An Introduction to Modern Mathematical Computing: With Mathematica®

by Jonathan M. Borwein

Thirty years ago mathematical, as opposed to applied numerical, computation was difficult to perform and so relatively little used. Three threads changed the emergence of the personal computer; the discovery of fiber-optics and the consequent development of the modern internet; and the building of the Three “M’s” Maple, Mathematica and Matlab. We intend to persuade that Mathematica and other similar tools are worth knowing, assuming only that one wishes to be a mathematician, a mathematics educator, a computer scientist, an engineer or scientist, or anyone else who wishes/needs to use mathematics better. We also hope to explain how to become an "experimental mathematician" while learning to be better at proving things. To accomplish this our material is divided into three main chapters followed by a postscript. These cover elementary number theory, calculus of one and several variables, introductory linear algebra, and visualization and interactive geometric computation.

240 pages, Paperback

Published August 23, 2016

Reviews

[Thomas Ray · Rating 3 out of 5]

An Introduction to Modern Mathematical Computing with Mathematica, Jonathan M. Borwein & Matthew P. Skerritt, 2012, for Mathematica version 8, 224 pages, ISBN 9781461442523, Library-of-Congress QA 76.95 B67 2012. Originally An Introduction to Modern Mathematical Computing with Maple, 2011, revised for Mathematica. Publisher's website:
https://link.springer.com/book/10.100...
Jonathan M. Borwein (1951-2016) wikipedia page: https://en.m.wikipedia.org/wiki/Jonat...

As of 2022, Mathematica version 13 is current.

This is a good tutorial, very readable, showing some unexpected ways of doing things. The focus is on using Mathematica to explore number theory, calculus, and linear algebra, at a college-freshman level. Many typographical errors.


See also:
Wolfram Mathematica online documentation:
https://reference.wolfram.com/language/

Sage is an open-source symbolic-computation system like Mathematica. p. vii

Internet math resources:
Wolfram MathWorld https://mathworld.wolfram.com/
PlanetMath
Wikipedia
p. viii

Online Encyclopedia of Integer Sequences http://oeis.org/
p. 99

Preface: see also:
Christian S. Claude, Randomness and Complexity, from Leibnitz to Chaitin, 2007. https://www.goodreads.com/book/show/2...



Integrate[E^(-x^2), {x, -Infinity, Infinity}]
= Sqrt[Pi] = 1.77245
https://www.wolframalpha.com/input?i=...
p. vii

Plot[x Sin[1/x], {x, -1/2, 1/2}]
https://www.wolframalpha.com/input?i=...
p. viii

A set operation on a single list deletes duplicates and sorts it:
Union[{3, 1, 2, 2, 1}]
{1, 2, 3}
https://www.wolframalpha.com/input?i=...
Intersection does the same, on a single list. Wolfram Alpha doesn't understand Complement as Mathematica does. pp. 9-10.

Wallis Product
Product[4 n^2/(4n^2-1), {n, 1, Infinity}]
= Pi/2
https://www.wolframalpha.com/input?i=...
https://en.m.wikipedia.org/wiki/Walli...
p. 13

ClearAll["Global`*"] clears all previous assignments. p. 49.

Sum[k^2, {k, 1, n}] = (1/6) n (n + 1) (2 n + 1)
https://www.wolframalpha.com/input?i=...
p. 49

Continued fractions:
GoldenRatio = Phi = (1 + Sqrt[5])/2
p. 56

ContinuedFraction[GoldenRatio] = {1, {1}}

ContinuedFraction[Sqrt[2]] = {1, {2}}

ContinuedFraction[Sqrt[5]] = {2, {4}}

ContinuedFraction[E, 36] = {2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14, 1, 1, 16, 1, 1, 18, 1, 1, 20, 1, 1, 22, 1, 1, 24}
https://www.wolframalpha.com/input?i=...

Plotting
Plot[1/(x^3 - 2 x^2), {x, -2, 4}]
https://www.wolframalpha.com/input?i=...
p. 89



"One must always keep one's wits about one when using a computer algebra system (of course this is true when reading a book or taking a bus too)." p. 92.



ERRATA (following are correct)
p. vii penultimate sentence: this book uses Mathematica.

Log[3, 2] is the base-3 log of 2. p. xiv
https://www.wolframalpha.com/input?i=...

Sum[k^(-2), {k, 1, Infinity}]
= (Pi^2)/6
https://www.wolframalpha.com/input?i=...
p. 12

p. 18: input 112 should read, Sum[k^(-2), {k, 1, Infinity}]

p. 18, penultimate line: 2^(17)

p. 20 1st paragraph, line 2: 17 times

p. 33, second paragraph: the function computes whether the number is perfect (sum of all divisors equals twice the number).

p. 33, output 208, and p. 34, output 222: False: 27 is not perfect. (28 is perfect.)

p. 36, input 236, the final function is AmicableQ[n, m].

p. 38, input 240 := findAmicable[100000]

p. 46, input 272 needs //Timing appended.

p. 48, inputs 282 need semicolons before the // to suppress output as shown.

p. 87, output 22 = (Pi^2)/6

p. 88, input 23, and p. 89, inputs 24 and 25: the = should be a minus sign.

p. 88, input 23: the upper limit should be +10.

pp. 88, 89 plots: Mathematica version 8 plotted an artificial vertical line at the x = 2 vertical asymptote. Mathematica 12.2 and wolframalpha.com correctly omit this artifact. https://www.wolframalpha.com/input?i=...

p. 89, input 25 must be written
With[{p = 1/(x^3 - 2 x^2)},
{Limit[p, x -> -Infinity],
Limit[p, x -> 0],
Limit[p, x -> 2],
Limit[p, {x -> Infinity}]
}]

p. 89, Mathematica version 8 output 25 gave a false limit as x -> 2, claiming Infinity, which holds only from the right; the limit from the left is -Infinity. Mathematica 12.2 correctly gives
{0, -Infinity, Indeterminate, 0}.

p. 91 says the plot is of both the function and the list. Only the list is shown.

p. 96 Integrate[x^2, {x, 1, 3}] = 26/3
p. 96 Integrate[x, {x, 1, 3}] = 4

p. 139 input 2 must be written
MatrixForm[{a, b, c}]
to give output 2.

p. 141, paragraph after [17] should read, "... in the case of matrices it will be matrix multiplication."

Same paragraph, change "begin" to "being."

p. 141, output 18 should read,
a + 2 b + 3 c

p. 143, penultimate paragraph, x and b are vectors.

p. 148, inputs 44 and 46: in each, the second r1 should be r2.


And many errors of English usage.