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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, kron, 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. <http://www.nr.com/aands/>. Arndt, J. (2010). Matters Computational: Ideas, Algorithms, Source Code. Springer-Verlag, Berlin Heidelberg Dordrecht. FXT: a library of algorithms: <http://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. <http://www.encyclopediaofmath.org/>. 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. <http://eom.springer.de/>. MathWorld.com (2011). Matlab Central: <http://www.mathworks.com/matlabcentral/>. Mathtools.net: <http://www.mathtools.net/>. 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, <http://dlmf.nist.gov/>; Dictionary of Algorithms and Data Structures, <http://www.nist.gov/>; Guide to Available Mathematical Software, <http://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. <http://www.nrbook.com/a/bookcpdf.php> [chapters], or <http://apps.nrbook.com/c/index.html> [pages]. 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: <http://www.cs.sunysb.edu/~algorith/>. 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. Matlab Codes: <http://www-math.mit.edu/cse/> Weisstein, E. W. (2003). CRC Concise Encyclopedia of Mathematics. Second Edition, Chapman & Hall/CRC Press. Wolfram MathWorld: <http://mathworld.wolfram.com/>. Zhang, S., and J. Jin (1996). Computation of Special Functions. John Wiley & Sons.