Introduction#
This book addresses the design and analysis of methods for computing numerical values for solutions to mathematical problems. Often, only accurate approximations are possible rather than exact solutions, so a key mathematical goal is to assess the accuracy of such approximations.
Given that most numerical methods allow any degree of accuracy to be achieved by working hard enough, the next level of analysis is assessing cost, or equivalently speed, or more generally the efficiency of resource usage. The most natural question then is how much time and other resources are needed to achieve a given degree of accuracy.
Topics#
The main areas of interest are:
Finding the zeros of a function: solving \(f(x) = 0\).
Solving systems of simultaneous linear equations: in matrix-vector notation, solving \(Ax = b\) for \(x\).
Fitting polynomials to a collection of data points, either exactly (collocation) or approximately (least-squares).
Approximating a function by a polynomial, or several polynomials.
Approximating derivatives and definite integrals.
Solving ordinary differential equations.
There is also briefer coverage of
Computing the eigenvalues and eigenvectors of a matrix.
Finding the minimum of a function or one or several variables. (These topics are being expanded.)
Although it is the last major topic, the numerical solution of differential equations will often be mentioned earlier as a motivation for other topics. However, we start in a simpler setting: the problem of finding the zeros of a real-valued function: solving \(f(x) = 0\).
Some References#
Online Resources#
[Chasnov, 2012] Numerical Methods by J. Chesnov, online at https://www.hlevkin.com/hlevkin/60numalgs/Common/Chasnov-Introduction to Numerical Methods.pdf and mirrored at https://lemesurierb.people.charleston.edu/Numerical-Methods-Chasnov.pdf (For a fairly basic introduction to and summary of various topics.)
[Sullivan, 2021] Numerical Methods (An Inquiry-Based Approach with Python) by Eric Sullivan, onlne at https://numericalmethodssullivan.github.io/
[Dionne, 2023] Numerical Analysis by Benoit Dionne, online at https://ruor.uottawa.ca/server/api/core/bitstreams/337a83da-97ca-4f46-a0b7-8a3a692d8cee/content
[Vuik et al., 2015]Numerical Methods for Ordinary Differential Equations by Kees Vuik, Fred Vermolen, Martin van Gijzen and Thea Vuik, online at https://books.open.tudelft.nl/home/catalog/view/165/285/536
[Strang and Herman, 2016-2025] Calculus, Volume 2, by Gilbert Strang and Edwin Herman, with contributions from many others.
[SciPy, 2025] online at Scientific Python Lectures; a free reference mainly on the SciPy package, but also with some general information on using Python for scientific computing. It is available both as a web-site or downloadable as PDF or HTML — for off-line access, I prefer the HTML version rather than the PDF.
Books#
[Sauer, 2022] Numerical Analysis by Timothy Sauer, Pearson, 2022.
[Burden et al., 2016] Numerical Analysis by Richard L. Burden, J. Douglas Faires and Annette M. Burden, Cengage 2016.
[Chenney and Kincaid, 2013] Numerical Mathematics and Computing by Ward Chenney and David Kincaid, Cengage 2013.
[Kincaid and Chenney, 1990] Numerical Analysis by David Kincaid and Ward Chenney, Brooks/Cole, 1990.
Books for Specific Topics#
[Trefethen and Bau, 2022] Numerical Linear Algebra by LLoyd “Nick” Trefethen and David Bau III, SIAM 2022.
[Dahlquist and Björck, 2008] Numerical Methods in Scientific Computing, Volume 1 by Germund Dahlquist and Åke Björck, SIAM, 2008.
[Dahlquist and Björck, 2009] Numerical Methods in Scientific Computing, Volume 2 by Germund Dahlquist and Åke Björck, SIAM, 2009.
[LeVeque, 2007] Finite Difference Methods for Ordinary and Partial Differential Equations by Randall LeVeque, SIAM, 2007.
[Johnson, 1994] Numerical solution of partial differential equations by the finite element method.
See also the Bibliography.