How to reduce the matrix size for Newton Computation for my problem ?

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Rohitashya
Rohitashya 2024-11-20，15:18
This code is computed from the formula :
Basically I convert the Farrow Coefficients to Newton coefficients then compute the filter transfer function .
This code is from dsp.FarrowRateConverter to compute Farrow Lagrange coefficients
%%Newton Converter function
function C = designModifiedFarrowCoefficients(N)
if N==0
C = 1;
return
end
% Nondegenerate form, explicitly derived in
% http://www.acoustics.hut.fi/~vpv/publications/vesan_vaitos/ch3_pt2_lagrange.pdf
T = eye(N+1);
for i = 1:N+1
for j = i:N+1
T(i,j) = nchoosek(j-1, i-1)*(floor(N/2))^(j-i);
end
end
C = flipud(T)/fliplr(vander(0:N));
% Force near-zeros and and near-ones to eliminate roundoff error
[Ns, Ds] = rat(C);
C(Ns == 0) = 0;
C(Ns == Ds) = 1;
% Note: below is a shorter alternative
% exponents = (0:N);
% taps = exponents' - floor(N/2);
% C = flipud(inv(taps.^exponents));
end
The solution for >> designModifiedFarrowCoefficients(3)
ans =
-0.1667 0.5000 -0.5000 0.1667
0.5000 -1.0000 0.5000 0
-0.3333 -0.5000 1.0000 -0.1667
0 1.0000 0 0
Its a 4cross4 matrix
Now for transformation computing Td,Td1,Td2 and Tz
which are given as
function Td1 = compute_Td1(M)
% Create a matrix of binomial coefficients
[I, J] = ndgrid(1:M, 1:M);
binomials = zeros(M, M);
for i = 1:M
binomials(i, 1:i) = arrayfun(@(ii, jj) nchoosek(ii-1, jj-1), I(i, 1:i), J(i, 1:i));
end
% Create a matrix for powers
powers = ((- (M - 1) / 2) .^ (I - J)) .* (J <= I);
% Element-wise multiplication to get Td1
Td1 = binomials .* powers;
end
function Td2 = compute_Td2(M)
% Initialize Td2 with identity
Td2 = eye(M+1);
for n = 2:M+1
for k = 2:n-1
Td2(n, k) = Td2(n-1, k-1) - (n-2) * Td2(n-1, k);
end
end
Td2 = Td2(2:M+1, 2:M+1);
end
function T_D = compute_TD(M)
% Compute Td1 using compute_Td1 function
Td1 = compute_Td1(M);
% Compute Td2 using compute_Td2 function
Td2 = compute_Td2(M);
% Matrix multiplication (cross product)
T_D = Td1 * Td2;
end
function T_z = calculate_Tz(M)
% Initialize the matrix T_z with zeros
T_z = zeros(M, M);
% Fill the matrix T_z using the given formula
for i = 1:M
for j = 1:i % j goes from 1 to i
T_z(i, j) = nchoosek(i-1, j-1) * (-1)^(j+1);
end
end
end
Then the function for calculating Newton Coefficients :
function Q = computeNewtonCoefficients(M)
% Compute Td (T_D in the description)
T_D = compute_TD(M);
% Compute Tz
T_z = calculate_Tz(M);
