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Note of
Numeric calculus |
Tables
for Legendre’s Polynomials
Legendre’s polynomials can be defined in a very
elegant and compact way by the Rodrigues formulas
The first
five Legendre’s polynomials are:
The above
formula is quite expensive. It is better to use the following iteration formula
This formula is very popular, but from the point of
view of numeric calculus has one disadvantage: its coefficients are decimal and
this causes round-off errors leading inaccuracy for higher polynomial degree.
It is convenient to rearrange the iterative formula to avoid fractional
coefficients.
Let’s assume that a Legendre’s polynomial can be
written as
Where kn is an
integer number and Ln(x) is a polynomial having integer coefficients
The Legendre’ polynomial Pn(x) is fully
defined by the couple of (kn , Ln(x))
Starting
with
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Substituting in the recurrent equation we have
Rearraging we have
Separating the numerator and denominator we obtain the
following two recurrence equations
We can show
that the following iterative process, with n ³ 2 , gives all the couples (kn , Ln(x))
Where the coef() operator returns the vector of all
coefficients of the polynomial Vn(x)., and the GCD is the greatest
common divisor.
Then, we
get, finally the next couple (kn , Ln(x))
An arrangement in VB of the above process was adapted
to build the following table
of the
Legendre’s polynomial coefficients for
0 £
n £
24.
18-12-2004