NUMERICAL AND STATICAL METHODS

CONTENTS

Chapter 1 Introduction

Introduction to Computers

Definitions

Introduction to “C” Language

Advantages/Features of ‘C’ language

‘C’ Character Set

‘C’ Constants

“C” Variables

‘C’ Key Words

“C Instructions”

Hierarchy of Operations

Escape Sequences

Basic Structure of ‘‘C’’ Program

Decision Making Instructions in “C”

Loop Control Structure

Arrays and String

Pointers

Structure and Unions

Storage Classes in ‘C’

Chapter 2 Errors

Errors and Their Analysis

Accuracy of Numbers

Errors

A General Error Formula

Errors in Numerical Computations

Inverse Problems

Error in a Series Approximation

Mathematical Preliminaries

Floating Point Representation of Numbers

Arithmetic Operations with Normalized Floating Point Numbers

Machine Computation

Computer Software

Chapter 3 Algebraic and Transcendental Equations

Bisection (or Bolzano) Method

Algorithm

Flow-Chart

Program Writing

Order of Convergence of Iterative Methods

Order of Convergence of Bisection Method

Convergence of a Sequence

Prove that Bisection Method Always Converges

Program to Implement Bisection Method

Iteration Method—(Successive Approximation Method)

Sufficient Condition for Convergence of Iterations

Theorem

Convergence of Iteration Method

Algorithm for Iteration Method

Flow-Chart for Iteration Method

Computer Program

The Method of Iteration for System of Non-Linear Equations

Method of False Position or Regula-Falsi Method

Algorithm

Flow-Chart

Convergence of Regula-Falsi Method

Secant Method

Lin-Bairstow’s Method or Method for Complex Root

Muller ’s Method

Algorithm of Muller ’s Method

Flow-Chart for Muller ’s Method

The Quotient-Difference Method

Horner ’s Method

Newton-Raphson Method

Convergence

Order of Convergence

Geometrical Interpretation

Algorithm of Newton-Raphson Method

Flow-Chart of Newton–Raphson Method

Newton’s Iterative Formulae for Finding Inverse, Square Root

Rate of Convergence of Newton’s Square Root Formula

Rate of Convergence of Newton’s Inverse Formula

Definitions

Methods for Multiple Roots

Nearly Equal Roots

Comparison of Newton’s Method with Regula-Falsi Method

Comparison of Iterative Methods

Graeffe’s Root-Squaring Method

Ramanujan’s Method

Chapter 4 Interpolation

Introduction

Assumptions for Interpolation

Errors in Polynomial Interpolation

Finite Differences

Other Difference Operators

Relation Between Operators

Differences of a Polynomial

Factorial Notation

To Show that (i) Δn[x]n= n! (ii) Δn+1[x]n= 0

Reciprocal Factorial

Missing Term Technique

Method of Separation of Symbols

Detection of Errors by Use of Difference Tables

Newton’s Formulae for Interpolation

Newton’s Gregory Forward Interpolation Formula

Newton’s Gregory Backward Interpolation Formula

Central Difference Interpolation Formulae

Gauss’ Forward Difference Formula

Gauss’ Backward Difference Formula

Stirling’s Formula

Bessel’s Interpolation Formula

Laplace-Everett’s Formula

Interpolation by Unevenly Spaced Points

Lagrange’s Interpolation Formula

Error in Lagrange’s Interpolation Formula

Expression of Rational Function as a Sum of Partial Fractions

Inverse Interpolation

Divided Differences

Properties of Divided Differences

Newton’s General Interpolation Formula or Newton’s Divided

Difference Interpolation Formula

Relation Between Divided Differences and Ordinary

Differences

Merits and Demerits of Lagrange’s Formula

Hermite’s Interpolation Formula

Chapter 5 Numerical Integration and Differentiation

Introduction

Numerical Differentiation

Formulae for Derivatives

Maxima and Minima of a Tabulated Function

Errors in Numerical Differentiation

Numerical Integration

Newton-cote’s Quadrature Formula

Trapezoidal Rule (n= 1)

Simpson’s One-third Rule (n= 2)

Simpson’s Three-Eighth Rule (n= 3)

Boole’s RuleWeddle’s Rule (n= 6)

Algorithm of Trapezoidal Rule

Flow-Chart for Trapezoidal Rule

Program to Implement Trapezoidal Method of

Numerical Integration

Output

Algorithm of Simpson’s 3/8th Rule

Flow-Chart of Simpson’s 3/8th Rule

Program to Implement Simpson’s 3/8th Method of

Numerical Integration

Output

Algorithm of Simpson’s 1/3rd Rule

Flow-Chart of Simpson’s 1/3rd Rule

Program to Implement Simpson’s 1/3rd Method

of Numerical Integration Output Euler

-Maclaurin’s Formula

Gaussian Quadrature Formula

Numerical Evaluation of Singular Integrals

Evaluation of Principal Value Integrals

Chapter 6 Numerical Solution of Ordinary Differential Equations

Introduction

Initial-Value and Boundary-Value Problems

Single Step and Multi-Step Methods

Comparison of Single-Step and Multi-Step Methods

Numerical Methods of Solution of O.D.E.

