the recurrence relation

for n = 1, 2, 3, . . . and u0(x) is equivalent to any selected real valuedfunction. The most commonly selected function for u0(x) are 0, 1 and x. Thus, atthe limit, the solution u(x) of the Volterra equation is obtained as,

so that the resulting solution u(x) is independent of the choice of the zerothapproximation u0(x). This process of approximation is extremely simple. However,if we follow the Picard’s successive approximation method, we need to set u0(x)= f (x), and determine u1(x) and other successive approximation as follows:


NOTES


The last equation is the recurrence relation. Consider

Where,

Thus, it can be easily observed that,

if 0(x)=f (x), and further that

where m=1, 2, 3, . . . and hence,


NOTES


The repeated integrals in may be considered as a double integral

over the triangular region; thus interchanging the order of integration, we obtain

Where,

Similarly,

Where the iterative kernels, aredefined by the recurrence formula given by,

Thus, the solution for un(x) can be written as,

Resolvent Kernel as a Series in L

It is also possible that we should be led to the solution of Volterra equation bymeans of the sum if it exists, of the infinite series defined by un(x). Thus, we haveusing m(x)

hence it is also possible that the solution of Volterra equation will be givenby as n


NOTES


Where,

is known as the resolvent kernel.

Laplace Transform Method for A Difference Kernel

Volterra integral equation of convolution type such as

where the kernel K(x–t) is of convolution type, can very easily be solvedusing the Laplace transform method. To begin the solution process, we first definethe Laplace transform of u(x)

Using the Laplace transform of the convolution integral, we have

Thus, taking the Laplace transform of Volterra integral equations ofconvolution type, we obtain

and the solution for L{u(x)} is given by

By inverting this transform, we obtain

where


NOTES


Example 1: Solve the following Volterra integral equation of the second kind ofthe convolution type using (a) the Laplace transform method and (b) successiveapproximation method

Solution: (a) Taking the Laplace transforms, we obtain

and solving for L{u(x)} yields

The Laplace inverse of the above can be written immediately as

where (x) is the Dirac delta.(b) Solution by successive approximationLet us assume that the zeroth approximation is,u0(x) = 0Then the first approximation can be obtained asu1(x) = f (x)Hence, the second approximation is given by

Proceeding in similar manner, the third approximation can be obtained as


NOTES


In the double integration the order of integration is changed to obtain thefinal result. In a similar manner, the fourth approximation u4(x) can be at oncewritten as

Now, as n

Here, the resolvent kernel is H(x, t; ) = e(1+ )(x–t).

Method of Successive Substitution to Solve Volterra IntegralEquations of Second Kind

In this method, we substitute successively for u(x) its value as given by Volterraintegral equation of the second kind. We find that

Where,


NOTES


is the remainder after n terms.Now,

Accordingly, the general series for u(x) can be written as

Example 2: Solve the integral

Solution: By the method of successive substitution, we get

Iterated Kernels and Neumann Series for Volterra Equations

The integral,

where is the

resolvent kernel is the solution of the Volterra integral equation of the second kind,given by

When both and f(x) are continuous then the resolvent kernel canbe constructed in terms of the Neumann series


NOTES


Where is the iterated kernel which is evaluated as,

and .

For showing this, assume the following infinite series form for the solutionu(x),

Substituting this in the Volterra integral equation of the second kind andassuming good convergence which allows the exchange of summation with theintegration operation, we get

Equating coefficients of on both sides, we have

By successive substitution, we get

And


NOTES


Similarly,

So we can now write,

as the general term of the

iterated kernel and

Therefore,


NOTES


Solution of a Volterra Integral Equation of the First Kind

The first kind Volterra equation is usually written as,

If the derivatives exist and arecontinuous, then the solution of this equation is found by reducing it to its secondkind and then proceeding with the methods discussed above.

Differentiating the above Volterra equation and applying Leibnitz rule, weget

If then

The second way to obtain the second kind Volterra integral equation fromthe first kind is by using integration by parts, if we set

Or

By integrating by parts, we have

which reduces to

Giving


NOTES


(0) = 0, and dividing out by K(x, x) we have

In this method the function f(x) is not required to be differentiable. But u(x)must finally be calculated by differentiating the function (x) given by the formula

where H(x, t : 1) is the resolvent kernel corresponding to . To do this

f (x) must be differentiable.

Boundary Value Problems Reduced to Fredholm Integral Equations

A boundary value problem can be converted to an equivalent Fredholm integralequation. But this method is complicated and so is rarely used. This method isdemonstrated with the help of following illustration:

Consider the differential equation

with boundary

conditions

Where and β are given constants. Make the transformation,

Integrating both sides from a to x, we get

Integrating with respect to x from a to x and applying the given boundarycondition at x = a, we get


NOTES


And using the boundary condition at x = b gives,

And the unknown constant is determined as

Hence the solution can be rewritten as,

Therefore,

where and so y(x) can be determined. It is a complicatedprocedure to determine the solution of a Boundary Value Problem (BVP) byequivalent Fredholm equation.

If a = 0 and b = 1, i.e., 0 x 1, then


NOTES


dtdttuyxxyx x

∫ ∫+′+=0 0

)()0()( α

And hence the unknown constant can be determined as

Thus,

Where the kernel is given by,,

It can be easily verified that confirming that the

kernel is symmetric. The Fredholm integral equation is given by u(x).


NOTES


Example 3: Consider the boundary value problem,

Solution: Integrating the equation with respect to x from 0 to x two times yields

To determine the unknown constant we use the condition at x = 1,i.e., y(1) = y1. Hence,

And

Therefore,

Where the kernel is given by,

If we specialize our problem with simple linear BVP y (x) = – y(x), 0 < x< 1 with the boundary conditions y(0) = y0, y(1) = y1, then y(x) reduces to thesecond kind Fredholm integral equation,

where F(x) = y0 + x(y1 – y0). It can be easily verified that K(x, t) =K(t, x) confirming that the kernel is symmetric.


NOTES


Method of Successive Approximation to Solve Fredholm Equations ofSecond Kind

The successive approximation method, which was successfully applied to Volterraintegral equations of the second kind, can also be applied to the basic Fredholmintegral equations of the second kind:

We set u0(x) = f (x). Note that the zeroth approximation can be any selectedreal valued function u0(x), a x b.

Accordingly, the first approximation u1(x) of the solution of u(x) is definedby

The second approximation u2(x) of the solution u(x) can be obtained byreplacing u0(x) by the previously obtained u1(x). Hence we find

This process can be continued in the same manner to obtain the nthapproximation. In other words, the various approximations can be put in a recursivescheme given by

Even though we can select any real valued function for the zerothapproximation u0(x), the most commonly selected functions for u0(x) are u0(x) =0, 1 or x. With the selection of u0(x) = 0, the first approximation u1(x) =f (x). The final solution u(x) is obtained by

so that the resulting solution u(x) is independent of the choice of u0(x). Thisis known as Picard’s method. The Neumann series is obtained if we set u0(x) = f(x) such that


NOTES


Where,

The second approximation u2(x) can be obtained as,

Where,

The final solution u(x) known as Neumann series can be obtained as,

Where,

Example 4: Solve the Fredholm integral equation

by using the successive approximation method.Solution: Let us consider the zeroth approximation is u0(x) = 1, and then the firstapproximation can be computed as


NOTES


Thus,

And

is the solution.

