Numerical matheatical l analysis/ Scarborough,James B.

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?