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# Package PRACMA

## Introduction

This package provides R implementations of more advanced functions in
numerical analysis, with a special view on on optimization and time
series routines. Uses Matlab/Octave function names where appropriate
to simplify porting.

Some of these implementations are the result of courses on Scientific
Computing (``Wissenschaftliches Rechnen'') and are mostly intended to
demonstrate how to implement certain algorithms in R/S. Others are
implementations of algorithms found in textbooks.

## Details

The package encompasses functions from all areas of numerical analysis,
for example:

* Root finding and minimization of univariate functions,  
  e.g. Newton-Raphson, Brent-Dekker, Fibonacci or `golden ratio' search.
* Handling polynomials, including roots and polynomial fitting,  
  e.g. Laguerre's and Muller's methods.
* Interpolation and function approximation,  
  barycentric Lagrange interpolation, Pade and rational interpolation,
  Chebyshev or trigonometric approximation.
* Some special functions,  
  e.g. Fresnel integrals, Riemann's Zeta or the complex Gamma function,
  and Lambert's W computed iteratively through Newton's method.
* Special matrices, e.g. Hankel, Rosser, Wilkinson
* Numerical differentiation and integration,  
  Richardson approach and ``complex step'' derivatives, adaptive
  Simpson and Lobatto integration and adaptive Gauss-Kronrod quadrature.
* Solvers for ordinary differential equations and systems,  
  Euler-Heun, classical Runge-Kutta, ode23, or predictor-corrector method 
  such as the Adams-Bashford-Moulton.
* Some functions from number theory,  
  such as primes and prime factorization, extended Euclidean algorithm.
* Sorting routines, e.g. recursive quickstep.
* Several functions for string manipulation and regular search,
  all wrapped and named similar to their Matlab analogues.

## Goals

It serves three main goals:

* Collecting R scripts that can be demonstrated in courses on
  Numerical Analysis or Scientific Computing using R/S as the chosen
  programming language.
* Wrapping functions with appropriate Matlab names to simplify
  porting programs from Matlab or Octave to R.
* Providing an environment in which R can be used as a full-blown
  numerical computing system.

Besides that, many of these functions could be called in R applications
as they do not have comparable counterparts in other R packages (at least
at this moment, as far as I know). 

All referenced books have been utilized in one way or another.
Web links have been provided where reasonable.

## Emulated MATLAB Functions

The following 220 functions are emulations of correspondingly named Matlab 
functions and bear the same signature as their Matlab cousins if possible:

    accumarray, acosd, acot, acotd, acoth, acsc, acscd, acsch, and, angle, ans,
    arrayfun, asec, asecd, asech, asind, atand, atan2d,  
    beep, bernoulli, blank, blkdiag, bsxfun,  
    cart2pol, cart2sph, cd, ceil, circshift, clear, compan, cond, conv,  
    cosd, cot, cotd, coth, cross, csc, cscd, csch, cumtrapz,  
    dblquad, deblank, deconv, deg2rad, detrend, deval, disp, dot,  
    eig, eigint, ellipj, ellipke, eps, erf, erfc, erfcinv, erfcx, erfi, erfinv,  
    errorbar, expint, expm, eye, ezcontour, ezmesh, ezplot, ezpolar, ezsurf,  
    fact, fftshift, figure, findpeaks, findstr, flipdim, fliplr, flipud,  
    fminbnd, fmincon, fminsearch, fminunc, fplot, fprintf, fsolve, fzero,  
    gammainc, gcd, geomean, gmres, gradient,  
    hadamard, hankel, harmmean, hilb, histc, humps, hypot,  
    idivide, ifft, ifftshift, inpolygon, integral, integral2, integral3,  
    interp1, interp2, inv, isempty, isprime,  
    legendre, linprog, linspace, loglog, logm, logseq, logspace, lsqcurvefit,  
    lsqlin, lsqnonlin, lsqnonneg, lu,  
    magic, meshgrid, mkpp, mldivide, mod, mrdivide,  
    nchoosek, ndims, nextpow2, nnz, normest, nthroot, null, num2str, numel,  
    ode23, ode23s, ones, or, orth,  
    pascal, pchip, pdist, pdist2, peaks, perms, piecewise, pinv, plotyy,  
    pol2cart, polar, polyfit, polyint, polylog, polyval, pow2, ppval,  
    primes, psi, pwd,  
    quad, quad2d, quadgk, quadl, quadprog, quadv, quiver,  
    rad2deg, randi, randn, randsample, rat, rats, regexp, regexpi,  
    regexpreg, rem, repmat, roots, rosser, rot90, rref, runge,  
    sec, secd, sech, semilogx, semilogy, sinc, sind, size, sortrows, sph2cart,  
    sqrtm, squareform, std, str2num, strcat, strcmp, strcmpi,  
    strfind, strfindi, strjust, subspace,  
    tand, tic, toc, trapz, tril, trimmean, triplequad, triu,  
    vander, vectorfield, ver,  
    what, who, whos, wilkinson,  
    zeros, zeta

