Numerical calculus; approximations, interpolation, finite differences, numerical integration and curve fitting.

406 Pages·1949·19.09 MB·English

by  Milne

Numerical Calculus

Numerical Calculus
Approximations, Interpolation,
Finite Differences,
Numerical Integration,
and Curve Fitting
By William Edmund urfilne
1949
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW JERSEY
COPYRIGHT, I 949, BY PRINCETON UNIVERSITY PRESS
LONDON: GEOFFREY CUMBERLEGE, OXFORD UNIVERSITY PRESS
SECOND PRINTING, 1950
PRINTED IN THE UNITED STATES OF AMERICA
PREFACE
The growth of computational facilities in the period
since World War I has already been phenomenal, and the
possibilities in the near future are beyond imagining.
The number of excellent calculating machines now avail
able in almost every office or laboratory, to say nothing
of the amazing Sequence Controlled Calculator, or the
equally marvelous Electronic Numerical Integrator and
Calculator, makes it possible to solve whole categories
of problems that only yesterday were prohibitively diffi
cult. In consequence the subject of numerical analysis
is surely destined to make enormous strides in the decades
to come.
On the other hand our traditional courses in college
and graduate mathematics too often turn out students
poorly trained in the art of translating theoretical
analysis into the concrete numerical results generally
required in practical applications. As a physicist friend
of mine once said, "You mathematicians know how to solve
this problem but you can't actually do it".
The aim of this book is to aid in bridging the
considerable gulf between classroom mathematics and the
numerical applications. It is designed to provide rudi
mentary instruction in such topics as solution of equa
tions, interpolation, numerical integration, numerical
solution of differential equations, finite differences,
approximations by least Squares, smoothing of data, and
simple equations in finite differences. The presentation
is intentionally elementary so that anyone with some
knowledge of calculus and differential equations can read
it understandingly. JVBthematical elegance and rigor have
frequently been sacrificed in favor of a purely naive
treatment.
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PREFACE
From the wealth of material available the decision
to include or to exclude specific topics has posed a
vexing problem, and personal interest rather than object
ive logic has doubtless influenced the choice. It is
hoped that enough has been included to meet a fairly
general need and to stimulate wider study of this
interesting and useful branch of mathematics.
The author is indebted to Mr. James Price for the
calculation of Table VI, to Professor William M. Stone
for some of the material in Article 82, and to Professor
Burns W. Brewer for criticisms of Chapter VIII.
William Edmund Milne, Ph.D., D.Sc.
Professor of PJBthema.tics
Oregon State College
Corvallis, Oregon
M9.rch 1948
vi
CONTENTS
Page
Chapter I .. SIMULTANEOUS LINEAR EQUATIONS .... 3
.1 . Determinants . . . . . . . . 3
2. Solution of Linear Equations by Determinants 7
3. Homogeneous Equations .... 1 1
4. The Method of Elimination. . 1 5
5. Numerical Solution .. of Linear Equations 17
6. Symmetrical Equations ..... . 24
7. Check on Accuracy of Computation 25
8. Evaluation of Determinants . 26
9. Calculation of Cofactors . . 27
1 0. IVl9.gnitude of I:nherent Errors 29
Chapter II. SOLUTION OF EQUATIONS BY SUCCESSIVE
APPROXIIW\TIONS ...• 36
11 . One Equation in One Unknown. 36
12. Exceptional Case.s. . . . 41
13. Simultaneous Equations . 44
14. Successive Substitutions 47
15. Exceptional Cases ....• 49
16. Complex Roots of Algebraic Equations 53
17. Solutions of A-determinants. 57
Chapter III. INTERPOLATION •. 63
18. Interpolating Functions. 63
19. Linear Interpolation 68
20. Aitken's Process of Iteration. 69
21. Neville's Method of Iteration •• 72
22. Inverse Interpolation. •.•.• 73
23. The Error in Polynomial Interpolation. 78
24. Lagrange's Interpolation Formula. • 83
25. Equally Spaced Points .•.•..... 89
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Page
Chapter N. NUMERICAL DIFFERENTIATION AND
INTEGRATION • . • • . • • . • 93
26. Numerical Differentiation . • . • . • . • . • 93
27. Differentiation Formulas for Equally Spaced
Points. . . , . • • • • . 96
28. Numerical Integration ••.• 100
29. Undetermined Coefficients .• 104
30. Investigation of the Error. 1 08
31 • Calculation of the Error. 114
32. The Trapezoidal Rule .• 116
33. Simpson's Rule ••.... 120
34. Newton-Cotes Quadrature Formulas, Closed Type • 122
35. Newton-Cotes Quadrature Formulas, Open Type 126
36. Summary • • • • • • • • • • • • • 127
Chapter V. NUMERICAL SOIDTION OF DIFFERENTIAL
EQUATIONS . 131
37. First Method .•.• 132
38. Second Method . . • 134
39. Equations of Second Order 138
40. Special Formulas for Second-Order Equations 140
41 . Simultaneous Equations. • 141
42. Use of Five-term Formulas .• 142
Chapter VI. FINITE DIFFERENCES. 145
43. Factorial Polynomials .•• 145
44. The Binomial Coefficients 150
45. Finite Differences ...• 154
46. Detection of Errors .•. 158
47. Newton's Binomial Interpolation Formula,.
Forward Differences . • . . . . • . • • 160
48. Newton's Binomial Interpolation Formula,
Backward Differences .•...•.•.•• 161
49. Gauss's Interpolation Formulas .•.•.• 163
50. Central Differences. Stirling's Formula .• 172
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