Mathematica for numerical analysis (was Re: A question on Newton's Method)

Roman Werpachowski wrote:

> Yes, but what's the point of discussing Mathematica in the context of
> NUMERICAL analysis? It's not its main strength, is it?

That is the repuation of Mathematica, but in recent releases they have
been emphasizing improvements in speed in numerical calculations. They
claim performance enhancements in Mathematica 5.0 in the areas of

Fast Dense Numerical Linear Algebra
High-Speed Sparse Linear Algebra
Large-Scale Linear Systems
Big-Number Arithmetic
64-Bit Platform Support

At
http://www.wolfram.com/products/mathematica/technologies/giganumerics.html
they say the following about their "giganumerics" initiative. I have
not used Mathematica much and cannot comment on their speed claims.

"The challenge Wolfram Research tackled with gigaNumerics was to
achieve exceptional raw computing speed while maintaining Mathematica's
generality and accuracy. This challenge was met successfully by the
following combination of gigaNumerics technologies developed at Wolfram
Research:

Precompilation
Compilation can speed up numerical calculations for certain types of
input. Mathematica optimizes its performance and efficiency by
preapplying compilation automatically as a transparent part of many
numerical calculations in cases in which Mathematica assesses that it
is feasible.

Packed Arrays
Computations to be performed on machine-precision matrices and arrays
are analyzed to decide whether packing them into a specialized format
will improve the performance of the computation. This process of
analysis and application occurs transparently, with outputs presented
the same way regardless of which methodology Mathematica chooses.

Automatic Algorithm Adaptation and Selection
Many Mathematica functions automatically choose between a variety of
algorithms and, in addition, adaptively adjust their sampling rate
throughout the calculation to optimize speed and accuracy.

Processor Optimization
Libraries are optimized for each processor, including the latest 64-bit
varieties.

Symbolic Preprocessing
In some cases the total calculation time is least if you simplify a
problem algebraically before evaluating the result numerically.
Mathematica employs this technique automatically where appropriate.

Vectorization
Certain Mathematica operations can work on an entire vector, matrix, or
array rather than on just a single element. Operating on all the data
at once reduces the number of top-level calls to Mathematica, replacing
them with optimized internal routines.

For the first time with Version 5.0, Mathematica outperforms
traditional dedicated numerical systems in terms of raw computational
speed alone. Advances in gigaNumerics technologies have achieved
this--they more than cancel out the speed deficit that might be
expected from the generality and accuracy that Mathematica delivers. In
the future, Mathematica's lead is expected to increase because
traditional numerical systems do not have integrated symbolic
capabilities with which to perform symbolic preprocessing."

.

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