There are some excellent code packages to solve specific problems. Since other people already spent time, efforts and applied their expertise, no need to reinvent a secondary wheel. so...

1. R packages,

1.5 MachineLearning

.... they're there

2. Python packages (for windows) and there are corresponding Linux-alike packages

3. convex optimization solver (CVX) Matlab based

4. Filters ReBel (kalman filter, ETK, particle filter,..., I have extended to LETKF)

5. CentPack (c++) for central scheme PDEs solution (for 1-D, and 2-D hyperbolic conservation laws, seems extra work for source term)

need 5.1 a package for discontinuous Galerkin Finite element methods (may take a look at DG-FEM)

5.2 a package for moving mesh finite volume solver for hyperbolic PDEs

If to do PDE solving in R, may take a look at this, beside those matlab/C++ routines.

6. Gibbs Sampling Bayes: OpenBugs, Stan, JAGS

7. lightspeed: for some optimized Matlab functions (only for windows sys)

8. still find the PMTK3 not easy to use (for probabilistic graphical models)

9. Parse PDF PDFMiner (python)

10. Other languages Erlang for concurrence, functional, maybe also for AI (instead of prolog?)

11. Compressive sensing

10. Other languages Erlang for concurrence, functional, maybe also for AI (instead of prolog?)

11. Compressive sensing

R is in fact a very good glue language to interface many mainstream high efficiency languages (read J. M. Chambers' Software for Data Analysis: Programming with R, chap 9 to chap 12).

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