Method of Successive Substitutions to Solve Fredholm Equations ofSecond Kind

This method is almost analogous to the successive approximation method exceptthat it concerns with the solution of the integral equation in a series form throughevaluating single and multiple integrals.

K(x, t) 0, is real and continuous in the rectangle R, for which a x band a t b; f (x) 0 is real and continuous in the interval I, for which a x b; and , a constant parameter.


NOTES


Substituting the value of u(t) in this equation, we get

or

Hence,

The unknown function u(x) is replaced by the known function f(x).Example 5: Use the successive substitutions to solve the Fredholm integral equation

Solution: Here, = 12, f (x) = cos x, and K(x, t) = sin x


NOTES


Iterated Kernels and Neumann Series for Fredholm Equations

The Liouville-Neumann series is defined as,

It is a unique continuous solution of a Fredholm integral equation of thesecond kind, defined as

If the nth iterated kernel is defined as

Then,

And the resolvent kernel is given by

.

Resolvent Kernel as a Sum Of Series

Let the Fredholm equation of the second kind be,

Where the range of the separable kernel

which consists of arbitrary linear combinations of the functions fi is given by,


NOTES


Therefore,

To find uj define,

Consider the algebraic problem . If we replace h by a kernel

cK instead of the separable K then the equation becomes,

where, I is the identity matrix. Let D(c) be the determinant of the matrix Mthen

If determinant D(c) is not zero then the matrix M has an inverse,

Then the solution of the algebraic equation becomes or .

Now by substituting these values of ui in the Fredholm integral equation andthen expressing hi in terms of h(x), we get

Where, is called the resolvent kernel.

Fredholm Resolvent Kernel as a Ratio of Two Series

If is a continuous kernel, not necessarily real then the resolvent kernel

can be expressed as the ratio of two infinite series of powers of such that both of these series converge for all values of .

Expressing the resolvent kernel as a ratio,


NOTES


Where,

And

Here the coefficients Ci and the function can be determined by,,

The solution of the equation given by,

now becomes

This method is preferable only when the kernel is separable.

Fredholm’s Equations with Separable Kernels

This section deals with the study of the homogeneous Fredholm integral equationwith separable kernel given by,

This equation is obtained from the second kind Fredholm equation

Setting f(x) = 0, it is easily seen that u(x) = 0 is a solution which is known asthe trivial solution. The homogeneous Fredholm integral equation with separablekernel may have nontrivial solutions. We shall use the direct computational methodto obtain the solution in this case. Without loss of generality, we assume that


NOTES


So that,

For

We note that = 0 gives the trivial solution u(x) = 0. However, to determinethe nontrivial solution, we need to determine the value of the parameter byconsidering 0. Therefore,

Or

which gives a numerical value for 0 by evaluating the definite integral.

Check Your Progress

1. What is an integral equation?2. Write the standard form of the Volterra equation.3. What is first step in the method of successive approximation to solve

Volterra equations?4. Write the Volterra integral equation of convolution type.5. What are the two methods to reduce a Volterra integral equation of the

first kind to a second kind?6. Express the resolvent kernel as a ratio of two series.

1.3 ANSWERS TO CHECK YOUR PROGRESSQUESTIONS

1. An integral equation is an equation in which an unknown function appearsunder an integral sign.


NOTES


2. The most standard form of Volterra linear integral equations is of the form

3. In successive method of approximation, we first replace the unknown functionu(x) under the integral sign of the Volterra equation by any selective realvalued continuous function u0(x).

4. Volterra integral equation of convolution type is

where the kernel K(x – t) is of convolution type.5. The two methods to reduce a Volterra integral equation of the first kind to a

second kind are differentiating the Volterra equation and applying Leibnitzrule, and by using integration by parts.

6. Expressing the resolvent kernel as a ratio,

where, and

1.4 SUMMARY

An integral equation in u(x) is given by,

where K(x, t) is called the kernel of the integral equation, and (x) and(x) are the limits of integration.

The most frequently used integral equations fall under two major classes,namely Volterra and Fredholm integral equations.In Volterra equation one of the limits of integration is variable while inFredholm equation both the limits are constant.


NOTES


1.5 KEY WORDS

Integral equation: It is an equation in which an unknown function appearsunder an integral sign.

1.6 SELF ASSESSMENT QUESTIONS ANDEXERCISES

Short-Answer Questions

1. Write the two kinds of Volterra integral equations.2. What is the basic difference between Volterra and Fredholm equations?3. List the methods used to solve Fredholm and Volterra integral equations of

the second kind.4. How can you find the solution of the Volterra integral equation of the first

kind?5. Define iterated kernel for Fredholm and Volterra integral equations.

Long-Answer Questions

1. Reduce the following initial value problem to an equivalent Volterra equation:

2. Solve the following Volterra integral equations using methods of successiveapproximation with five approximations with u0(x) = 0:

3. Find the solution of the following Volterra integral equations of the first kind:

(a)


NOTES


(b)

(c)

(d)

4. Reduce the following initial value problem into an equivalent Fredholmequation:

5. Solve the following linear Fredholm integral equations:

(a)

(b)

(c)

(d)

(e)

1.7 FURTHER READINGS

Jain, M. K., S. R. K. Iyengar and R. K. Jain. 2007. Numerical Methods forScientific and Engineering Computation. New Delhi: New AgeInternational (P) Limited.

Atkinson, Kendall E. 1989. An Introduction to Numerical Analysis, 2nd Edition.US: John Wiley & Sons.

Jain, M. K. 1983. Numerical Solution of Differential Equations. New Delhi:New Age International (P) Limited.

Conte, Samuel D. and Carl de Boor. 1980. Elementary Numerical Analysis:An Algorithmic Approach. New York: McGraw-Hill.

Skeel, Robert. D and Jerry B. Keiper. 1993. Elementary Numerical Computingwith Mathematica. New York: McGraw-Hill.

Balaguruswamy, E. 1999. Numerical Methods. New Delhi: Tata McGraw-Hill.Datta, N. 2007. Computer Oriented Numerical Methods. New Delhi: Vikas

Publishing House Pvt. Ltd.

Methods for FindingComplex Roots andPolynomial Equations

NOTES


UNIT 2 METHODS FOR FINDINGCOMPLEX ROOTS ANDPOLYNOMIAL EQUATIONS

Structure2.0 Introduction2.1 Objectives2.2 Methods for Finding Complex Roots2.3 Polynomial Equations2.4 Answers to Check Your Progress Questions2.5 Summary2.6 Key Words2.7 Self Assessment Questions and Exercises2.8 Further Readings

2.0 INTRODUCTION

In mathematics and computing, a root-finding algorithm is an algorithm forfinding zeroes, also called roots, of continuous functions. A zero of a function f,from the real numbers to real numbers or from the complex numbers to the complexnumbers, is a number x such that f(x) = 0. As, generally, the zeroes of a functioncannot be computed exactly nor expressed in closed form, root-finding algorithmsprovide approximations to zeroes, expressed either as floating point numbers oras small isolating intervals, or disks for complex roots (an interval or disk outputbeing equivalent to an approximate output together with an error bound).

In this unit, you will study about the methods for finding complex roots andpolynomial equations.

