Numerical matheatical l analysis/
Scarborough,James B.
- New Delhi: Oxford & IBH,
- 600
2. Approximate Numbers and Significant Figures 2 3. Bounding of Numbers 2 4. Absolute, Relative, and Percentage Errors t 6. Relation between Relative Error and the Number of Significant Figures 4 6. The General Formula for Errors 8 7. Application of the Error Formulas to the Fundamental Opera tions of Arithmetic and to Logarithms 10 8. The Impossibility, in General, of Obtaining a Result More Accurate than the Data Used 20 9. Further Considerations on the Accuracy of a Computed Result 23 10. Accuracy in the Evaluation of a Formula or Complex Ex pression 24 11. Accuracy in the Determination of Arguments from a Tabulated Function 28 12. Accuracy of Series Approximations 32 13. Errors in Determinants 39 14. A Final Remark ^ Exercises I CHAPTER II INTERPOLATION differences. NEWTON'S FORMULAS OF INTERPOLATION 15. Introduction 16. Differences • • • 17. Effect of an Error in a Tabular Value 52 18. Relation between Differences and-Derivatives 54 19. Differences of a Polynomial 54 20. Newton's Formula for Forward Interpolation 56 21. Newton's Formula for Backward Interpolation 59
ziT CONTENTS CHAPTER III INTERPOLATION WITH UNEQUAL INTERVALS OF THE ARGUMENT ARTICLE PAGE 22. Divided Differences 66 23. Tables of Divided Differences 66 24. Symmetry of Divided Differences 67 25. Relation between Divided Differences and Simple Differences.. 68 26. Newton's General Interpolation Formula 70 27. Lagrange's Interpolation Formula 74 Exercises III CHAPTER IV CENTRAL-DIFFERENCE INTERPOLATION FORMULAS 28. Introduction 29. Gauss's Central-Difference Formulas 79 30. Stirling's Interpolation Formula 82 31. Bessel's Interpolation Formulas 84 Exercises IV 32. Definition CHAPTER V INVERSE INTERPOLATION 33. By Lagrange's Formula 34. By Successive Approximations •. 93^ 36. By Reversion of Series 96 Exercises V CHAPTER VI THE ACCURACY OF INTERPOLATION FORMULAS 36. Introduction 102 37. Remainder Term in Newton's Formula (I) and in Lagrange's Formula 38. Remainder Term in Newton's Formula (II) 104 39. Remainder Term in Stirling's Formula 105 40. Remainder Terms in Bessel's Formulas 106 ARTICLE CONTENTS XV 41. Recapitulation of Formulas for the Remainder 107 42. Accuracy of Linear Interpolation from Tables 112 Exercises VT O CHAPTER VII INTERPOLATION WITH TWO INDEPENDENT VARIABLES 43. Introduction TRIGONOMETRIC INTERPOLATION 44. Double Interpolation by a Double Application of Single Inter polation 45. Double or Two-Way Differences 46. A General Formula for Double Interpolation • 122 47. Trigonometric Interpolation 130 Exercises VII 13 CHAPTER VII NUMERICAL DIFFERENTIATION AND INTEGRATION I. NUMERICAL DIFFERENTIATION 48. Numerical Differentiation 133 II. NUMERICAL INTEGRATION 49. Introduction 136 50. A General Quadrature Formula for Equidistant Ordinates.... 136 61. Simpson's Rule 137 52. Weddle's Rule 138 62A. The Trapezoidal Rule /. 142 53. Central-Difference Quadrature Formulas 144 54. Gauss's Quadrature Formula • • 152 55. Lobatto's Formula. 159 56. Tchebycheff's Formula 16? 