Chebyshev approximation puzzle

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2 total messages Started by Ajay Shah Thu, 06 Aug 1992 12:23

#198

Author: Ajay Shah
Date: Thu, 06 Aug 1992 12:23

33 lines
1446 bytes

I was playing with the Numerical Recipes code which calculates
the chebyshev approximation of a user-specified function.
To convince myself it works, I tried to approximate the function 
f(x) = T_0(x) - 2T_1(x) + 3T_2(x) - T_4(x) on (-1, 1).  I expect him
to recover the true fourier coefficients (1, -2, 3, -1, 0, 0, 0, ...).

The puzzle is: the approximation function (named chebft) correctly
recovers the coefficients -2, 3, -1.  It fails on the first one (the
constant).  It returns 2 instead of 1.  Higher order terms are
correctly set to 0.

At the same time, when I compare the approximation-evaluation vs. the
function evaluation, there is no error.  

What may I be missing?

        -ans.

In case you would like to play with it:

double f(double x)
{
    int i;
    double t[4], constants[4] = {1,-2,3,-1}, sum=0;
    t[0] = 1;
    t[1] = x;
    t[2] = 2*x*x - 1;
    t[3] = 4*x*x*x - 3*x;
    for (i=0; i<4; i++) sum += t[i]*constants[i];
    return sum;
}
-- 
Ajay Shah, (213)749-8133, ajayshah@usc.edu

Re: Chebyshev approximation puzzle

#232

Author: Andrew Haley
Date: Tue, 11 Aug 1992 11:29

42 lines
1834 bytes

ajayshah@almaak.usc.edu (Ajay Shah) writes:
: I was playing with the Numerical Recipes code which calculates
: the chebyshev approximation of a user-specified function.
: To convince myself it works, I tried to approximate the function 
: f(x) = T_0(x) - 2T_1(x) + 3T_2(x) - T_4(x) on (-1, 1).  I expect him
: to recover the true fourier coefficients (1, -2, 3, -1, 0, 0, 0, ...).
: 
: The puzzle is: the approximation function (named chebft) correctly
: recovers the coefficients -2, 3, -1.  It fails on the first one (the
: constant).  It returns 2 instead of 1.  Higher order terms are
: correctly set to 0.
: 
: At the same time, when I compare the approximation-evaluation vs. the
: function evaluation, there is no error.  
: 
: What may I be missing?
: 
:         -ans.
: 
: In case you would like to play with it:
: 
: double f(double x)
: {
:     int i;
:     double t[4], constants[4] = {1,-2,3,-1}, sum=0;
:     t[0] = 1;
:     t[1] = x;
:     t[2] = 2*x*x - 1;
:     t[3] = 4*x*x*x - 3*x;
:     for (i=0; i<4; i++) sum += t[i]*constants[i];
:     return sum;
: }
: -- 
: Ajay Shah, (213)749-8133, ajayshah@usc.edu

The Numerical recipes code includes a kluge to improve the efficiency
of the Chebyshev evaluation function.  The first term of a Chebyshev
series is conventinally multiplied by two.  To allow the evaluation
function to ignore this fact, the first coefficient in the output
array is doubled.

Andrew.

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