2.1 OBJECTIVES

After going through this unit, you will be able to:Explain the various methods for finding complex rootsAnalyse the polynomial equations

2.2 METHODS FOR FINDING COMPLEX ROOTS

In this section, we consider numerical methods for computing the roots of anequation of the form,

f (x) = 0 (2.1)

Methods for FindingComplex Roots and

Polynomial Equations

NOTES


Where f (x) is a reasonably well-behaved function of a real variable x. The functionmay be in algebraic form or polynomial form given by,

011

1 ...)( axaxaxaxfn

nn

n ++++=−

− (2.2)

It may also be an expression containing transcendental functions such as cosx, sin x, ex, etc. First, we would discuss methods to find the isolated real roots ofa single equation. Later, we would discuss methods to find the isolated roots of asystem of equations, particularly of two real variables x and y, given by,

f (x, y) = 0 , g (x, y) = 0 (2.3)A root of an equation is usually computed in two stages. First, we find the

location of a root in the form of a crude approximation of the root. Next we usean iterative technique for computing a better value of the root to a desired accuracyin successive approximations/computations. This is done by using an iterativefunction.

Methods for Finding Location of Real RootsThe location or crude approximation of a real root is determined by the use of anyone of the following methods, (i) graphical and (ii) tabulation.

Graphical Method: In the graphical method, we draw the graph of the functiony = f (x), for a certain range of values of x. The abscissae of the points where thegraph intersects the x-axis are crude approximations for the roots of the Equation(2.1). For example, consider the equation,

f (x) = x2 + 2x – 1 = 0From the graph of the function y = f (x), shown in Figure 2.1 we find that it

cuts the x-axis between 0 and 1. We may take any point in [0, 1] as the crudeapproximation for one root. Thus, we may take 0.5 as the location of a root. Theother root lies between – 2 and – 3. We can take – 2.5 as the crude approximationof the other root.

Fig. 2.1 Graph of 122 −+= xxy

In some cases, where it is complicated to draw the graph of y = f (x), we mayrewrite the equation f (x) = 0, as f1(x) = f2(x), where the graphs of y = f1 (x) andy = f2(x) are standard curves. Then we find the x-coordinate(s) of the point(s) of


NOTES


intersection of the curves y = f1(x), and y = f2(x), which is the crude approximationsof the root (s).

For example, consider the equation,

02.132.153 =−− xx

This can be rewritten as,

2.132.153 += xx

Where it is easy to draw the graphs of y = x3 and y = 15.2 x + 13.2. Then, theabscissa of the point(s) of intersection can be taken as the crude approximation(s)of the root(s).

10

20

y x = 3

y x = 15.2 + 13.2

Fig. 2.2 Graph of y = x3 and y = 15.2x + 13.2

Example 1: Find the location of the root of the equation 10log 1.x x =

Solution: The equation can be rewritten as 10

1log .xx

=

Now the curves 101log , and ,y x yx

= = can be easily drawn and are shown inFigure 2.3.

O 1 2 3

y =

y x = log 10

Y

X

1x

Fig. 2.3 Graph of 101 logy and y xx

= =

The point of intersection of the curves has its x-coordinates value 2.5approximately. Thus, the location of the root is 2.5.

Tabulation Method: In the tabulation method, a table of values of f (x) is madefor values of x in a particular range. Then, we look for the change in sign in thevalues of f (x) for two consecutive values of x. We conclude that a real root lies



NOTES


between these values of x. This is true if we make use of the following theorem oncontinuous functions.Theorem 1: If f (x) is continuous in an interval (a, b) and f (a) and f(b) are ofopposite signs, then there exists at least one real root of f (x) = 0, between aand b.

Consider for example, the equation f (x) = x3 – 8x + 5 = 0Constructing the following table of x and f (x)

83251213227)(32101234

−−−−−−−

xfx

We observe that there is a change in sign of f (x) in each of the sub-intervals (–3,–4), (0, 1) and (2, 3). Thus we can take the crude approximation for the three realroots as – 3.2, 0.2 and 2.2.

Methods for Finding the Roots—Bisection and Simple Iteration Methods

Bisection Method: The bisection method involves successive reduction of theinterval in which an isolated root of an equation lies. This method is based upon animportant theorem on continuous functions as stated below.Theorem 2: If a function f (x) is continuous in the closed interval [a, b] and f (a)and f (b) are of opposite signs, i.e., f (a) f (b) < 0, then there exists at least onereal root of f (x) = 0 between a and b.

The bisection method starts with two guess values x0 and x1. Then this interval

[x0, x1] is bisected by a point ),(21

102 xxx += where .0)()( 10


NOTES


Step 5:Check if y0 y1 < 0, then go to Step 6else go to Step 2

Step 6:Compute x2 = (x0 + x1)/2Step 7:Compute y2 = f (x2)Step 8:Check if y0 y2 > 0, then set x0 = x2

else set x1 = x2Step 9:Check if |/)(| 101 xxx − > epsilon, then go to Step 3Step 10: Write x2, y2Step 11: EndNext, we give the flowchart representation of the above algorithm to get a

better understanding of the method. The flowchart also helps in easy implementationof the method in a computer program.

Flow Chart for Bisection AlgorithmBegin

Define ( )f x

Read epsilon

Read , x x0 1

Compute = ( )y f x0 0


No

Yes

Compute = ( + )/2x x x2 0 1


x x0 2 =

x x1 2 =

print ‘root’ = x2

End

No

Yes

YesNo

Is > 1y y0 2

Is|( – ) / |

> epsilonx x x1 0 0

Is > 0y y0 1



NOTES


Example 2: Find the location of the smallest positive root of the equation x3 – 9x+ 1 = 0 and compute it by bisection method, correct to two decimal places.Solution: To find the location of the smallest positive root we tabulate the functionf (x) = x3 – 9x + 1 below:

1921)(3210

−−xfx

We observe that the smallest positive root lies in the interval [0, 1]. Thecomputed values for the successive steps of the bisection method are given in theTable.

053.011718.0125.0109375.07016933.0109375.0125.009375.06

155.009375.0125.00625.05437.00625.0125.004123.0125.025.00323.125.05.00237.35.0101

)( 2210

−

−−−

xfxxxn

From the above results, we conclude that the smallest root correct to twodecimal places is 0.11.

Simple Iteration Method: A root of an equation f (x) = 0, is determined usingthe method of simple iteration by successively computing better and betterapproximation of the root, by first rewriting the equation in the form,

x = g(x) (2.4)Then, we form the sequence {xn} starting from the guess value x0 of the root

and computing successively,

1 0 2 1 1( ), ( ),.., ( )n nx g x x g x x g x −= = =

In general, the above sequence may converge to the root ,as ∞→nξ or itmay diverge. If the sequence diverges, we shall discard it and consider anotherform x = h(x), by rewriting f (x) = 0. It is always possible to get a convergentsequence since there are different ways of rewriting f (x) = 0, in the form x = g(x).However, instead of starting computation of the sequence, we shall first test whetherthe form of g(x) can give a convergent sequence or not. We give below a theoremwhich can be used to test for convergence.Theorem 3: If the function g(x) is continuous in the interval [a, b] which containsa root ξ of the equation f (x) = 0, and is rewritten as x = g(x), and 1|)(| ≤≤′ lxg inthis interval, then for any choice of ],[0 bax ∈ , the sequence {xn} determined bythe iterations,


NOTES


1 ( ), for 0, 1, 2,...k kx g x k+ = = (2.5)

This converges to the root of f (x) = 0.Proof: Since ,ξ=x is a root of the equation x = g(x), we have

)(ξξ g= (2.6)The first iteration gives x1 = g(x0) (2.7)Subtracting Equation (2.7) from Equation (2.6), we get

)()( 01 xggx −=− ξξ

Applying mean value theorem, we can write

ξξξ



NOTES


Step 1: Input x0, epsilon, maxit, where x0 is the initial guess of root, epsilon isaccuracy desired, maxit is the maximum number of iterations allowed.

