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====== Computer Algebra Systems (CAS) ======

Computer algebra systems, such as Mathematica, Maple, SageMath, Matlab's Symbolic Math Toolbox, etc. are very useful for the following tasks (ranked in importance): 

  * 1 = Deriving and manipulating exact closed-form quantities or special functions, including pattern recognition
  * 2 = Numerical scratchpad for exploration
  * 2 = High precision computations and variable precision arithmetic
  * 3 = (Floating-point) Numerical analysis and numerical solutions 

In terms of alternatives to the above tasks, there is no real alternative to item 1 in terms of conventional software you will use. Items ranked 2-3 all have alternatives, e.g. Python, Matlab, C++, etc. 

===== Choosing a software =====

  * Unfortunately, as far as I'm aware, there is no good competitor to Mathematica (education license \$1k; student license \$70/year). Maple is probably the closest, but it was previously unsatisfactory for my needs. 
  * There is no good open source alternative...as far as I know. Perhaps we should do a benchmark test?

===== Example 1: using Mathematica as a scratchpad =====

  * Simple 2D, 3D, and contour plotting.  
  * Use of special functions 
  * Obtaining precision 
  * FullSimplify and identities 
  * DSolve, NDSolve, and RSolve

===== Example 2: quadrature =====

I will do an example showing quadrature using Mathematica. This will look at the [[https://en.wikipedia.org/wiki/Hundred-dollar,_Hundred-digit_Challenge_problems|SIAM 100-digit challenge]] problem 1. 

===== Example 3: series expansions =====

I will do a basic asymptotic expansion example to show the series functionality

===== Example 4: contour plotting =====

A script from before showing contour plotting

===== The philosophy of CAS vs floating point coding =====