Coefficient of best-fit polynomial

From: Ken Honda (Honda_Kiai_at_hotmail.com)
Date: 03/08/05 

Date: 8 Mar 2005 10:43:28 -0800

Hi,
I have been struggling for the last few days over a problem in
Luenberger's "Optimization by Vector Space Methods" and I cannot seem
to get anywhere. Can anybody help me with this problem?

Suppose randomly varying data in the form of a function x(t) is
observed from t = 0 to t = T. We can predict the future data (t>T) by
fitting a polynomial of degree (n-1) to the past data and using the
extrapolated values of the polynomial as the estimate.

The best fit polynomial, for our purposes, will be the polynomial
p(T,t) that minimizes the integral of [x(t) = p(T,T)]^2 as t varies
from 0 to T (so it's the closest in the L^2 norm). Show that the
coefficients of p(T,t), call them a_i(T), do not need to be completely
recomputed for each T but rather can be continuously updated according
to a formula of the form

d/dT [a_i(T)] = b_i*e(T)/T^i

where the b_i are fixed constants and e(T) is the instantaneous error
x(T) - p(T,T).

The section that precedes this discusses the Legendre polynomials as an
orthonormal basis for the space of polynomials, so I'm sure the first
step is to take x(t) and project it to the space of (n-1) degree
polynomials with teh first (n-1) Legendres as a basis. Then we have to
move from the Legendres as a basis to the standard basis
{1,t,t^2,...,t^(n-1)}. But I can't figure out where on earth that
differential equation comes from. Any suggestions?

Thanks!
KH

Relevant Pages

  • Re: linear map question
    ... X i.e. find a matrix representing differentiation, ... the standard basis of this polynomial is 1,x,x^2. ... Representing the linear mapping ... polynomials of degree two or less, ...
    (sci.math)
  • Re: Exchange algorithm (Remez or Parks-McClellan)
    ... for the second remez there is no matrix: ... whatever basis I like to represent the polynomials. ... The worst basis to use it the power basis, ... if you use the chebyshev basis ...
    (sci.math.num-analysis)
  • base for infinite dimensional space
    ... continuous function so the statement doesn't bothered me first. ... gurantees approximation with polynomial.) ... was the basis. ... many polynomials and expso I convince it's infinite dimensional ...
    (sci.math)
  • Re: base for infinite dimensional space
    ... > continuous function so the statement doesn't bothered me first. ... > many polynomials and expso I convince it's infinite dimensional ... > then what the heck is the basis for this space?? ... > spans the whole space with FINITE linear combination. ...
    (sci.math)
  • Re: base for infinite dimensional space
    ... >continuous function so the statement doesn't bothered me first. ... >gurantees approximation with polynomial.) ... >many polynomials and expso I convince it's infinite dimensional ... >then what the heck is the basis for this space?? ...
    (sci.math)