Picard’s Method of Successive Approximations

Picard’s Method for Simultaneous First Order Differential

Equations

Euler ’s Method

Algorithm of Euler ’s Method

Flow-Chart of Euler ’s Method

Program of Euler ’s Method

Modified Euler ’s Method

Algorithm of Modified Euler ’s Method

Flow-Chart of Modified Euler ’s Method

Program of Modified Euler ’s Method

Taylor ’s Method

Taylor ’s Method for Simultaneous I Order Differential

Equations

Runge-Kutta Methods

Fourth Order Runge-Kutta Method

Runge-Kutta Method for Simultaneous First Order Equations

Predictor-Corrector Methods

Milne’s Method

Adams-Moulton (or Adams–Bashforth) Formula

Stability

Stability in the Solution of Ordinary Differential Equations

Stability of I Order Linear Differential Equation ofFormdy/dx=AywithInitial Condition y(x0) = y

Chapter 7 Statistical Computation

The Statistical Methods

Limitation of Statistical Methods

Frequency Charts

Graphical Representation of a Frequency Distribution

Types of Graphs and Diagrams

Histograms

Frequency Polygon

Frequency Curve

Cumulative Frequency Curve or the Ogive

Types of Frequency Curves

Diagrams

Curve Fitting

Principle of Least Squares

Fitting a Straight Line

Algorithm for Fitting a Straight Line of the Form y= a+bx

for a Given Set of Data Points

Flow-Chart for Fitting a Straight Line y= a+ bx for a Given

Set of Data Points

Program to Implement Curve Fitting to Fit a Straight Line

Fitting of an Exponential Curve y= aebx Fitting of the Curve y=axb

Fitting of the Curve y= abxFitting of the Curve pvr= k

Fitting of the Curve of Type xy= b+ ax

Fitting of the Curve y= ax2+b/x

Fitting of the Curve y= ax+ bx2

Fitting of the Curve y= ax+ b/x

Fitting of the Curve y= a+ b/x+ c/x2

Fitting of the Curve y= c0/x+ c1x

Fitting of the Curve 2x= ax2+ bx+ c

Most Plausible Solution of a System of Linear Equations

Curve-Fitting by Sum of Exponentials

Spline Interpolation

Spline Function

Cubic Spline Interpolation

Steps to Obtain Cubic Spline for Given Data

Approximations

Legendre and Chebyshev Polynomials

Legendre Polynomials

Chebyshev Polynomials

Special Values of Chebyshev Polynomials

Orthogonal Properties

Recurrence Relations

Aliter to Find Chebyshev PolynomialsExpression of Powers of xin terms of Chebyshev Polynomials

Properties of Chebyshev Polynomials

Chebyshev Polynomial Approximation

Lanczos

Economization of Power Series for a General Function

Regression Analysis

Curve of Regression and Regression Equation

Linear Regression

Lines of Regression

Derivation of Lines of Regression

Use of Regression Analysis

Comparison of Correlation and Regression Analysis

Pr

operties of Regression Co-efficients

Angle between Two Lines of Regression

Algorithm for Linear Regression

Program to Implement Least Square Fit of a Regression Lineof yon x

Program to Implement Least Square Fit of a Regression Lineof xon y

Polynomial Fit: Non-linear Regression

Multiple Linear Regression

Statistical Quality Control

Advantages of Statistical Quality Control

Reasons for Variations in the Quality of a Product

Techniques of Statistical Quality Control

Control Chart

Objectives of Control Charts

Construction of Control Charts for Variables

Control Charts for Attributes

Chapter 8 Testing of Hypothesis

Population or Universe

Sampling

Parameters of Statistics

Standard Error

Test of Significance

Critical Region

Level of Significance

Errors in Sampling

Steps in Testing of Statistical Hypothesis

Test of Significance for Large Samples

Test of Significance of Small Samples

Student’s t-DistributionTest I: t-test of Significance of the Mean of a Random SampleTest II: t-test for Difference of Means of Two Small Samples

(From a Normal Population)

Snedecor’s Variance Ratio Test or F-test

Chi-square (χ2) Test The χ2Distribution

χ2Test as a Test of Goodness of Fit

χ2Test as a Test of Independence

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