% Compute the modified Lagrange coefficient matrix P
P = designModifiedFarrowCoefficients(M);
% Compute Q using the transformation Q = (T_D^(-T)) * P * (T_z^(-1))
% Q = inv(T_D') .* P .* inv(T_z);
O = kron(inv(T_D'),P);
Q = kron(O,inv(T_z));
end
solution > computeNewtonCoefficients(3)
ans =
Columns 1 through 10
-0.1667 0 0 0.5000 0 0 -0.5000 0 0 0.1667
-0.1667 0.1667 0 0.5000 -0.5000 0 -0.5000 0.5000 0 0.1667
-0.1667 0.3333 -0.1667 0.5000 -1.0000 0.5000 -0.5000 1.0000 -0.5000 0.1667
0.5000 0 0 -1.0000 0 0 0.5000 0 0 0
0.5000 -0.5000 0 -1.0000 1.0000 0 0.5000 -0.5000 0 0
0.5000 -1.0000 0.5000 -1.0000 2.0000 -1.0000 0.5000 -1.0000 0.5000 0
-0.3333 0 0 -0.5000 0 0 1.0000 0 0 -0.1667
-0.3333 0.3333 0 -0.5000 0.5000 0 1.0000 -1.0000 0 -0.1667
-0.3333 0.6667 -0.3333 -0.5000 1.0000 -0.5000 1.0000 -2.0000 1.0000 -0.1667
0 0 0 1.0000 0 0 0 0 0 0
0 0 0 1.0000 -1.0000 0 0 0 0 0
0 0 0 1.0000 -2.0000 1.0000 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Columns 11 through 20
0 0 -0.3333 0 0 1.0000 0 0 -1.0000 0
-0.1667 0 -0.3333 0.3333 0 1.0000 -1.0000 0 -1.0000 1.0000
-0.3333 0.1667 -0.3333 0.6667 -0.3333 1.0000 -2.0000 1.0000 -1.0000 2.0000
0 0 1.0000 0 0 -2.0000 0 0 1.0000 0
0 0 1.0000 -1.0000 0 -2.0000 2.0000 0 1.0000 -1.0000
0 0 1.0000 -2.0000 1.0000 -2.0000 4.0000 -2.0000 1.0000 -2.0000
0 0 -0.6667 0 0 -1.0000 0 0 2.0000 0
0.1667 0 -0.6667 0.6667 0 -1.0000 1.0000 0 2.0000 -2.0000
0.3333 -0.1667 -0.6667 1.3333 -0.6667 -1.0000 2.0000 -1.0000 2.0000 -4.0000
0 0 0 0 0 2.0000 0 0 0 0
0 0 0 0 0 2.0000 -2.0000 0 0 0
0 0 0 0 0 2.0000 -4.0000 2.0000 0 0
0 0 -0.1667 0 0 0.5000 0 0 -0.5000 0
0 0 -0.1667 0.1667 0 0.5000 -0.5000 0 -0.5000 0.5000
0 0 -0.1667 0.3333 -0.1667 0.5000 -1.0000 0.5000 -0.5000 1.0000
0 0 0.5000 0 0 -1.0000 0 0 0.5000 0
0 0 0.5000 -0.5000 0 -1.0000 1.0000 0 0.5000 -0.5000
0 0 0.5000 -1.0000 0.5000 -1.0000 2.0000 -1.0000 0.5000 -1.0000
0 0 -0.3333 0 0 -0.5000 0 0 1.0000 0
0 0 -0.3333 0.3333 0 -0.5000 0.5000 0 1.0000 -1.0000
0 0 -0.3333 0.6667 -0.3333 -0.5000 1.0000 -0.5000 1.0000 -2.0000
0 0 0 0 0 1.0000 0 0 0 0
0 0 0 0 0 1.0000 -1.0000 0 0 0
0 0 0 0 0 1.0000 -2.0000 1.0000 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
Columns 21 through 30
0 0.3333 0 0 -0.8333 0 0 2.5000 0 0
0 0.3333 -0.3333 0 -0.8333 0.8333 0 2.5000 -2.5000 0
-1.0000 0.3333 -0.6667 0.3333 -0.8333 1.6667 -0.8333 2.5000 -5.0000 2.5000
0 0 0 0 2.5000 0 0 -5.0000 0 0
0 0 0 0 2.5000 -2.5000 0 -5.0000 5.0000 0
1.0000 0 0 0 2.5000 -5.0000 2.5000 -5.0000 10.0000 -5.0000
