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2Math

Thursday, March 31, 2005

Two More FREE softwares for Computational Algebra

A few weeks back we talked about 2 Free softwares for computational
algebra
. Here we introduce two more. These were developed in France:

Coq: Coq is used for proving theorems in number theory.Coq is based on a logical
framework called "Calculus of Inductive Constructions" extended by a modular
development system for theories. It is a formal proof management system.
A proof done with Coq is mechanically checked by the machine. Coq allows, for
example, the definition of functions or predicates, stating mathematical
theorems and software specifications, developing interactively formal proofs of
these theorems as well as checking these proofs by a small certification "kernel".
For details and download, click here.

PARI/GP: PARI is a C library for computer algebra system and is designed for
fast computations in number theory (factorizations, algebraic number theory,
elliptic curves...) . It was developed by Henri Cohen and his co-workers at
Universit├ę Bordeaux I, France. It also contains a large number of other useful
functions to compute with mathematical entities such as matrices, polynomials,
power series, algebraic numbers, transcendental functions, etc.
GP is an easy-to-use interactive shell giving access to the PARI functions.
The package can be downloaded from here.

Sunday, March 27, 2005

Peter Lax receives 2004 Abel Prize

The 2005 Abel Prize in mathematics has been awarded to Peter D.
Lax of the Courant Institute of Mathematical Sciences at New York
University.

The Abel prize is a mathematics prize of the Norwegian Academy of
Science and Letters, dedicated to the memory of Niels Henrik Abel
(1802-1829) on the occasion of the bicentenary of his birth. It is
modeled after the Nobel Prize, and developed from a proposal by the
mathematics department at the University of Oslo in fulfillment of a
request forwarded by the Norwegian mathematician Sophus Lie
towards the end of the 19th century.

From 2003 onward the Abel Prize has been awarded annually.
Previous Abel Prize recipients include Jean-Pierre Serre in 2003,
and Sir Michael Francis Atiyah and Isadore M. Singer in 2004.

Lax was awarded the Abel Prize "for his groundbreaking contributions
to the theory and application of partial differential equations and to the
computation of their solutions." In particular, Lax laid the foundations
for the modern theory of nonlinear hyperbolic systems in the 1950s and
1960s. He constructed explicit solutions, identified classes of especially
well-behaved systems, and studied of how solutions behave over a long
period of time.

For more details visit the Abel Prize Site.

Thursday, March 24, 2005

Femlab 3.1 : Adds 64 bit computing support & much more ...

The latest release of the Stockholm, Sweden based Comsol's multiphysics
modeling software, Femlab 3.1 (read our introduction to Femlab ) adds
access to 64 bit computing platforms, such as Linux on AMD64 and the
Itanium, or HP-UX and Solaris system.

It also offers a new solver technology. The multi-grid preconditioner
optimizes routines which are otherwise computationally intensive and
reduces solution times and memory consumption.

These two significant additions enables Femlab to handle problems even
with millions of degrees of freedom. Their website gives example of a
problem of computing the electric field in a radar reflector which can have
more than 20 million degrees of freedom to take care.

Another feature they introduce in this version is the structured meshing
technique which allows users to select from quadrilateral, hexahedral and
prism elements for the optimal discretization of a given geometrical
structure. With innovative coupling of field variables, the simulation of
multiple structures has also become much easier. These mesh options
improve both the solution's accuracy and the execution speed.

Tuesday, March 15, 2005

Fractint: The Most Versatile Fractal Generator

Fractint is a popular freeware fractal generator. It is probably the most
versatile, extensive and well-established fractal program readily available
to enthusiasts who wish to roam inside the beautiful world of fractals,
although many users consider it to be an art toy rather than a serious
mathematical tool.

Fractint plots and manipulates images of "objects" -- actually, sets of
mathematical points -- that have fractal dimension. Its originators were
members of the Stone Soup Group. The group derives its name from an
Eastern European folklore in which a peddler persuades startving villagers
to share pieces of hoarded food to augment his 'stone soup' until all of them
have a nourishing meal to share. In a similar way, the four original
contributors developed Fractint by integrating by taking help and codes
from other users.

You will get download-links to all versions of Fractint (and many other
related useful links) by clicking here.
If you wish to read a detailed introduction or to know more about the
authors, click on this: Introduction .

Saturday, March 12, 2005

Matlab's Optimization Toolbox

The Matlab Optimization Toolbox (latest version is 3.0) is an add-on for
the Matlab package. It contains routines for many types of optimization
including constrained and unconstrained nonlinear minimization including
goal attainment problems, minimax problems, and semi-infinite
minimization problems, quadratic and linear programming.

