🚀 go-pugleaf

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.