Re: Mathematica Vs. Matlab

From: Paul Abbott (paul_at_physics.uwa.edu.au)
Date: 08/19/04 

Date: Thu, 19 Aug 2004 14:43:04 +0800

In article <cfvmpd$u$1@fred.mathworks.com>,
 "Steven Lord" <slord@mathworks.com> wrote:

> Now rather than having an emotional and subjective discussion about the two
> products (see my previous comment about a religious war) let's take a look
> at a benchmark that I believe is fairly well-known and impartial:
>
> http://www.scientificweb.com/ncrunch/
>
> This is Stefan Steinhaus's report comparing Mathematica, MATLAB, and a
> number of other software packages. Now since there have been changes to
> both Mathematica and MATLAB in the two years since the report was last
> compiled, it's probably about time for it to be rerun -- but as of the last
> time it ran MATLAB scored higher than Mathematica in every category except
> platform support, where they were tied with 100% each.
> Look at the summary in section 8.2.

Actually, Mathematica topped the mathematical functions, just ahead of
Gauss, with Matlab in third place (see section 2.8). It appears that
Mathematica's result in this table were incorrectly transcribed to the
summary table in section 8.2. However, the computation of the weighted
overall result is correct.

> I'd be surprised if the figures in that table changed so
> wildly as to cause Mathematica to jump significantly higher than MATLAB, as
> your comments seem to indicate would be your expectation.

I would be most interested to see a revised comparison. Certainly, the
numerical speed of Mathematica improved dramatically with the release of
version 5. Since this was the worst area of Mathematica's performance in
the Steinhaus survey, a significant change here would make a large
difference to the overall outcome.

I have not read all of the Steinhaus survey but here are a few code
one-liners and pointers to freely available Notebooks and Packages that
provide some of the missing mathematical functionality:

Hadamard Matrices: Notebook available at
http://library.wolfram.com/infocenter/MathSource/499/

HouseHolderMatrix:

  HouseHolderMatrix[v_?VectorQ] :=
    IdentityMatrix[Length[v]]-2 Transpose[{v}].{v}/(v.v)

PascalMatrix:

  PascalMatrix[n_] :=
    Table[Binomial[i+j,i], {i, 0, n - 1}, {j, 0, n - 1}]
 
  PascalMatrix[n_, 1] :=
    (-1)^(Range[n]-1) CholeskyDecomposition[PascalMatrix[n]]

  PascalMatrix[n_, 2] := Reverse /@ PascalMatrix[n, 1]

ToeplitzMatrix:

  ToeplitzMatrix[c_?VectorQ, r_?VectorQ] :=
    Reverse[Partition[Join[Reverse[c], Rest[r]], Length[c], 1]] /;
      0 < Length[c] == Length[r]

  ToeplitzMatrix[r_?VectorQ] := Transpose[ToeplitzMatrix[r, r]]

Dulmage-Mendelsohn decomposition: See

  A Debugging Scheme for Declarative Equation Based Modeling Languages
  Peter Bunus and Peter Fritzson
  URL: http://www.ida.liu.se/~petbu/publications/peterBPADL2002.pdf

and

 Bunus P. and Fritzson P.
 Linköping University, Sweden
 Methods for Structural Analysis and Debugging of Modelica Models
 URL: http://www.modelica.org/Conference2002/papers.shtml

Smith Normal Forms: Notebook available at
http://library.wolfram.com/infocenter/MathSource/682/

Quadratic optimization is built-in in version 5.0.

Contingency-table data: See RiskQ 4.2 <http://www.exetersoftware.com/cat/loehle/riskq.html>

Gumbel distribution function: Notebook available at
http://mathworld.wolfram.com/GumbelDistribution.html

Logarithmic distribution: Notebook available at
http://mathworld.wolfram.com/LogarithmicDistribution.html

S distribution: Package available at
http://aksenov.freeshell.org/sdist.html

von Mises distribution: Notebook available at
http://mathworld.wolfram.com/vonMisesDistribution.html

Loess statistics: Package available at
http://www.verbeia.com/mathematica/mathecon/othercode.html

Logistic regression: Package available at
http://library.wolfram.com/infocenter/MathSource/706/

Pearson and Kolmogorov-Smirnov tests: Package available at
http://library.wolfram.com/infocenter/Articles/1379/

I expect that many more of the holes identified in the current survey
have already been filled.

Cheers,
Paul 

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
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Crawley WA 6009                      mailto:paul@physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul

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