Solution procedures for large-scale problems that are sparse and
structured are a special feature of this toolbox. One such algorithm, called
'linprog' is very fast and can solve a model of about, say, 2000 variables
and, say, 500 linear equality constraints in a matter of seconds.

Nonlinear least squares and curve-fitting and nonlinear system of
equation solving are also part of this wonderful package.

Matlab's functions can find a global minimum if it's the only minimum
and the function is continuous. However, the toolbox provides a wide
variety of local optimization functions and in most of the application areas,
such locally optimal solutions are more than adequate. However, one can
transfer the local optimum point of one run to the next one until the
global optimum is found.

One can extend the capabilities of the Optimization Toolbox by writing
his/her own Matlab-files, or by using the toolbox in combination with
other toolboxes, or with MATLAB or Simulink®.

Matlab's wonderful interface and the ready-availability of a
comprehensive range of other toolboxes and very-efficient functionalties
make this the choice for many in the world of system modeling.

Wednesday, March 09, 2005

Finite Element Method

In the late 18th century, mathematicians such as Lagrange recognized
that a variety of physical phenomenon were governed by partial
differential equations (PDE). 'Finite Element Method' (FEM) is one of the
central techniques using partial differential equations and is applied to a
variety of problems in physics simulation or to study engineering
components.

The generally acknowledged originator of FEM was the German
Mathematician, Richard Courant who used this technique in 1943 to
solve a torsion problem on a cylinder. Actually, he referred to this
technique long back in 1922 in his book on function theory. Courant, of
course, drew heavily from the work of stalwarts like Lord Raleigh, Boris
Galerkin, Walter Ritz etc.

The main driving force behind developing FEM was the insolubility of
PDEs for all but simple geometries. FEM is basically a transformation to
discretize the system. The target region is divided into a number of
geometrical elements. In each element the continuous field of PDE
variables is modeled by a local polynomial approximation controlled by
a few coefficients.

Then all these elements are linked by the values at shared nodes. This
results in a set of simultaneous algebraic equations which can be solved
numerically by optimization techniques and matrix algorithms.

Monday, March 07, 2005

Software Packages for Optimization Problems

The universe is full of problems which need to find out the maxima and
minima of lines, surfaces or multidimensional entities - we always need
to know the least difficult, the most profitable or the fastest possible
way of arriving at a solution.

You are given some functions of some variables and some constraints
on those variables and, may be, some explicit data and your job is to
find out the optimum solution. We guess professional people in all areas
of Science, Engineering and management encounter such problems
very frequently in their lifetime.

No single optimization software package can solve all optimization
problems. The space that the problem covers could be vast, littered
with discontinuities and noisy. For solving complex problems, you may
need to try searching the space with very good resolution, which may
not be possible in a reasonable time on a reasonable computer. The
inherent correlation among variables may easily lead to degeneracy of
possible solutions.

In near future we will introduce the following two widely used
Optimization packages: Mathematica's "Math Optimizer" add-on
package and Matlab's Optimization toolbox, "Global Optimizer".

Interested readers may also refer to the following widely-acclaimed
books on optimization methods:
"Global Optimization in Action" - J.D. Pinter (Kluwer, 1996)
"Global Optimization using Interval Analysis" - Eldon Hansen,
G. William Walster (Marcel Dekkar, 2003)
"Computational Global Optimization in Nonlinear Systems"
- J. D. Pinter (Lionhart Pub., 2001)

Wednesday, March 02, 2005

Octave : A Clone of Matlab

In this world there are a number of clones of Matlab. Even though
Matlab really grew up to take over majority of high-tech industries,
these clones also could manage to survive. There could be two major
reasons behind this: (i) Matlab is relatively high priced (ii) Some people
in academic world felt the urge to reinvent the wheel or may be they
just could not avoid the intellectual urge for continuing a parallel
evolution.

These parallel developments date back to 1980s when the power of
matrix based numerical computation was realised. The general
viewpoint was that the matrix programming could be developed with
an interface that could get away with Fortran syntax with its
declaration of variables of different kinds.

Octave is the closest clone of Matlab. It was developed at University
of Wisconsin by John W Eaton as a companion program of a textbook,
so that students could solve Chemical Engineering problems without
wasting time in debugging ugly-looking Fortran codes. It has a popular
user-base in a number of universities in USA - especially where no or
only a limited number of licenses are available to students.

Octave is available as both source code and binaries for Windows,
Linux, Unix and is freely distributable under the GNU General Public
License (GPL). Click here to download Octave.