Numerical Methods

Numerical Methods......Page 2 Copyright......Page 3 Contents......Page 6 Preface......Page 10 Approximate Numbers and Significant Figures......Page 12 Classical Theorems Used in Numerical Methods......Page 13 Types of Errors......Page 15 General Formula for Errors......Page 16 Order of Approximation......Page 18 Approximate Values of The Roots......Page 22 Bisection Method......Page 23 Regula–Falsi Method......Page 26 Convergence of Regula–Falsi Method......Page 27 Newton–Raphson Method......Page 31 Square Root of a Number Using Newton–Raphson Method......Page 34 Order of Convergence of Newton–Raphson Method......Page 35 Fixed Point Iteration......Page 36 Convergence of Iteration Method......Page 37 Square Root of a Number Using Iteration Method......Page 38 Sufficient condition for the convergenceof newton–raphson method......Page 39 Newton’s Method for Finding Multiple Roots......Page 40 Newton–Raphson Method for Simultaneous Equations......Page 43 Graeffe’s Root Squaring Method......Page 48 Muller’s Method......Page 52 Bairstow Iterative Method......Page 56 Direct Methods......Page 62 Iterative Methods for Linear Systems......Page 81 The Method of Relaxation......Page 90 Ill-Conditioned System of Equations......Page 93 Eigenvalues and Eigenvector......Page 96 The Power Method......Page 99 Jacobi’s Method......Page 105 Given’s Method......Page 112 Eigenvalues of a Symmetric Tri-Diagonal Matrix......Page 126 Bounds on Eigenvalues......Page 128 Finite Differences......Page 133 Factorial Notation......Page 141 Some More Examples of FiniteDifferences......Page 143 Error Propagation......Page 150 Interpolation......Page 154 Use of Interpolation Formulae......Page 173 Interpolation with Unequal-SpacedPoints 163......Page 174 9 Newton’s Fundamental (Divided Difference)Formula......Page 175 Error Formulae......Page 179 Lagrange’s Interpolation Formula......Page 182 Error in Lagrange’s Interpolation Formula......Page 190 Hermite Interpolation Formula......Page 191 Throwback Technique......Page 196 Inverse Interpolation......Page 199 Chebyshev Polynomials......Page 206 Approximation of a Functionwith a Chebyshev Series......Page 209 Interpolation by Spline Functions......Page 211 Existence of Cubic Spline......Page 213 Least Square Line Approximation......Page 224 The Power Fit y = a x m 219......Page 230 Least Square Parabola......Page 232 Centered Formula of Order O(h2)......Page 239 Centered Formula of Order O(h4)......Page 240 Error Analysis......Page 241 Richardson’s Extrapolation......Page 242 Central Difference Formula of Order O(h4) forf ′′(x)......Page 245 General Method for Deriving DifferentiationFormulae......Page 246 Differentiation of a Function Tabulated inUnequal Intervals......Page 255 Differentiation of Lagrange’sPolynomial......Page 256 Differentiation of Newton Polynomial......Page 257 General Quadrature Formula......Page 263 Cote’s Formulae......Page 267 Error Term in Quadrature Formula......Page 269 Richardson Extrapolation......Page 274 Simpson’s Formula with End Correction......Page 276 Romberg’s Method......Page 278 Euler–Maclaurin Formula......Page 288 Double Integrals......Page 290 Definitions and Examples......Page 299 Homogeneous Difference Equation with ConstantCoefficients......Page 300 Particular Solution of a DifferenceEquation......Page 304 Taylor Series Method......Page 312 Euler’s Method......Page 316 Picard’s Method of Successive Integration......Page 323 Heun’s Method......Page 326 Runge–Kutta Method......Page 328 Runge–Kutta Method for System of First Order Equations......Page 337 Runge–Kutta Method for Higher Order Differential Equations......Page 339 Explicit Multistep Methods......Page 341 Implicit Multistep Methods......Page 343 Milne–Simpson’s Method......Page 347 Stability of Methods......Page 355 Stability of Milne’s Method......Page 356 Second Order Differential Equation......Page 360 Solution of Boundary Value Problems byFinite Difference Method......Page 363 Eigenvalue Problems......Page 368 Use of the Formula......Page 366 Formation of Difference Equation......Page 374 Geometric Representation of PartialDifference Quotients......Page 375 Standard Five Point Formula and DiagonalFive Point Formula......Page 376 Gauss–Seidel Method......Page 377 Solution of Elliptic Equation by RelaxationMethod......Page 387 Poisson’s Equation......Page 390 Eigenvalue Problems......Page 394 Parabolic Equations......Page 400 Iterative Method to Solve Parabolic Equations......Page 410 Hyperbolic Equations......Page 413 C Tokens......Page 423 Library Functions......Page 427 Input Operation......Page 428 Output Operation......Page 429 Control (Selection) Statements......Page 430 Structure of a C Program......Page 434 Program in C Demonstrating Bisection Method......Page 436 Program in C Demonstrating Newton–Raphson Method......Page 437 Program in C Demonstrating Gauss Elimination Method......Page 439 Program in C Demonstrating Gauss–Jordan Method......Page 440 Program in C demonstrating Gauss–Seidel Method......Page 441 Program in C Demonstrating Lagrange’s Interpolation Method......Page 443 Program in C Demonstrating Least Square Method to Fit aStraight Line to a Given Data......Page 444 Program in C Demonstrating Least Square Method to Fit a Parabolato a Given Data......Page 446 Program in C Demonstrating Trapezoidal Rule......Page 448 Program in C Demonstrating Simpson’s 1/3 Rule......Page 449 Program in C Demonstrating Simpson’s 3/8 Rule......Page 450 Program in C Demonstrating Euler’s Method......Page 451 Program in C Demonstrating Runge–Kutta Method......Page 452 Program in C Demonstrating Milne–Simpson’s Method......Page 453 Program Demonstrating solution of Laplace’s Equation......Page 454 Program in C Demonstrating Bender–Schmidt Method to SolveOne-Dimensional Heat Equation......Page 457 Model Paper I......Page 460 Model Paper II......Page 470 Model Paper III......Page 477 Model Paper IV......Page 487 Model Paper V......Page 497 Bibliography......Page 508 Index......Page 510 Numerical Methods is a mathematical tool used by engineers and mathematicians to do scientific calculations. It is used to find solutions to applied problems where ordinary analytical methods fail. This book is intended to serve for the needs of courses in Numerical Methods at the Bachelors' and Masters' levels at various universities