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
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
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
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
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
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
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
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
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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