Step 2:Set i = 0Step 3:Set x1 = g (x0)Step 4:Set i = i + 1Step 5:Check, if |(x1 – x0)/ x1| < epsilon, then print ‘root is’, x1

else go to Step 6Step 6:Check, if i < n, then set x0 = x1 and go to Step 3Step 7:Write ‘No convergence after’, n, ‘iterations’Step 8:End

Example 3: In order to compute a real root of the equation x3 – x – 1 = 0, nearx = 1, by iteration, determine which of the following iterative functions can beused to give a convergent sequence.

(i) x = x3 – 1 (ii) 21

xxx += (iii)

xxx 1+=

Solution:

(i) For the form ,13 −= xx g(x) = x3 – 1, and 23)( xxg =′ . Hence, ,1|)(| >′ xgfor x near 1. So, this form would not give a convergent sequence ofiterations.

(ii) For the form .1)(,1 22 xxxg

xxx +=+= Thus, .13|)1(|and21)( 32 >=′−−=′ gxx

xg

Hence, this form also would not give a convergent sequence of iterations.

(iii)For the form, .11

21)(,1)( 2

21

−⋅

+=′

+=

−

xxxxg

xxxg

.122

1|)1(|


NOTES


For the form (i), 322)(,11)(x

xgx

xg −=′+−= and for x in (0, 1), 1|)(| >′ xg .

So, this form is not suitable. For the form

−−

−

=′ 11

111.

21)( 2x

x

xg (ii) and

1|)(| >′ xg for all x in (0, 1). Finally, for the form (iii)

).1,0(infor1)(and

)1(

1.21)(

23 xxg

x

xg



NOTES


In case of form (ii) 32292)(xx

xxg +−=′ and for x in [2, 4], 1|)(| >′ xg

In case of form (iii) 1|)(|and2

119)( 221


NOTES


)()(

.........

1n

nnn xf

xfxx

′−=+ (2.13)

If the sequence }{ nx converges, we get the root.Algorithm: Computation of a root of f (x) = 0 by Newton-Raphson method.

Step 0:Define f (x), )(xf ′

Step 1: Input x0, epsilon, maxit[x0 is the initial guess of root, epsilon is the desired accuracy of theroot and maxit is the maximum number of iterations allowed]

Step 2:Set i = 0Step 3:Set f0 = f (x0)Step 4:Compute df0 = f ′ (x0)Step 5:Set x1 = x0 – f0/df0Step 6:Set i = i + 1Step 7:Check if |(x1 – x0) |x1| < epsilon, then print ‘root is’, x1 and stop

else if i < n, then set x0 = x1 and go to Step 3Step 8:Write ‘Iterations do not converge’Step 9:End

Example 7: Use Newton-Raphson method to compute the positive root of theequation x3 – 8x – 4 = 0, correct to five significant digits.Solution: Newton-Raphson iterative scheme is given by,

1( ) , for 0, 1, 2, ...( )

nn n

n

f xx x nf x+

= − =′

For the given equation f (x) = x3 – 8x – 4First we find the location of the root by the method of tabulation. The table for

f (x) is,

28112134)(43210

−−−−xfx

Evidently, the positive root is near x = 3. We take x0 = 3 in Newton-Raphsoniterative scheme.

3

1 2

8 43 8

n nn n

n

x xx xx+− −

= −−

We get, 127 24 43 3.0526

27 8x − −= − =

−



NOTES


Similarly, x2 = 3.05138, and x3 = 3.05138Thus, the positive root is 3.0514, correct to five significant digits.

Example 8: Find a real root of the equation 3 27 9 0,x x+ + = correct to fivesignificant digits.Solution: First we find the location of the real root by tabulation. We observe thatthe real root is negative and since f (–7) = 9 > 0 and f (–8) = – 55 < 0, a root liesbetween –7 and – 8.

For computing the root to the desired accuracy, we take x0 = –8 and useNewton-Raphson iterative formula,

3 2

1 2

7 9, for 0, 1, 2, ...

3 14n n

n nn n

x xx x nx x++ +

= − =+

The successive iterations give,x1 = –7.3125x2 = –7.17966x3 = –7.17484x4 = –7.17483

Hence, the desired root is –7.1748, correct to five significant digits.

Example 9: For evaluating ,a deduce the iterative formula ,21

1

+=+

nnn x

axx

by using Newton-Raphson scheme of iteration. Hence, evaluate 2 using this,correct to four significant digits.

Solution: We observe that a is the solution of the equation x2 – a = 0.Now, using f (x) = x2 – a in the Newton-Raphson iterative scheme

2

1 2n

n nn

x ax xx+−

= −

We have,2

1 2n

n nn

x ax xx+−

= −

2

1 2n

n nn

x ax xx+−

= −

i.e., 11 , for 0, 1, 2,...2n n n

ax x nx+

= + =


NOTES


Now, for computing ,2 we assume x0 = 1.4. The successive iterations give,

41421.1414.12414.1

21

414.18.2

96.34.1

24.121

2

1

=

+=

==

+=

x

x

Hence, the value of 2 is 1.414 correct to four significant digits.

Example 10: Prove that k a can be computed by the iterative scheme,

.)1(1 11

+−=

−+ kn

nnx

axkk

x Hence evaluate ,23 connect to five significant digits.

Solution: The value k a is the positive root of xk – a = 0. Thus, the iterative

scheme for evaluating k a is,

1 1

kn

n n kn

x ax xkx

+ −

−= −

1 1

1or, ( 1) , for 0, 1, 2,...n n kn

ax k x nk x+ −

= − + =

Now, for evaluating ,23 we take x0 = 1.25 and use the iterative formula,

1 2

1 223n n n

x xx+

= +

1 2

2 3

1 2We have, 1.25 2 1.263 (1.25)

1.259921, 1.259921

x

x x

= × + =

= =

Hence, 3 2 = 1.2599, correct to five significant digits.

Example 11: Find by Newton-Raphson method, the real root of3x – cos x – 1 = 0, correct to three significant figures.Solution: The location of the real root of f (x) = 3x – cos x – 1 = 0, is [0, 1] sincef (0) = – 2 and f (1) > 0.

We choose x0 = 0, and use Newton-Raphson scheme of iteration.

13 cos 1, 0, 1, 2,...

3 sinn n

n nn

x xx x nx+

− −= − =

+



NOTES


The results for successive iterations are,x1 = 0.667, x2 = 0.6075, x3 = 0.6071

Thus, the root is 0.607 correct to three significant figures.Example 12: Find a real root of the equation xx + 2x – 6 = 0, correct to foursignificant digits.Solution: Taking f (x) = xx + 2x – 6, we have f (1) = –3 < 0 and f (2) = 2 > 0.Thus, a root lies in [1, 2]. Choosing x0 = 2, we use Newton-Raphson iterativescheme given by,

12 6 , for 0, 1, 2,...

