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Re: [Axiom-developer] choose the better expand


From: Martin Rubey
Subject: Re: [Axiom-developer] choose the better expand
Date: 01 Apr 2006 08:45:23 +0200
User-agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4

Dear Francois,

 
> 5/ Do you prefer :
> a-expand (sin (2*x+y)) gives an Expression in sin x, sin y, cos x... of corse.
> b-expand (sin (3)) remains sin (3) [or do you prefer with sin 1 and cos 1]
> c-expand (sin (3*expressions without variables)) remains the same. 

To be honest, I find it very difficult to answer your question. I guess that in
general there is no such thing as a "normal form" for trigonometric
expressions. However, for special cases there may well be.

I vaguely remember that maxima was very good at simplifying that stuff, they
have trigsimp, trigrat, trigreduce,...

See

http://www.ma.utexas.edu/maxima/maxima_14.html

I think it would be very useful if you would browse the web a little and see
what has been done on the subject.

For example, for the case of no variables, MathSciNet gave me

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MR0981072 (90c:68040) 
 Schorn, Peter(1-NC-C) 
 A canonical simplifier for trigonometric expressions in the kinematic
 equation. 
 Inform. Process. Lett. 29 (1988), no. 5, 241--246.
 68Q40

Summary: "We present a new canonical simplification algorithm for trigonometric
expressions in the kinematic equation. The expressions consist of rational
numbers and the function symbols $+, -, *, /, \sin$ and $\cos$. Variables are
not allowed and the arguments of $\sin$ and $\cos$ are limited to rational
multiples of $\pi$. We present an algorithm which computes a simple normal form
in the real domain, in contrast to the known methods which either produce
unnecessarily complicated normal forms or are inefficient due to large integer
arithmetic. The simplifier has successfully been applied in the field of robot
kinematics."

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ScienceDirect told me that within ScienceDirect this article was cited once,
namely by:

Trigonometric polynomials with simple roots 

Achim Schweikard* 
 
Technische Universitt Berlin, Berlin, Germany 
 

Abstract


Trigonometric polynomials frequently occur applications in physics, numerical
analysis and engineering, since each periodic function can be approximated by a
trigonometric polynomial. Additionally, there are many analogies between
trigonometric and standard algebraic polynomials. Algorithms in computer
algebra depend on methods for the square-free decomposition of
polynomials. These methods use polynomial division and cannot be applied
directly to trigonometric polynomials. Let P denote the set of odd multiples of
?. A trigonometric polynomial T* is a reduced representation of a trigonometric
polynomial T if the set of zeros of T in P is the same as the set of zeros of
T* in P, and if all zero s of T* are simple zeros. It is shown that a reduced
representation of a trigonometric polynomial with rational or algebraic
coefficients can be found in polynomial time. 


Author Keywords: Analysis of algorithms; computer algebra; trigonometric and
exponential polynomial; root multiplicity; greatest square-free divisor 

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it might also be helpful to send a message to Richard Fateman, or to
math.sci.symbolic.

I truly appreciate your work. I find it's a great pity that you cannot come to
the workshop.

Martin





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