57. Euler's Formula of Summation and Quadrature 165 58. Caution in the Use of Quadrature Formulas 168 59. Mechanical Cubature 172 60. Prismoids and the Prismoidal Formula 176 Exercises VIII 180 xvi CONTENTS CHAPTER IX THE ACCURACY OF QUADRATURE FORMULAS ARTICLE PAOE 61. IntRoduction 183 62. Formulas for the Inherent Error in Simpson's Rule 183 63. The Inherent Error in Weddle's Rule 189 64. The Remainder Terms in Central-Difference Formulas (53.1) and (53.3)...; 189 65. The Inherent Errors in the Formulas of Gauss, Lobatto, and Tchebycheff 191 66. The Remainder Term in Euler's Formula 192 Exercises IX 193 CHAPTER X THE SOLUTION OF NUMERICAL ALGEBRAIC AND TRANSCENDENTAL EQUATIONS I. EQUATIONS IN ONE UNKNOWN 67. Introduction 194 68. Finding Approximate Values of the Roots 194 68A. Finding Roots by Repeated Application of Location Theorem 195 69. The Method of Interpolation, or of False Position (Regula Falsi) 197 70. Solution by Repeated Plotting on a Larger Scale 199 71. The Newton-Raphson Method 201 72. Geometric Significance of the Newton-Raphson Method 203 73. The Inherent Error in the Newton-Raphson Method 205 74. A Special Procedure for Algebraic Equations 207 75. The Method of Iteration 208 76. Geometry of the Iteration Process 210 77. Convergence of the Iteration Process 211 78. Convergence of the Newton-Raphson Method 212 79. Errors in the Roots due to Errors in the Coefficients and Con stant Term 213 II. SIMULTANEOUS EQUATIONS IN SEVERAL UNKNOWNS 80. The newton-Raphson Method for Simultaneous Equations 215 81. This Method of Iteration for Simultaneous Equations 219 ^^2. Convergence of the Iteration Process in the *Case of Several. Unknowns 221 Exercises X 223 CONTENTS xvii CHAPTER XI GRAEFFE'S ROOT-SQUARING METHOD FOR SOLVING ABTIOLB 83. Introduction ALGEBRAIC EQUATIONS 84. Principle of the Method 85. The Root-Squaring Process 86. Case I. Roots Real and Unequal 228 87. A Check on the Coefficients in the Root-Squared Equation 232 88. Case II. Complex Roots 89. Case III. Roots Real and Numerically Equal 243 90. Brodetsky and SmeaPs Improvement of Graeffe's Method 245 91. Improving the Accuracy of the Roots 257 Exercises XI CHAPTER XII NUMERICAL SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS L SOLUTION BY DETERMINANTS 92. Evaluation of Numerical Determinants 260 93. Cramer's Rule 266 II. SOLUTION BY SUCCESSIVE ELIMINATION OP THE UNKNOWNS 94. The Method of Division by the Leading Coefficients 269 95. The Method of Gauss 272 96. Another Version of the Gauss Method 274 in. SOLUTION BY INVERSION OP MATRICES 97. Definitions 277 98. Addition and Subtraction of Matrices 278 99. Multiplication of Matrices 279 100. Inversion of Matrices ' 284 101. Solution of Equations by Matrix Methods 296 IV. SOLUTION BY ITERATION 102. Systems Solvable by Iteration 297 103. Conditions for the Convergence of the Iteration Process 301 jcviii CONTENTS ARTICLE PAGE 104. Errors in the Solutions when the Coefficients and Constant Terms are Subject to Errors 303 Exercises XII 307 CHAPTER XIII THE NUMERICAL SOLUTION OF ORDINARY 105. Introduction DIFFERENTIAL EQUATIONS I. EQUATIONS OP THE FIRST ORDER 106. Euler's Method and Its Modification 310 107. Picard's Method of Successive Approximations 316 108. Use of Approximating Polynomials 320 109. Methods of Starting the Solution 327 110. Halving the Interval for 334 Exercises XIII II. EQUATIONS OP THE SECOND ORDER AND SYSTEMS .OP SIMULTANEOUS EQUATIONS 111. Equations of the Second Order ' 337 112. Second-Order Equations with First Derivative Absent 342 113. Systems of Simultaneous Equations 348 114. Conditions for Convergence 350 III. OTHER METHODS OP SOLVING DIFFERENTIAL Milne's Method EQUATIONS NUMERICALLY The Runge-Kutta Method 35B Checks, Errors, and Accuracy i - 367 Some General Remarks Exercises XIV IV. THE DIFFERENTIAL EQUATIONS OP EXTERIOR BALLISTICS 119. The Simplest Case—Flat EartH with Constant Acceleration of Gravity 120. The General Case, Allowing for Variation in Air Density with Altitude 121. Methods of Finding the Starting Values 375 exerciSE XV ARTICLE 00NTE1JT8 CHAPTER XIV THE NUMERICAL SOLUTION OF PARTIAL 122. Introduction DIFFERENTIAL EQUATIONS I. DIFFERENCE QUOTIENTS AND DIFFERENCE EQUATIONS 123. Difference Quotients 392 124. Difference Equations 394 II. THE METHOD OF ITERATION 125. Solution of Difference Equations by Iteration 396 1216. The Inherent Error in the Solution by Difference Equations.. 