The following Matlab function names have been capitalized in `pracma' to
avoid shadowing functions from R base or one of its recommended packages
(on request of Bill Venables and because of Brian Ripley's CRAN policies):

    Diag, factos, finds, Fix, Imag, Lcm, Mode, Norm, nullspace (<- null),
    Poly, Rank, Real, Reshape, strRep, strTrim, Toeplitz, Trace, uniq (<- unique).

To use `ans` instead of `ans()` -- as is common practice in Matlab -- 
type (and similar for other Matlab commands):

    makeActiveBinding("ans", function() .Last.value, .GlobalEnv)
    makeActiveBinding("who", who(), .GlobalEnv)

### Note

The R package `matlab' contains some of the basic routines from Matlab,
but unfortunately not any of the higher math routines.

## References

  Abramowitz, M., and I. A. Stegun (1972). Handbook of Mathematical Functions
  (with Formulas, Graphs, and Mathematical Tables). Dover, New York.
  URL: www.math.ubc.ca/~cbm/aands/notes.htm

  Arndt, J. (2010). Matters Computational: Ideas, Algorithms, Source Code.
  Springer-Verlag, Berlin Heidelberg Dordrecht.
  FXT: a library of algorithms: <https://www.jjj.de/fxt/>.

  Cormen, Th. H., Ch. E. Leiserson, and R. L. Rivest (2009). Introduction
  to Algorithms. Third Edition, The MIT Press, Cambridge, MA.

  Encyclopedia of Mathematics (2012). Editor-in-Chief: Ulf Rehmann.

  Gautschi, W. (1997). Numerical Analysis: An Introduction.
  Birkhaeuser, Boston.

  Gentle, J. E. (2009). Computational Statistics.
  Springer Science+Business Media LCC, New York.

  Hazewinkel, M., Editor (2002). Encyclopaedia of Mathematics.
  Springer-Verlag, Berlin Heidelberg New York.

  MathWorld.com (2011).
  Wolfram MathWorld: <https://mathworld.wolfram.com/>.
  Matlab Central: <https://www.mathworks.com/matlabcentral/>.

  NIST: National Institute of Standards and Technology.
  Olver, F. W. J., et al. (2010). NIST Handbook of Mathematical Functions.
  Cambridge University Press. 
  Internet: NIST Digital Library of Mathematical Functions, 
  Dictionary of Algorithms and Data Structures,
  Guide to Available Mathematical Software, <https://gams.nist.gov/>

  Press, W. H., S. A. Teukolsky, W. T Vetterling, and B. P. Flannery (2007).
  Numerical Recipes: The Art of Numerical Computing. Third Edition, incl.
  Numerical Recipes Software, Cambridge University Press, New York.
  URL: numerical.recipes/book/book.html

  Quarteroni, A., R. Sacco, and F. Saleri (2007). Numerical Mathematics.
  Second Edition, Springer-Verlag, Berlin Heidelberg.

  Skiena, St. S. (2008). The Algorithm Design Manual. Second Edition,
  Springer-Verlag, London. The Stony Brook Algorithm Repository:

  Stoer, J., and R. Bulirsch (2002). Introduction to Numerical Analysis.
  Third Edition, Springer-Verlag, New York.

  Strang, G. (2007). Computational Science and Engineering.
  Wellesley-Cambridge Press.

  Weisstein, E. W. (2003). CRC Concise Encyclopedia of Mathematics.
  Second Edition, Chapman & Hall/CRC Press.

  Zhang, S., and J. Jin (1996). Computation of Special Functions.
  John Wiley & Sons.
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