Legendre Expansion?

[SilverSprite]

Anybody know what a Legendre expansion is? Any help appreciated.

[Guv]

There are polynomials called legendre polynomials, and also some called Chebychev polynomials.

They are used in creating approximations to various functions like sine, cosine, expontials, et cetera.

I am not sure if this is the answer you are looking for. The idea is as follows.

A transcendental function is defined by some infinite series. For example, the following.

sin(x) = x - x³/3! + x⁵/5! + . . .

You would like to use some finite approximation, pehaps an 11th order polynomial instead of the infinite series.

There is a method for expressing the transcendental function as the sum of some Lengendre or Chebychev polynomials. If you drop the higher order Lengendre or Chebychev polynomials you will get a better approximation than you get by dropping the higher order terms of the ordinary power series.

You end up with a polynomial approximation for which the first coefficient is slightly different than 1, the second is slightly different from 1/3!, the third is not quite equal to 1/5!, et cetera. The slight differences partially adjust for dropping the higher order terms.

[SilverSprite]

Well i've heard of transcendental numbers, and they are irrational numbers that cannot be roots of equations, for example e and pi? That is correct if my memory serves its purpose. But what is a transcendental function?

You said that legendre polynomials are polynomials that approximate various functions. Could you give me an example of how this works or direct me to a website that gives an example?

Thanks

[bugzpodder]

thought it was called taylor's expansion

[bugzpodder]

Suppose that if is a piecewise smooth function on the interval [-1,1]. Then f has a Legendre series expansion:
f(x) = SUM(A_j,P_j(x))from j=0 to infinite

where
A_j = ((2j+1)/2)*integral(f(x)*P_j(x))from -1..1)

Where P_j is the jth Legendre polynomial.

[Guv]

Try mathworld.wolfram.com

Chebychev is not correct spelling, so do not search for it. The above site has some articles on Legendre polynomials.

Sine, cosine, e^x, and others are called transcendental functions.

[Guv]

A long time ago I forgot most of what I knew about these special polynomials, so I did a bit of research. Chebychev is an OK spelling and I think that Tchebychev is also. The following is an abbreviated summary of what I found.

It is not difficult to understand this concept, but it is tedious and requires paying attention.

As far as I know, either Legendre and Chevychev polynomials can be used to approximate any function known to be equal to a Taylor (or other) series using powers of x. This technique was useful in the prehistoric era prior to computers and in the early computer era when computers were slow and had small memories. In those days it was worthwhile to do a lot of analytical and tedious algebraic work in order to minimize the amount of computational work required for various tasks. In modern times, it is likely to be obsolete for practical purposes, but is probably being taught at many colleges.

The basic concept is as follows, using Chebychev polynomials. The same concept works with Legendre polynomials (I think). I do not know if there is any advantage to using one over the other.

One form of Chebychev polynomials is as follows (there are others).

P0(x) = 1
P1(x) = x
P2(x) = 2x² - 1
P3(x) = 4x³ - 3x
P4(x) = 8x⁴ - 8x² + 1
P5(x) = 16x⁵ -20x³ + 5x
P6(x) = 32x⁶ - 48x⁴ + 18x² -1

The above have been doctored so that P_n(1) = 1, which has advantages when using them in the interval -1 < x < 1

There is a recursive relationship, which is valid for the above and (I think) for other variations of these polynomials.

P_n+1(x) = 2xP_n - P_n-1

I hope there are no typo’s in the above. I am pretty sure I got the recursive equation correct. It can be used to check the other equations. I also hope there are no typo’ in the following.

Now, with a lot of one-time work, you can obtain the following.

1 = P1(x)
x = P2(x)
x² = P2(x)/2 + P0(x)/2
x³ = P3(x)/4 - 3P1(x)/4
and equations for x⁴, x⁵, et cetera.

Using the above, you can substitute into the first 10 (for example) terms of a Taylor series and obtain an expression in Chebychev polynomials which is equivalent to the Taylor series. For example, consider the series for e^x

1 + x + x²/2! + x³/3! + x⁴/4! + . . . .

With a lot of work, you work out the coefficients in the following series, which is identical to the first eleven terms of the Taylor series for e^x

A0P0(x) + A1P1(x) + A2P2(x) + . . . + A10P10(x)

Now if you drop the last term, the resulting expression in Chebychev polynomials will be a more accurate approximation than you would get by dropping the eleventh term of the Taylors series.

With some more algebraic work, you can get a modified Taylor series with ten terms from the truncated Chebychev series. Each coefficient will differ slightly from the corresponding term of the true Taylor series. The modified series will be more accurate in the interval -1 , x < 1 than the Taylor series with the same number of terms.

When computers were painfully slow, the above was a worthwhile method of obtaining approximating functions for sine, cosine, and other transcendental functions. With the speed of modern computers, I would expect that the standard programs use as many terms of a Taylor series as necessary to obtain the desired precision.

BTW: When computers were slow and still today, there are various simple tricks used to cut down on the amount of computations required for evaluation of transcendental functions. For example, instead of evaluating e⁵, you can evaluate e^5/8 using the standard Taylor series. With an exponent less that one, a lot of precision is obtained with relatively few terms of the Taylor series compared to the number of terms required for an exponent a lot greater than one. When you get the result, you can calculate e⁵ with three multiplications.

e^5/8e^5/8 = e^5/4
e^5/4e^5/4 = e^5/2
e^5/2e^5/2 = e⁵

[SilverSprite]

Wow, interesting/complicated stuff.