406 127. Applications of Conformal Transformation to Certain Problems 407 III. THE METHOD OF RELAXATION 128. Solution of Difference Equations by Relaxation 410 129. Triangular Networks 415 130. Block Relaxation 416 131. The Iteration and Relaxation Methods Compared 420 lY. THE RAYLEIGH-RITZ METHOD 132. Introduction 422 133. The Vibrating String 423 134. Vibration of a Rectangular Membrane 430 135. Comments on the Three Methods 435 CHAPTER XV NUMERICAL SOLUTION OF INTEGRAL EQUATIONS 136. Integral Equations—Definitions 437 137. Boundary-Value Problems of Ordinary Differential Equations. Green's Functions "• > 438 138. Linear Integral Equations 445 139. Non-Linear Integral Equations and Boundary-Value Problems 4 XX CONTENTS CHAPTER XVI THE NORMAL LAW OF ERROR AND THE PRINCIPLE OF LEAST SQUARES ARTICLE PAOE 140. Errors of Observations and Measurements 460 141. The Law of Accidental Errors 460 142. The Probability of Errors Lying between Given Limits 462 143. The Probability Equation 464 144. The Law of Error of a Linear Function of Independent Quan tities 468 145. The Probability Integral and Its Evaluation 473 146. The Probability of Hitting a Target 476 147. The Principle of Least Squares 481 148. Weighted Observations 482 149. Residuals 484 150. The Most Probable Value of a Set of Direct Measurements 485 151. Law of Error for Residuals 487 152. Agreement between Theory and Experience 491 Exercises XVI 492 CHAPTER XVII THE PRECISION OF MEASUREMENTS 153. Measurement, Direct and Indirect • 493 154. Precision and Accuracy 493 I. DIRECT MEASUREMENTS 155. Measures of Precision 494 156. Relations between the Precision Measures 496 157. Geometric* Significance of /*, r, and 497 158. Relation between Probable Error and Weight, and the Probable Error of the Arithmetic^ and Weighted Means 499 159. Computation of the Precision Measures from the Residuals.... 500 160. The Combination of Sets of Measurements when the p.e/s of Sets Are Given 503 Exercises XVII 609 II. INDIRECT MEASUREMENTS 161. The Probable Error of any Function of Independent Quantities ' Whose P.B.'s are Known 510 ABTIO LB CONTENTS 162. The Two Fundamental Problems of Indirect Measurements... 513 163. Rejection of Observations and Measurements 519 Exercises XVIII 520 164. Introduction CHAPTER XVII EMPIRICAL FORMULAS 165. The Graphic Method, or Method of Selected Points 522 166. The Method of Averages. 5^® 166. 167. The Method of Least Squares 5^^ 168. Weighted Residuals 541 169. Non-Linear Formulas—The General Case 545 170. Determination of the Constants when Both Variables Are Sub ject to Error... 551 171. Finding the Best Type of Formula 554 172. Smoothing of Observational and Experimental Data 556 Exercises XIX 562 CHAPTER XIX HARMONIC ANALYSIS OF EMPIRICAL FUNCTIONS 178. Introduction 174. Case of 12 Ordinates 564 176. Case of 24 Ordinates 176. Periods other than Chapter 1: Global Prospects and Policies Chapter 2: Country and Regional Perspectives Chapter 3: The Dog That Didn't Bark: Has Inflation Been Muzzled or Was It Just Sleeping? Chapter 4: Breaking through the Frontier: Can Today's Dynamic Low-Income Countries Make It?