First, the momentum fraction X and the scale MU of the point one is interested in are converted to the normalized scales LX and LMU as defined earlier. These are then subsequently translated into the A and B scales in such a way that the point (X,MU) is now (IA+A,IB+B) where IA and IB are integers and A and B have values between 0 and 1.
If the given X is outside the range of the lattice, LOG(PARTON) is extrapolated as a linear function of LOG(X/(1-X)).
The interpolation is performed using cubic splines. At each point of the lattice one knows the value of the parton distribution H, the first derivatives of the parton distribution with respect to A and B, i.e dH/dA and dH/dB, and finally the second derivative with respect to both A and B, i.e. d**2H/dAdB; a total of four values for each point. The point (IA+A,IB+B) at which we wish to find the value of the parton distribution is enclosed in a square with vertices (IA,IB), (IA+1,IB), (IA,IB+1), (IA+1,IB+1). At each of these points, we have the four values decribed above, for a total of sixteen values. The cubic spline given below
y = sum( kij * (A**i) * (B**j), i=0..3, j=0..3 )
or explicitly,
y = k1 + k2*A + k3*B + k4*(A**2) + k5*A*B + k6*(B**2) + k7*(A**3) + k8*(A**2)*B+ k9*A*(B**2)+k10*(B**3) + k11*(A**3)*B + k12*(A**2)*(B**2) + k13*A*(B**3) + k14*(A**3)*(B**2) + k15*(A**2)*(B**3) + k16*(A**3)*(B**3)
has sixteen constants and there are sixteen conditions: the constants can therefore be found.
Magic functions that satisfy
f(0)=1 f'(0)=0 f(1)=0 f'(1)=0
f(0)=0 f'(0)=1 f(1)=0 f'(1)=0
f(0)=0 f'(0)=0 f(1)=1 f'(1)=0
f(0)=0 f'(0)=0 f(1)=0 f'(1)=1
are constructed as functions of A, and as functions of B. These are then used to evaluate the parton distribution at the point in question. For optimal machine efficiency, only addition and multiplication operations are used in the interpolation, and that is where the magic functions play a crucial role. The value at the point in question is first approximated by a linear interpolation of the value at H(N,IA,IB). Successive refinements of this approximation are then made by adding terms calculated as the product of the first and second order derivatives at the vertices of the square and the magic functions. The net sum of all these contributions to the original approximation is the final interpolated value.
anandam@darkwing.uoregon.edu