(log 1) 2

n

n

xn n

n n xn e n

x xx x nx x+

+ −= − =

+ +

The computed results for successive iterations are,

72308.172321.1

72238.12)12(log4

6442

3

2

21

=

=

=++×

−+−=

xx

xx

e

Hence, the root is 1.723 correct to four significant figures.Order of Convergence: We consider the order of convergence of the Newton-Raphson method given by the formula,

)()(

1n

nnn xf

xfxx

′−=+

Let us assume that the sequence of iterations {xn} converges to the root ξ .Then, expanding by Taylor’s series about xn, the relation f (ξ ) = 0, gives

0...)()(21)()()( 2 =+′′−+′−+ nnnnn xfxxfxxf ξξ

)()(

.)(21

...)(')(

.)(21

)()(

21

2

n

nnn

n

nnn

n

n

xfxf

xx

xfxf

xxxfxf

′

′′−≈−∴

+′′

−+−=′

−∴

+ ξξ

ξξ

Taking n as the error in the nth iteration and writing n = n – , we have,

)()(

21 2

1 ξξεε

ff

nn ′′′

≈+ (2.14)

Thus, ,21 nkn εε =+ where k is a constant.

This shows that the order of convergence of Newton-Raphson method is 2.In other words, the Newton-Raphson method has a quadratic rate of convergence.


NOTES


The condition for convergence of Newton-Raphson method can easily bederived by rewriting the Newton-Raphson iterative scheme as 1 ( )n nx xϕ+ = with

( )( )( )

f xx xf x

ϕ = −′

Hence, using the condition for convergence of the linear iteration method, we

can write 2)]([)()()(

xfxfxfx

′

′′=′ϕ

Thus, the sufficient condition for the convergence of Newton-Raphson methodis,

,1)]([

)()(2



NOTES


(Line AB meets x-axis alone)

Fig. 2.4 Secant Method

Algorithm: To find a root of f (x) = 0, by Secant method.Step 1:Define f (x).Step 2: Input x0, x1, error, maxit. [x0, x1, are initial guess values, error is the

prescribed precision and maxit is the maximum number of iterationsallowed].

Step 3:Set i = 1Step 4:Compute f0 = f (x0)Step 5:Compute f1 = f (x1)Step 6:Compute x2 = (x0 f1 – x1 f0)/(f1 – f0)Step 7:Set i = i + 1Step 8:Compute accy = |x2 – x1| / |x1|Step 9:Check if accy < error, then go to Step 14Step 10: Check if i ≥ maxit then go to Step 16Step 11: Set x0 = x1Step 12: Set x1 = x2Step 13: Go to step 6Step 14: Print “Root =”, x2Step 15: Go to Step 17Step 16: Print ‘iterations do not converge’Step 17: Stop

Regula-Falsi MethodRegula-Falsi method is also a bracketing method. As in bisection method, westart the computation by first finding an interval (a, b) within which a real root lies.Writing a = x0 and b = x1, we compute f (x0) and f (x1) and check if f (x0) and f(x1) are of opposite signs. For determining the approximate root x2, we find the


NOTES


point of intersection of the chord joining the points (x0, f (x0)) and (x1, f (x1)) withthe x-axis, i.e., the curve y = f (x0) is replaced by the chord given by,

)()()(

)( 001

010 xxxx

xfxfxfy −−−

=− (2.16)

Thus, by putting y = 0 and x = x2 in Equation (2.16), we get

)()()(

)(01

01

002 xxxfxf

xfxx −−

−= (2.17)

Next, we compute f (x2) and determine the interval in which the root lies in thefollowing manner. If (i) f (x2) and f (x1) are of opposite signs, then the root lies in(x2, x1). Otherwise if (ii) f (x0) and f (x2) are of opposite signs, then the root liesin (x0, x2). The next approximate root is determined by changing x0 by x2 in thefirst case and x1 by x2 in the second case.

The aforesaid process is repeated until the root is computed to the desiredaccuracy , i.e., the condition

1( ) / ,k k kx x x ε+ − < should be satisfied.

Regula-Falsi method can be geometrically interpreted by the following Figure2.5.

X

Y

x f x1 1, ( )

x f x2 2, ( )

x f x0 2, ( )

O

Fig. 2.5 Regula-Falsi Method

Algorithm: Computing root of an equation by Regula-Falsi method.Step 1:Define f (x)Step 2:Read epsilon, the desired accuracyStep 3:Read maxit, the maximum no. of iterationsStep 4:Read x0, x1 two initial guess values of rootStep 5:Compute f0 = f (x0)Step 6:Compute f1 = f (x1)Step 7:Check if f0 f1 < 0, then go to the next step

else go to Step 4



NOTES


Step 8:Compute x2 = (x0 f1 – x1 f0) / (f1 – f0)Step 9:Compute f2 = f (x2)Step 10: Check if |f2| < epsilon, then go to Step 18Step 11: Check if f2 f0 < 0 then go to the next Step

else go to Step 15Step 12: Set x1 = x2Step 13: Set f1 = f2Step 14: Go to Step 7Step 15: Set x0 = x2Step 16: Set f0 = f2Step 17: Go to Step 7Step 18: Write ‘root =’ , x2, f3Step 19: End

Example 13: Use Regula-Falsi method to compute the positive root of x3 – 3x –5 = 0, correct to four significant figures.Solution: First we find the interval in which the root lies. We observe that f (2) =–3 and f (3) = 13. Thus, the root lies in [2, 3]. For using the Regula–Falsi method,we use the formula,

)()()(

)(01

01

002 xxxfxf

xfxx −−

−=

With x0 = 2, and x1 = 3, we have

1875.2)23(313

322 =−++=x

Again, since f (x2) = f (2.1875) = –1.095, we consider the interval[2.1875, 3]. The next approximation is x3 = 2.2461. Also, f (x3) = – 0.4128. Hence,the root lies in [2.2461, 3]

Repeating the iterations, we getx4 = 2.2684, f (x4) = – 0.1328x5 = 2.2748, f (x5) = – 0.0529x6 = 2.2773, f (x6) = – 0.0316x7 = 2.2788, f (x7) = – 0.0028x8 = 2.2792, f (x8) = – 0.0022

The root correct to four significant figures is 2.279.


NOTES


Check Your Progress

1. How will you compute the roots of the form f (x) = 0?2. Define tabulation method.3. Explain bisection method.4. How is order of convergence determined?5. Explain Newton-Raphson method.6. Define secant method.7. Explain Regula-Falsi method.

2.3 POLYNOMIAL EQUATIONS

Polynomial equations with real coefficients have some important characteristicsregarding their roots. A polynomial equation of degree n is of the form pn(x) =anxn + an–1xn–1 + an–2xn–2 + ... + a2x2 + a1x + a0 = 0.

(i) A polynomial equation of degree n has exactly n roots.(ii) Complex roots occur in pairs, i.e., if βα i+ is a root of pn(x) = 0, then

βα i− is also a root.(iii) Descarte’s rule of signs can be used to determine the number of possible

real roots (positive or negative).(iv) If x1, x2,..., xn are all real roots of the polynomial equation, then we can

express pn(x) uniquely as,))...()(()( 21 nnn xxxxxxaxp −−−=

(v) pn(x) has a quadratic factor for each pair of complex conjugate roots.Let, βα i+ and βα i− be the roots, then )}(2{ 222 βαα ++− xx is thequadratic factor.

