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