0 -0.3333 0 0 -1.6667 0 0 -2.5000 0 0
0 -0.3333 0.3333 0 -1.6667 1.6667 0 -2.5000 2.5000 0
2.0000 -0.3333 0.6667 -0.3333 -1.6667 3.3333 -1.6667 -2.5000 5.0000 -2.5000
0 0 0 0 0 0 0 5.0000 0 0
0 0 0 0 0 0 0 5.0000 -5.0000 0
0 0 0 0 0 0 0 5.0000 -10.0000 5.0000
0 0.1667 0 0 -0.8333 0 0 2.5000 0 0
0 0.1667 -0.1667 0 -0.8333 0.8333 0 2.5000 -2.5000 0
-0.5000 0.1667 -0.3333 0.1667 -0.8333 1.6667 -0.8333 2.5000 -5.0000 2.5000
0 0 0 0 2.5000 0 0 -5.0000 0 0
0 0 0 0 2.5000 -2.5000 0 -5.0000 5.0000 0
0.5000 0 0 0 2.5000 -5.0000 2.5000 -5.0000 10.0000 -5.0000
0 -0.1667 0 0 -1.6667 0 0 -2.5000 0 0
0 -0.1667 0.1667 0 -1.6667 1.6667 0 -2.5000 2.5000 0
1.0000 -0.1667 0.3333 -0.1667 -1.6667 3.3333 -1.6667 -2.5000 5.0000 -2.5000
0 0 0 0 0 0 0 5.0000 0 0
0 0 0 0 0 0 0 5.0000 -5.0000 0
0 0 0 0 0 0 0 5.0000 -10.0000 5.0000
0 0 0 0 -0.1667 0 0 0.5000 0 0
0 0 0 0 -0.1667 0.1667 0 0.5000 -0.5000 0
0 0 0 0 -0.1667 0.3333 -0.1667 0.5000 -1.0000 0.5000
0 0 0 0 0.5000 0 0 -1.0000 0 0
0 0 0 0 0.5000 -0.5000 0 -1.0000 1.0000 0
0 0 0 0 0.5000 -1.0000 0.5000 -1.0000 2.0000 -1.0000
0 0 0 0 -0.3333 0 0 -0.5000 0 0
0 0 0 0 -0.3333 0.3333 0 -0.5000 0.5000 0
0 0 0 0 -0.3333 0.6667 -0.3333 -0.5000 1.0000 -0.5000
0 0 0 0 0 0 0 1.0000 0 0
0 0 0 0 0 0 0 1.0000 -1.0000 0
0 0 0 0 0 0 0 1.0000 -2.0000 1.0000
Columns 31 through 36
-2.5000 0 0 0.8333 0 0
-2.5000 2.5000 0 0.8333 -0.8333 0
-2.5000 5.0000 -2.5000 0.8333 -1.6667 0.8333
2.5000 0 0 0 0 0
2.5000 -2.5000 0 0 0 0
2.5000 -5.0000 2.5000 0 0 0
5.0000 0 0 -0.8333 0 0
5.0000 -5.0000 0 -0.8333 0.8333 0
5.0000 -10.0000 5.0000 -0.8333 1.6667 -0.8333
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
-2.5000 0 0 0.8333 0 0
-2.5000 2.5000 0 0.8333 -0.8333 0
-2.5000 5.0000 -2.5000 0.8333 -1.6667 0.8333
2.5000 0 0 0 0 0
2.5000 -2.5000 0 0 0 0
2.5000 -5.0000 2.5000 0 0 0
5.0000 0 0 -0.8333 0 0
5.0000 -5.0000 0 -0.8333 0.8333 0
5.0000 -10.0000 5.0000 -0.8333 1.6667 -0.8333
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
-0.5000 0 0 0.1667 0 0
-0.5000 0.5000 0 0.1667 -0.1667 0
-0.5000 1.0000 -0.5000 0.1667 -0.3333 0.1667
0.5000 0 0 0 0 0
0.5000 -0.5000 0 0 0 0
0.5000 -1.0000 0.5000 0 0 0
1.0000 0 0 -0.1667 0 0
1.0000 -1.0000 0 -0.1667 0.1667 0
1.0000 -2.0000 1.0000 -0.1667 0.3333 -0.1667
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
How can I reduce this size as it will cause problem in interpolation to 4*4 or in general less .

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