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CONTENTS 1 FIRST STEP TO NUMERICAL METHODS 1 1 INTRODUCTION 1 2 WHAT ARE THE NUMERICAL METHODS? 1 3 NEED FOR STUDYING NUMERICAL METHODS 1 4 COMPUTER HARDWARE AND SOFTWARE 1 5 ERRORS 1 6 CLOSURE EXERCISES 2 ROOT OF EQUATIONS 2 1 INTRODUCTION 2 2 GRAPHICAL METHOD 2 3 BISECTION METHOD 2 4 FALSE-POSITION METHOD 2 5 ONE-POINT ITERATION METHOD 2 6 NEWTON-RAPHSON METHOD 2 7 SECANT METHOD 2 8 MATLAB FUNCTIONS FOR FINDING ROOT OF EQUATION 2 9 ROOTS OF SYSTEM OF NON-LINEAR EQUATIONS 2 9 1 DIRECT ITERATION METHOD 2 9 2 NEWTON-RAPHSON ITERATION METHOD 2 10 CLOSURE 3 SYSTEM OF LINEAR EQUATIONS 3 1 INTRODUCTION 3 2 CRAMER’S RULE 3 3 GAUSS ELIMINATION METHOD 3 4 PROBLEMS OF GAUSS ELIMINATION METHOD 3 4 1 DIVISION BY ZERO 3 4 2 ROUND-OFF ERROR 3 4 3 ILL-CONDITIONED SYSTEM 3 5 IMPROVED GAUSS ELIMINATION METHOD 3 5 1 PIVOTING 3 5 2 SCALING 3 5 3 TRIDIAGONAL SYSTEM 3 6 GAUSS-JORDAN METHOD 3 7 MATRIX INVERSION METHOD 3 8 SOLVING SYSTEM OF LINEAR EQUATIONS BY MATLAB 3 9 LU DECOMPOSITION METHOD 3 10 MATLAB FUNCTION FOR LU DECOMPOSITION 3 11 CHOLESKY DECOMPOSITION METHOD 3 12 MATLAB FUNCTION FOR CHOLESKY DECOMPOSITION 3 13 JACOBI ITERATION METHOD 3 14 GAUSS-SEIDEL ITERATION METHOD 3 15 SUCCESSIVE OVER-RELAXATION METHOD 3 16 CONJUGATE GRADIENT METHOD 3 17 CLOSURE 4 INTERPOLATION AND EXTRAPOLATION 4 1 INTRODUCTION 4 2 NEWTON’S DIVIDED DIFFERENCES 4 2 1 LINEAR INTERPOLATION 4 2 2 QUADRATIC INTERPOLATION 4 2 3 NTH-ORDER POLYNOMIAL INTERPOLATION 4 3 LAGRANGE INTERPOLATING POLYNOMIALS 4 3 1 LINEAR INTERPOLATION 4 3 2 QUADRATIC INTERPOLATION 4 3 3 POLYNOMIAL INTERPOLATION 4 4 SPLINE INTERPOLATIONS 4 4 1 LINEAR SPLINE 4 4 2 QUADRATIC SPLINE 4 4 3 CUBIC SPLINE 4 5 MATLAB FUNCTIONS FOR INTERPOLATIONS 4 6 EXTRAPOLATION 4 7 CLOSURE 5 LEAST-SQUARES REGRESSION 5 1 INTRODUCTION 5 2 LINEAR REGRESSION 5 3 LINEAR REGRESSION FOR NONLINEAR DATA 5 4 POLYNOMIAL REGRESSION 5 5 MATLAB FUNCTIONS FOR LEAST-SQUARES REGRESSION 5 6 MULTIPLE REGRESSION 5 6 1 LINEAR 5 6 2 POLYNOMIAL 5 7 CLOSURE 6 NUMERICAL INTEGRATION AND DIFFERENTIATION 6 1 INTRODUCTION 6 2 TRAPEZOIDAL RULE 6 3 COMPOSITE TRAPEZOIDAL RULE 6 4 SIMPSON’S RULE 6 5 COMPOSITE SIMPSON’S RULE 6 6 NEWTON-COTES FORMULAS 6 7 ROMBERG INTEGRATION 6 8 GAUSS INTEGRATION 6 9 MULTIPLE INTEGRATION 6 10 MATLAB COMMANDS FOR INTEGRATION 6 11 DIFFERENTIATION 6 12 MATLAB COMMANDS FOR DIFFERENTIATION 6 13 CLOSURE 7 ORDINARY DIFFERENTIAL EQUATIONS 7 1 INTRODUCTION 7 2 EULER’S METHOD 7 3 HEUN’S METHOD 7 4 MODIFIED EULER’S METHOD 7 5 RUNGE-KUTTA METHOD 7 5 1 SECOND-ORDER 7 5 2 THIRD-ORDER 7 5 3 FOURTH-ORDER 7 6 SYSTEM OF EQUATIONS 7 7 MATLAB COMMANDS 7 8 MULTISTEP METHODS 7 8 1 NON-SELF-STARTING HEUN’S METHOD 7 8 2 ADAMS-BASHFORTH METHOD 7 8 3 ADAMS-MOULTON METHOD 7 9 CLOSURE 8 PARTIAL DIFFERENTIAL EQUATIONS 8 1 INTRODUCTION 8 1 1 DEFINITIONS 8 1 2 TYPES OF EQUATIONS 8 1 3 BOUNDARY AND INITIAL CONDITIONS 8 2 ELLIPTIC EQUATION 8 2 1 DIFFERENTIAL EQUATION 8 2 2 COMPUTATIONAL PROCEDURES 8 2 3 EXAMPLE 8 3 PARABOLIC EQUATION 8 3 1 DIFFERENTIAL EQUATION 8 3 2 EXPLICIT METHOD 8 3 3 IMPLICIT METHOD 8 3 4 CRANK-NICOLSON METHOD 8 4 HYPERBOLIC EQUATION 8 4 1 DIFFERENTIAL EQUATION 8 4 2 COMPUTATIONAL PROCEDURES 8 4 3 EXAMPLE 8 5 CLOSURE