(vi) There is a special method, known as Horner’s method of syntheticsubstitution, for evaluating the values of a polynomial and its derivativesfor a given x.

Descarte’s RuleThe number of positive real roots of a polynomial equation is equal to the numberof changes of sign in pn(x), written with descending powers of x, or less by aneven number.

Consider for example, the polynomial equation,5 4 3 23 2 2 2 0x x x x x+ + − + − =



NOTES


Clearly there are three changes of sign and hence the number of positive realroots is three or one. Thus, it must have a real root. In fact, every polynomialequation of odd degree has a real root.

We can also use Descarte’s rule to determine the number of negative roots byfinding the number of changes of signs in pn(–x). For the above equation,

;02223)( 2345 =−−−−+−=− xxxxxxpn and it has two changes of sign. Thus, ithas either two negative real roots or none.

Check Your Progress

8. Define polynomial equations.9. Give the statement of Descarte’s rule.


1. We consider numerical methods for computing the roots of an equation of theform,

f (x) = 0Where f (x) is a reasonably well-behaved function of a real variable x.

2. In the tabulation method, a table of values of f (x) is made for values of x ina particular range. Then, we look for the change in sign in the values of f (x)for two consecutive values of x. We conclude that a real root lies betweenthese values of x.

3. The bisection method involves successive reduction of the interval in whichan isolated root of an equation lies.

The sub-interval in which the root lies is again bisected and the aboveprocess is repeated until the length of the sub-interval is less than the desiredaccuracy.

The bisection method is also termed as a bracketing method, since themethod successively reduces the gap between the two ends of an intervalsurrounding the real root, i.e., brackets the real root.

4. The order of convergence of an iterative process is determined in terms ofthe errors en and en+1 in successive iterations.

5. Newton-Raphson method is a widely used numerical method for finding aroot of an equation f (x) = 0, to the desired accuracy. It is an iterative methodwhich has a faster rate of convergence and is very useful when the expressionfor the derivative f (x) is not complicated. To derive the formula for thismethod, we consider a Taylor’s series expansion of f (x0 + h), x0 being aninitial guess of a root of f (x) = 0 and h is a small correction to the root.


NOTES


6. Secant method can be considered as a discretized form of Newton-Raphsonmethod. The iterative formula for this method is obtained from formula ofNewton-Raphson method on replacing the derivative )( 0xf ′ by the gradientof the chord joining two neighbouring points x0 and x1 on the curve y = f (x).

7. Regula-Falsi method is also a bracketing method. As in bisection method, westart the computation by first finding an interval (a, b) within which a real rootlies. Writing a = x0 and b = x1, we compute f (x0) and f (x1) and check if f (x0)and f (x1) are of opposite signs. For determining the approximate root x2, wefind the point of intersection of the chord joining the points (x0, f (x0)) and (x1,f (x1)) with the x-axis, i.e., the curve y = f (x0) is replaced by the chord givenby,

)()()(

)( 001

010 xxxx

xfxfxfy −−−

=−

8. A polynomial equation of degree n is of the form pn(x) = anxn + an–1xn–1 +an–2xn–2 + ... + a2x2 + a1x + a0 = 0.

9. The number of positive real roots of a polynomial equation is equal to thenumber of changes of sign in pn(x), written with descending powers of x, orless by an even number.

2.5 SUMMARY

A root of an equation is usually computed in two stages. First, we find thelocation of a root in the form of a crude approximation of the root. Nextwe use an iterative technique for computing a better value of the root to adesired accuracy in successive approximations/computations.

Tabulation Method: In the tabulation method, a table of values of f (x) ismade for values of x in a particular range.The bisection method involves successive reduction of the interval in whichan isolated root of an equation lies.If a function f (x) is continuous in the closed interval [a, b] and f (a) and f(b) are of opposite signs, i.e., f (a) f (b) < 0, then there exists at least onereal root of f (x) = 0 between a and b.The bisection method is also termed as a bracketing method, since themethod successively reduces the gap between the two ends of an intervalsurrounding the real root, i.e., brackets the real root.If the function g(x) is continuous in the interval [a, b] which contains a rootξ of the equation f (x) = 0, and is rewritten as x = g(x), and 1|)(| ≤≤′ lxg inthis interval, then for any choice of ],[0 bax ∈ , the sequence {xn} determinedby the iterations,



NOTES


1 ( ), for 0, 1, 2,...k kx g x k+ = =

This converges to the root of f (x) = 0.

Order of Convergence: The order of convergence of an iterative process isdetermined in terms of the errors en and en+1 in successive iterations. An

iterative process is said to have kth order convergence if ,lim 1 Me

ekn

nn


NOTES


3. Define Newton-Rapshon method.4. What is meant by secant method?5. Explain Regula-Falsi method.

Long-Answer Questions

1. Use graphical method to find the location of a real root of the equation x3 +10x – 15 = 0.

2. Draw the graphs of the function f (x) = cos x – x, in the range [0, /2) andfind the location of the root of the equation f (x) = 0.

3. Compute the root of the equation x3 – 9x + 1 = 0 which lies between 2 and 3correct upto three significant digits using bisection method.

4. Compute the root of the equation x3 + x2 – 1 = 0, near 1, by the iterativemethod correct upto two significant digits.

5. Use iterative method to find the root near x = 3.8 of the equation 2x – log10x= 7 correct upto four significant digits.

6. Compute using Newton-Raphson method the root of the equation ex = 4x,near 2, correct upto four significant digits.

7. Use an iterative formula to compute 7 125 correct upto four significant digits.8. Find the real root of x log10x – 1.2 = 0 correct upto four decimal places using

Regula-Falsi method.9. Use Regula-Falsi method to find the root of the following equations correct

upto four significant figures:(i) x3 – 4x – 1 = 0, the root near x = 2

(ii) x6 – x4 – x3 – 1 = 0, the root between 1.4 and 1.510. Compute the positive root of the given equation correct upto four places of

decimals using Newton-Raphson method: x + loge x = 2

2.8 FURTHER READINGS

Jain, M. K., S. R. K. Iyengar and R. K. Jain. 2007. Numerical Methods forScientific and Engineering Computation. New Delhi: New AgeInternational (P) Limited.

Atkinson, Kendall E. 1989. An Introduction to Numerical Analysis, 2nd Edition.US: John Wiley & Sons.

Jain, M. K. 1983. Numerical Solution of Differential Equations. New Delhi:New Age International (P) Limited.



NOTES


Conte, Samuel D. and Carl de Boor. 1980. Elementary Numerical Analysis:An Algorithmic Approach. New York: McGraw-Hill.

Skeel, Robert. D and Jerry B. Keiper. 1993. Elementary Numerical Computingwith Mathematica. New York: McGraw-Hill.

Balaguruswamy, E. 1999. Numerical Methods. New Delhi: Tata McGraw-Hill.Datta, N. 2007. Computer Oriented Numerical Methods. New Delhi: Vikas

Publishing House Pvt. Ltd.

