How to reconstruct the polynomial form of a multi-dimensional spline at each interval using csaps?

I am using csaps to fit a 3-dimensional cubic spline, and need to reconstruct the polynomial form for each interval. while pp.coefs provides the coefficients (in an array format), I am not sure about correct assignment of provided coefficients to different terms of the cubic spline at each interval. Here is my sample code:

x={-1:1:1;-1:1:1;-1:1:1} [xx,yy,zz] = ndgrid(x{1},x{2},x{3}); v=2*xx.^2+4*yy.^2-3*xx.*zz; [pp,p]=csaps(x,v);

Answer 1

[sanidhyak]

Hi @Adel,

I understand that you are trying to reconstruct the polynomial form of a multi-dimensional spline at each interval using MATLAB’s “csaps” function.

When working with multi-dimensional splines, we need to extract and assign the coefficients from “pp.coefs” to their respective polynomial terms.

The approach would be as follows:

• Use “csaps” to generate a 3-dimensional cubic spline.
• Extract coefficients using “pp.coefs”.
• Iterate through each interval, reconstructing and displaying the polynomial terms using “syms”.

This will make sure that the correct polynomial form for each interval is obtained from the “csaps” output.

Below is the MATLAB code for the same:

x = {-1:1:1, -1:1:1, -1:1:1}; 
[xx, yy, zz] = ndgrid(x{1}, x{2}, x{3}); 
v = 2*xx.^2 + 4*yy.^2 - 3*xx.*zz; 

[pp, p] = csaps(x, v); 

% Extract coefficients 
coefs = pp.coefs; 

% Display coefficients for each dimension 
for i = 1:size(coefs, 1) 
    fprintf('Interval %d:\n', i); 
    disp(coefs(i, :)); 
end 

% Reconstruct polynomial terms manually 
syms X Y Z 
for i = 1:size(coefs, 1) 
    poly = coefs(i, 1)*X^3 + coefs(i, 2)*Y^3 + coefs(i, 3)*Z^3 + ... 
        coefs(i, 4)*X^2 + coefs(i, 5)*Y^2 + coefs(i, 6)*Z^2 + ... 
        coefs(i, 7)*X*Y + coefs(i, 8)*X*Z + coefs(i, 9)*Y*Z + ... 
        coefs(i, 10)*X + coefs(i, 11)*Y + coefs(i, 12)*Z + coefs(i, 13); 
    fprintf('Polynomial for interval %d:\n', i); 
    disp(vpa(poly, 4)); 
end 

For further reference on “csaps”, kindly refer to the following documentation:

• https://www.mathworks.com/help/curvefit/csaps.html

I hope this helps!

(As can be seen in the screenshot below, the error has been resolved)