Introduction to Computational Mathematics: An Outline

213 Pages·2022·10.257 MB·English

by  William C. Bauldry

Introduction to
Computational Mathematics
Textbooks in Mathematics
Series editors:
Al Boggess, Kenneth H. Rosen
ModelingChangeandUncertainty
MachineLearningandOtherTechniques
WilliamP.FoxandRobertE.Burks
AbstractAlgebra
AFirstCourse,SecondEdition
StephenLovett
MultiplicativeDifferentialCalculus
SvetlinGeorgiev,KhaledZennir
AppliedDifferentialEquations
ThePrimaryCourse
VladimirA.Dobrushkin
IntroductiontoComputationalMathematics: AnOutline
WilliamC.Bauldry
https://www.routledge.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH
Introduction to
Computational Mathematics:
An Outline
William C. Bauldry
Firsteditionpublished2022
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DOI:10.1201/9781003299257
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Introduction to Computational Mathematics: An Outline
“1+1 = 3
for large enough
values of 1.”
Photo©WmCB,2015
Introduction to Computational Mathematics: An Outline
Contents
Preface ...................................................viii
I. Computer Arithmetic .......................................1
II. Control Structures .........................................29
S I. Special Topics: Computation Cost and Horner’s Form .......64
III. Numerical Differentiation ..................................70
IV. Root-Finding Algorithms ...................................84
S II. Special Topics: Modified Newton’s Method ................113
V. Numerical Integration ....................................117
VI. Polynomial Interpolation ..................................140
S III. Case Study: TI Calculator Numerics .......................162
VII. Projects .................................................175
VIII. References ...............................................200
Note: Light blue text links inside this Outline; grey text links to a web page.
ICMvi
Introduction to Computational Mathematics: An Outline
Photo©WmCB,2015
On two occasions I have been asked, “Pray, Mr. Babbage, if you put into the
machine wrong figures, will the right answers come out?” ... I am not able
rightly to apprehend the kind of confusion of ideas that could provoke such a
question. — Charles Babbage Passages from the Life of a Philosopher, p. 67.
ICMvii
Preface: Is Computation Important?
A simple computational error can cause very serious problems.
The Mars Climate Orbiter Crash
Two teams used different units of measure when writing the Orbiter’s software:
NASA used the metric system, and Lockheed Martin used the Imperial system.
Read NASA’s Mars Climate Orbiter page.
The Ariane 5 Explosion
The European Space Agency’s Ariane 5 rocket exploded 37 seconds after
takeoff; the explosion was caused by an integer overflow in the software used
for launching the rocket. Watch the launch video.
The Gulf War Dhahran “Scud” Missile Attack
The Patriot missile software measured time in binary and decimal based on the
system clock that used 10th’s of a second, a non-terminating fraction in binary.
After running for over 100 hours, the system was inaccurate. Twenty-eight
U.S. soldiers died when the system failed to track a Scud missile fired from
Iraq. Read Michael Barr’s “Lethal Software Defects: Patriot Missile Failure.”
ICMviii
Preface: Three Factors of Computation
There are three aspects to keep in mind while studying computation. The
first two are always in tension with each other:
Accuracy versus Efficiency
Accuracy concerns how much error occurs
Accuracy Efficiency
and how to control the error.
Efficiency concerns how much computation
Error
is needed to produce a result. Balance
Stability is the third aspect to always keep in mind. In a stable
computation, a small change in inputs produces only a small change in
outputs. Asa counterpoint, investigateinstability ina “Lorenz attractor.”
Startingat[1,1,1]. Startingat[ 1, 2, 3].
− − −
ICMix