pl_lft_direct_2d — [For comparison only] 2D Legendre-Fenchel conjugate, direct computation
Conj = pl_lft_direct_2d (Xr, Xc, f, Sr, Sc)
column vector of length n.
column vector of length m.
matrix of size nxm. The function f is sampled on a grid Xr x Xc so f(i,j)=ff(Xr(i),Xc(j)) for some function ff.
column vector of size m1.
column vector of size m2. The Conjugate is computed on a grid SrxSc.
matrix of size m1xm2 containing the Conjugate of the function f.
Warning: This function is provided only for comparison purposes and unit testing, use the more efficient pl_lft_llt_2d for faster computation.
Numerically compute the discrete Legendre transform on the grid Sr x Sc, given a function f(x,y) defined on a grid Xr x Xc, using the pl_lft_direct algorithm to compute the conjugate in one dimension, then to compute it in the other dimension. If n==length(Xr)==length(Xc)==length(Sr)==length(Sc) and N=n^2, this function calls pl_lft_direct n times in one dimension and then n times in the other dimension (2*n^3), giving a O(N^(3/2)) running time with respect to the O(n^2) input size.
The conjugate of a function in R^2 can be factored to several conjugates which are elements of R.
f*(s(1),s(2)) = Sup [x(1)y(1) + x(2)y(2) - f(x(1),x(2))] x(1),x(2) = Sup [s(1)x(1) + Sup[s(2)x(2) - f(x(1),x(2))]] x(1) x(2)
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