Birge – Vieta, Bairstow’sand Graeffe’s RootSquaring Methods

NOTES


UNIT 3 BIRGE – VIETA,BAIRSTOW’S ANDGRAEFFE’S ROOTSQUARING METHODS

Structure3.0 Introduction3.1 Objectives3.2 Birge – Vieta Method3.3 Bairstow’s Method3.4 Graeffe’s Root Squaring Method3.5 Answers to Check Your Progress Questions3.6 Summary3.7 Key Words3.8 Self-Assessment Questions and Exercises3.9 Further Readings

3.0 INTRODUCTION

In mathematics, a polynomial is an expression consisting of variables (also calledindeterminate) and coefficients that involves only the operations of addition,subtraction, multiplication, and non-negative integer exponents of variables. Thepolynomial of a single indeterminate, x, is x2 – 4x + 7 while in three variables it isx3 + 2xyz2 – yz + 1. A polynomial equation is, therefore, an equation that hasmultiple terms made up of numbers and variables. The degree tells us how manyroots can be found in a polynomial equation. For example, if the highest exponentis 3, then the equation has three roots. The roots of the polynomial equation arethe values of x where y = 0. Principally, the polynomial equation is an equation ofthe form f(x) = 0 where f(x) is a polynomial in x.

The sixteenth century French mathematician Francois Vieta was the pioneerto develop methods for finding approximate roots of polynomial equations. Later,several other methods were developed for solving polynomial equations. Innumerical analysis, Bairstow’s method is an efficient algorithm for finding the rootsof a real polynomial of arbitrary degree. The algorithm first appeared in the appendixof the 1920 book ‘Applied Aerodynamics’ by Leonard Bairstow. The algorithmfinds the roots in complex conjugate pairs using only real arithmetic. The Graeffe’sroot squaring method is a direct method to find the roots of any polynomial equationwith real coefficients. Polynomials are used to form polynomial equations, whichencode a wide range of problems, from elementary word problems to complicatedscientific problems.


NOTES


In this unit, you will study about the Birge-Vieta method, the Bairstow’smethod, and the Graeffe’s root squaring method.

3.1 OBJECTIVES

After going through this unit, you will be able to:Discuss the Birge-Vieta methodUnderstand the Bairstow’s methodElaborate on the Graeffe’s root squaring method

3.2 BIRGE – VIETA METHOD

Birge-Vieta method is used for finding the real roots of a polynomial equations.This method is based on an original method developed by the two Englishmathematicians Birge and Vieta. Finding and approximating the derivation of allroots of a polynomial equation is a very significant. In the field of science andengineering, there are numerous applications which require the solutions of allroots of a polynomial equations for a particular problem.

Newton-Raphson method is fundamentally used for finding the root of analgebraic and transcendental equations. Since the rate of convergence of this methodis quadratic, hence the Newton-Raphson method can be used to find a root of apolynomial equation as polynomial equation is an algebraic equation. Birge-Vietamethod is based on the Newton-Raphson method or this method is a modifiedform of Newton-Raphson method.

Consider the given polynomial equation of degree n, which has the form,Pn(x) = anxn + . . . + alx + a0 = 0.Let x0 be an initial approximation to the root . The Newton-Raphson

iterated formula for improving this approximation is,

To apply this formula, first evaluate both Pn(x) and P n(xi) at any xi. Theutmost natural method is to evaluate,


NOTES


However, this is stated as the most inefficient method of evaluating apolynomial, because of the amount of computations involved and also because ofthe possible growth of round off errors. Thus there must be some proficient andeffective method for evaluating Pn(x) and P n(x).

Vieta’s formula is used for the coefficients of polynomial to the sum andproduct of their roots, along with the products of the roots that are in groups.Vieta’s formula defines the association of the roots of a polynomial by means of itscoefficients. Following example will make the concept clear that how to find apolynomial with given roots.

Here we will discuss about the real-valued polynomials, i.e., the coefficientsof polynomials are real numbers.

Consider a quadratic polynomial. If the given two real roots are r1 and r2,then find a polynomial.

Let the polynomial is a2x2 + a1x + a0. When the roots are given, then wecan also write the polynomial equation in the form, k (x – r1) (x – r2).

Since both the equations denotes the same polynomial, therefore equateboth polynomials as,

a2x2 + a1x + a0 = k (x – r1) (x – r2) (3.1)On simplifying the Equation (3.1), we have the following form of equation,

a2x2 + a1x + a0 = kx2 – k (r1 + r2) x + k (r1r2)Comparing the coefficients of both the sides of the above equation, we

have,For x2, a2 = kFor x, a1 = – k (r1 + r2)For constant term, a0 = k r1r2Which gives,

a2 = kTherefore,

(3.2)

(3.3)

Equations (3.2) and (3.4) are termed as Vieta’s formulas for a second degreepolynomial.

As a general rule, for an nth degree polynomial, there are n different Vieta’sformulas which can be written in a condensed form as,


NOTES


For

Example 1: Find all the roots of the given polynomial equation P3(x) = x3 + x – 3= 0 rounded off to three decimal places. Stop the iteration whenever {xi+1 – xi} <0.0001.Solution: The equation P3(x) = 0 has three roots. Since there is only one changein the sign of the coefficients, the equation can have as a maximum one positivereal root. The equation has no negative real root since P3(–x) = 0 has no change ofsign of coefficients. Since P3(x) = 0 is of odd degree it has at least one real root.Hence the given equation x3 + x – 3 = 0 has one positive real root and a complexpair. Since P(1) = –1 and P(2) = 7, as per the intermediate value theorem theequation has a real root lying in the interval ]1,2[.

Now we will find the real root using Birge-Vieta method. Let the initialapproximation be 1.1.

First Iteration

Therefore x1 = 1.1 – (–0.569)/4.63 = 1.22289Similarly,

x2 = 1.21347x3 = 1.21341

Since, x2 x3 < 0.0001, we stop the iteration here. Hence the requiredvalue of the root is 1.213, rounded off to three decimal places.

Next we will find the deflated polynomial of P3(x). To obtain the deflatedpolynomial, we have to first find the polynomial q2(x) by using the final approximationx3 = 1.213, as shown in the following table.

Here, P3 (1.213) = 0.0022, i.e., the magnitude of the error in satisfyingP3(x3) = 0 is 0.0022.

We then find q2(x) = x2 + 1.213 x + 2.4714 = 0


NOTES


This is a quadratic equation and its roots are given by,

x =

=

= 0.6065 1.4505 i Hence the three roots of the equation rounded off to three decimal places

are 1.213, 0.6065 + 1.4505 i and 0.6065 1.4505 i.

3.3 BAIRSTOW’S METHOD

In numerical analysis, Bairstow’s method is an efficient algorithm for finding theroots of a real polynomial of arbitrary degree. The algorithm was formulated byLeonard Bairstow and first appeared in the appendix of the book ‘AppliedAerodynamics’ (1920). The algorithm finds the roots in complex conjugate pairsusing only real arithmetic.

Bairstow’s approach is to use Newton’s method to adjust the coefficients uand v in the quadratic x2 + ux + v until its roots are also roots of the polynomialbeing solved. The roots of the quadratic may then be determined, and the polynomialmay be divided by the quadratic to eliminate those roots. This process is theniterated until the polynomial becomes quadratic or linear, and all the roots havebeen determined.

