Numerical matheatical l analysis/ James B.Scarborough

1. Introduction
2. Approximate Numbers and Significant Figures 2
3. Bounding of Numbers
4. Absolute, Relative, and Percentage Errors
6. Relation between Relative Error and the Number of Significant
Figures ^
6. The General Formula for Errors • , 8
7. Application of the Error Formulas to the Fundamental Opera
tions of Arithmetic and to Logarithms V10 10
8. The Impossibili^, in General, of Obtaining a Result More
Accurate than the Data Used
9. Further Considerations on the Accuracy of a Computed Result 23
10. Accuracy in the Evaluation of a Formula or Complex
Ex
11. Accuracy in the Determination of Arguments from a Tabulated
Function
12. Accuracy of Series Approximations 82
13. Errors in Determinants
14. A Final Remark
Exercises I
CHAPTER H
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 64
19. Differences of a Polynomial.
20. Newton's Formula for Forward Interpolation 66
21. Newton's Formula for Backward Interpolation 59
Exercises II
xiv 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 77
CHAPTER IV
CENTRAL-DIFFERENCE INTERPOLATION FORMULAS
28. Introduction 79
29. Gauss's Central-Difference Formulas 79
30. Stirling's Interpolation Formula 82
31. Bessel's Interpolation Formulas 84
Exercises IV 90
CHAPTER V
INVERSE INTERPOLATION
32. Definition 93
33. By Lagrange's Formula 93
34. By Successive Approximations 9J
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 • 102
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
41. Recapitulation of Formulas for the Remainder
42. Accuracy of Linear Interpolation from Tables
Exercises ^
CHAPTER VII
INTERPOLATION WITH TWO INDEPENDENT VARIABLES
TRIGONOMETRIC INTERPOLATION
43. Introduction t i *
44. Double Interpolation by a Double Application of Single Inter
polation
45. Double or Two-Way Differences
46. A General Formula for Double Interpolation '
47. Trigonometric Interpolation ^
Exercises
CHAPTER VII
NUMERICAL DIFFERENTIATION AND INTEGRATION
I, NUMERICAL DIFFERENTIATION
48. Numerical Differentiation
II. NUMERICAL INTEGRATION
49. Introduction
50. A General Quadrature Formula for Equidistant Ordinates.... 136
51. Simpson's Rule
52. Weddle's Rule
52A. The Trapezoidal Rule
53. Central-Difference Quadrature Formulas 144
54. Gauss's Quadrature Formula
55. Lobatto's Formula 159
56. Tchebycheff's Formula
57. Euler's Formula of Summation and Quadrature 165
58. Caution in the Use of Quadrature Formulas 168
59. Mechanical Cubature
60. Prismoids and the Prismoidal Formula. 176