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This book is intended to help advanced undergraduate, graduate, and postdoctoral students in their daily work by o ering them a compendium of numerical methods. The choice of methods pays significant attention to error estimates, stability and convergence issues, as well as optimization of program execution speeds. Numerous examples are given throughout the chapters, followed by comprehensive end-of-chapter problems with a more pronounced physics background, while less stress is given to the explanation of individual algorithms. The readers are encouraged to develop a certain amount of skepticism and scrutiny instead of blindly following readily available commercial tools.

 

The second edition has been enriched by a chapter on inverse problems dealing with the solution of integral equations, inverse Sturm-Liouville problems, as well as retrospective and recovery problems for partial di erential equations. The revised text now includes an introduction to sparse matrix methods, the solution of matrix equations, and pseudospectra of matrices; it discusses the sparse Fourier, non-uniform Fourier and discrete wavelet transformations, the basics of non-linear regression and the Kolmogorov-Smirnov test; it demonstrates the key concepts in solving sti di erential equations and the asymptotics of Sturm-Liouville eigenvalues and eigenfunctions. Among other updates, it also presents the techniques of state-space reconstruction, methods to calculate the matrix exponential, generate random permutations and compute stable derivatives.

Computational Methods in Physics: Compendium for Students

Since this is a 2^nd Edition, we are giving below the topics we wish to add/update/revise in roughly the same chapter sequence as we had in the existing 1st Edition of the book. In addition to a general revision of the text, we propose the following major modifications (the asterisks denote the amount of text added/modified and/or or the difficulty level of the topics being discussed):

 

Chapter 2

 

- Subsection 2.1.2: add discussion on how to find all zeros by means

  of the Newton method (*)

 

Chapter 3

 

- Expand Subsection 3.2.7 on solving the A*x = b equations with sparse

  matrices to a full Section (**)

- Expand Subsection 3.4.5 on solving the eigenvalue problem A*x = lambda*x

  to a full Section (**)

- Discuss the exponentiation of a matrix, exp(A) (*)

- Add a Subsection on Pseudospectra (**)

- in general

, enhance the "sparse" aspect of the chapter

 

Chapter 4

 

- Add a new Section on Sparse FFT (following present Sec. 4.2),

  add corresponding Exercise (**)

- Expand Subsection 4.6.2 on the Discrete Wavelet Transform

  to a full Section, add Exercise (***)

- Add a Section on image denoising (**)

- Add Section on Radon transformation (**)

 

Chapter 5

 

- Rewrite Sections 5.1-5.5 to better distinguish between general

  discussion of distributions and the techniques involving samples,

  and to bring the notation in line with the book "Probability

  for Physicists" (***)

- Introduce Bayesian data analysis and inference (***)

- Expand Subsection 5.5.8 on Non-linear Regression to a full Section,

  add Exercises (**)

 

Chapter 6

 

- Expand Section 6.5

on Noise, add Exercise (**)

- Add Section on Takens Theorem and its applications: phase space

  reconstruction and optimal size determination (**)

- Add discussion on signal entropies (**)

- Update discussion on autoregressive models (optimal order) (*)

- Add discussion on signal directionality / causality (**)

 

Chapter 7

 

- Expand Section 7.10 on Stiff Problems of ODE, add Exercise (**)

 

Chapter 8

 

- Expand Subsection 8.7.4 on Singular SL Problems to a Section,

  add Exercise (**)

- Motivated by Section 8.8, write a new chapter on Inverse Problems (***)

 

Chapter 10

 

- Expand Section 10.8, add Exercise (**)

 

Chapter 11

 

- Expand Sections 11.7 and 11.8, add Exercises (**)

 

New Chapter on Inverse Methods (***)

 

New short Chapter or Appendix on minimization (**)

- with derivatives or without them

- with constraints or without them

- deterministic and quasi-deterministic (MC methods)

 

New Appendix on spline methods: B-splines, Bezier splines (**)