Long division of the polynomial to be solved as,

By x2 + ux + v yields a quotient,

And a remainder cx + d such that,

A second division of Q(x) by x2 + ux + v is performed to yield a quotient,


NOTES


And a remainder gx + h with,

The variables c, d, g, h and the {bi}, {fi} are functions of u and v. They canbe found recursively as follows,

The quadratic evenly divides the polynomial when, c (u, v) = d (u, v) = 0Values of u and v for which this occurs can be discovered by picking starting

values and iterating Newton’s method in two dimensions as,

This continues until convergence occurs. This method to find the zeroes ofpolynomials can thus be easily implemented with a programming language or evena spreadsheet.Example 2: The task is to determine a pair of roots of the polynomial, f (x) = 6 x5 + 11 x4 33 x3 33 x2 + 11 x + 6Solution: As first quadratic polynomial we can use the normalized polynomialformed from the leading three coefficients of f(x),

The iteration then produces as shown in the following table.

Iteration Steps of Bairstow’s Method


NOTES


After eight iterations the method produced a quadratic factor that containsthe roots –1/3 and –3 within the represented precision. The step length from thefourth iteration on demonstrates the superlinear speed of convergence.

3.4 GRAEFFE’S ROOT SQUARING METHOD

In mathematics, Graeffe’s method or Dandelin–Lobachesky–Graeffe method isan algorithm typically used for finding all of the roots of a polynomial. It wasdeveloped independently by Germinal Pierre Dandelin in 1826 and Lobachevskyin 1834. In 1837 Karl Heinrich Gräffe also discovered the principal idea of themethod. The method separates the roots of a polynomial by squaring themrepeatedly. This squaring of the roots is done implicitly, that is, only working onthe coefficients of the polynomial. Finally, Viète’s formulas are used in order toapproximate the roots.

Dandelin–Graeffe IterationLet p(x) be a polynomial of degree n,

Then,

Let q(x) be the polynomial which has the squares as itsroots,

Then we can denote as,

Next q(x) can now be computed by algebraic operations on the coefficientsof the polynomial p(x) alone.

Let,


NOTES


Then the coefficients are related by,

Graeffe observed that if one separates p(x) into its odd and even parts,then

We now obtain a simplified algebraic expression for q(x) of the form,

This expression involves the squaring of two polynomials of only half thedegree, and is therefore used in most implementations of the method.

Iterating this procedure several times separates the roots with respect totheir magnitudes. Repeating k times gives a polynomial of degree n, we have:

With roots,

If the magnitudes of the roots of the original polynomial were separated by

some factor >1, that is, , then the roots of the k-th iterate are

separated by a fast growing factor,

Next the Vieta relations are used as Classical Graeffe’s method as shownbelow:

If the roots are sufficiently separated, say by a factor

, then the iterated powers of the


NOTES


roots are separated by the factor , which quickly becomes very big. The

coefficients of the iterated polynomial can then be approximated by their leadingterm,

Implying,

Finally, logarithms are used in order to find the absolute values of the rootsof the original polynomial. These magnitudes alone are already useful to generatemeaningful starting points for other root-finding methods.

Check Your Progress

1. Why the Birge-Vieta method is used?2. How the Bairstow’s approach uses Newton’s method for adjusting the

coefficients u and v in the quadratic x2 + ux + v?3. Explain Graeffe’s method.


1. Birge-Vieta method is used for finding the real roots of a polynomialequations.

2. Bairstow’s approach is to use Newton’s method to adjust the coefficients uand v in the quadratic x2 + ux + v until its roots are also roots of thepolynomial being solved. The roots of the quadratic may then be determined,and the polynomial may be divided by the quadratic to eliminate those roots.This process is then iterated until the polynomial becomes quadratic orlinear, and all the roots have been determined.

3. Graeffe’s method or Dandelin–Lobachesky–Graeffe method is an algorithmtypically used for finding all of the roots of a polynomial. It was developedindependently by Germinal Pierre Dandelin in 1826 and Lobachevsky in1834. The method separates the roots of a polynomial by squaring themrepeatedly. This squaring of the roots is done implicitly, that is, only workingon the coefficients of the polynomial. Finally, Viète’s formulas are used inorder to approximate the roots.


NOTES


3.6 SUMMARY

Birge-Vieta method is used for finding the real roots of a polynomial equations.This method is based on an original method developed by the two Englishmathematicians Birge and Vieta.Finding and approximating the derivation of all roots of a polynomial equationis a very significant. In the field of science and engineering, there are numerousapplications which require the solutions of all roots of a polynomial equationsfor a particular problem.Newton-Raphson method is fundamentally used for finding the root of analgebraic and transcendental equations.Since the rate of convergence of this method is quadratic, hence the Newton-Raphson method can be used to find a root of a polynomial equation aspolynomial equation is an algebraic equation.Birge-Vieta method is based on the Newton-Raphson method or this methodis a modified form of Newton-Raphson method.The most inefficient method of evaluating a polynomial, because of the amountof computations involved and also because of the possible growth of roundoff errors. Thus there must be some proficient and effective method forevaluating Pn(x) and P n(x).Vieta’s formula is used for the coefficients of polynomial to the sum andproduct of their roots, along with the products of the roots that are in groups.Vieta’s formula defines the association of the roots of a polynomial by meansof its coefficients. Following example will make the concept clear that howto find a polynomial with given roots.As a general rule, for an nth degree polynomial, there are n different Vieta’sformulas which can be written in a condensed form as,

For

In numerical analysis, Bairstow’s method is an efficient algorithm for findingthe roots of a real polynomial of arbitrary degree.The algorithm was formulated by Leonard Bairstow. The algorithm findsthe roots in complex conjugate pairs using only real arithmetic.Bairstow’s approach uses Newton’s method to adjust the coefficients uand v in the quadratic x2 + ux + v until its roots are also roots of thepolynomial being solved.


NOTES


The roots of the quadratic may then be determined, and the polynomialmay be divided by the quadratic to eliminate those roots. This process isthen iterated until the polynomial becomes quadratic or linear, and all theroots have been determined.In mathematics, Graeffe’s method or Dandelin–Lobachesky–Graeffemethod is an algorithm typically used for finding all of the roots of apolynomial. It was developed independently by Germinal Pierre Dandelinin 1826 and Lobachevsky in 1834. In 1837 Karl Heinrich Gräffe alsodiscovered the principal idea of the method.The Graeffe’s method separates the roots of a polynomial by squaring themrepeatedly. This squaring of the roots is done implicitly, that is, only workingon the coefficients of the polynomial. Finally, Viète’s formulas are used inorder to approximate the roots.In Graeffe’s method, the logarithms are used in order to find the absolutevalues of the roots of the original polynomial. These magnitudes alone arealready useful to generate meaningful starting points for other root-findingmethods.

3.7 KEY WORDS

Birae-Vieta method: This method is used for finding the real roots of apolynomial equations.Bairstow’s method: This is an efficient algorithm for finding the roots of areal polynomial of arbitrary degree. The algorithm was formulated by LeonardBairstow for finding the roots in complex conjugate pairs using only realarithmetic.Graeffe’s method or Dandelin–Lobachesky–Graeffe method: It isan algorithm typically used for finding all of the roots of a polynomial.

3.8 SELF-ASSESSMENT QUESTIONS ANDEXERCISES

Short-Answer Questions

1. Why is Birge-Vieta metho

numerical analysis162.241.27.72/siteadmin/dde-admin/uploads/4/__pg_m.sc._mathematics.pdfmethods for...

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