Customer Reviews

An Introduction to Numerical Analysis

5 star:		(2)
4 star:		(1)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

This book has emphasis on analysis of numerical methods, including
error bound, consistency, convergence, stability. In most cases, a
numerical method is introduced, followed by analysis and proofs. For
engineering students, who like to know more algorithms and a little
bit of analysis, this book may not be the best choice.

Although this book is mainly about analysis, it does include clear
presentation of many numerical methods, including topics in nonlinear
equations solving, numerical linear algebra, polynomial interpolation
and integration, numerical solution of ODE. In numerical linear
algebra, it includes LU factorization with pivoting, Gerschgorin's
theorem of eigenvalue positions, Calculating eigenvalues by Jacobi
plane rotation, Householder tridiagonalization, Sturm sequence
property for tridiagonal symmetric matrix. Interpolation includes
Lagrange polynomial, Hermite polynomial, Newton-Cotes integration,
Improved Trapezium integration through Romberg method, Oscillation
theorem for minimax approximation, Chebyshev polynomial, least square
polynomial approximation to a known function, Gauss quadrature using
Hermite polynomial, Piecewise linear/cubic splines. Ordinary
ddifferential equations section includes initial value problems with
one-step and multiple steps, boundary value problems using finite
difference and shooting method, Galerkin finite element method.
The book gives basic definitions including norms, matrix condition
numbers, real symmetric positive definite matrix, Rayleigh quotient,
orthogonal polynomials, stiffness, Sobolev space.

One place that is not clear is about QR algorithm for tridiagonal
matrix.

In summary, the book is written clearly. Every numerical method is
presented based on mathematics. There are many proofs (there is one
proof with more than 3 pages), most of them that I decided to read are
pretty easy to follow. There are not much implementation details and
tricks. But this book will tell you when a method will converge and
when a method is better. As a non-math major reader, I wish it could
present more algorithms, such as algorithms for eigenvalues of
nonsymmetric matrix, more details in finite difference method, a
little bit of partial differential equations etc.

Many people naively believe that with the growing power of symbolic mathematics packages such as Mathematica?, knowledge in numerical analysis is increasingly irrelevant. That is not true, all programmers should have some knowledge of numerical techniques and even power users of a symbolic mathematics package should have some theoretical knowledge in numerical techniques. The packages will perform operations such as polynomial interpolation and at the very least a user should know when to use a specific type.
This book is a solid text in the basics of numerical mathematics, using more of a theoretical background than most. There are a large number of theorem-proof instances, so in that respect, it resembles a math book. The material covered is:

* Solutions of linear equations and systems of linear equations.
* Matrices, eigenvalues and eigenvectors.
* Polynomial interpolation and polynomial approximation.
* Numerical integration.
* Initial value and boundary value problems for ODEs.
* The finite element method.

There are a small number of exercises at the end of each chapter and no solutions are included.
If you are looking for a book to use in a course in numerical analysis where there is an emphasis on the theoretical background, then this one will serve your needs.

This book was used for a one semester course in numerical analysis. The companion
book was Numerical Linear Algebra. Together, these books make an
outstanding start to a personal numerical analysis reference shelf.

The first half of the book is where pure math students may find trouble since the
presentation of linear algebra in math departments varies widely. A practical working
knowledge of basic linear algebra is necessary. A poor linear algebra background will
require remediation before starting this book.

Topics covered in this text that I have found particularly useful over the years
include polynomial interpolation and quadrature. The presentation is perfect and easy
to understand.

Of course, no text can be everything to everyone. It helps to have an enthusiastic and
knowledgeable professor leading you through the material. That said, the dedicated
student should have no problem navigating this text.

I had to use this textbook for a first course in Numerical Analysis. It might be one of the worst textbooks I have ever used. It contains a lot of material, but I do not feel it is well suited for a first course in Numerical Analysis at an undergraduate university.

The main problem is the textbook assumes too much previous knowledge. I had difficulty understanding even the "simplest" explanations in this textbook because even those were too complicated. I often found myself searching the web and other texts for the concepts that I needed to learn, and I almost always found descriptions that I perfectly understood. Sometimes I could then go back and understand it from this textbook, but only after reading material elsewhere on the subject.

Simply put, this textbook might be a decent resource for someone who already knows Numerical Analysis and has a really strong background in Math (I took the class my final semester as part of a BS in Mathematics degree, and my background was not strong enough). However, for "An intro to Numerical Analysis" as this title states, this is NOT the textbook to use.