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

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Rohitashya on 20 Nov 2024 at 15:18

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https://uk.mathworks.com/matlabcentral/answers/2167508-how-to-reduce-the-matrix-size-for-newton-computation-for-my-problem

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

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

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

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

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

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

-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

-0.5000 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

How can I reduce this size as it will cause problem in interpolation to 4*4 or in general less .

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

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