pl_lft_llt_2d — 2D Legendre-Fenchel conjugate, LLT algorithm
Conj = pl_lft_llt_2d (Xr, Xc, f, Sr, Sc) Conj = pl_lft_llt_2d (Xr, Xc, f, Sr, Sc, isConvex)
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.
Boolean, optional. Whether or not the given function is known to be true. Defaults to false. It is passed to pl_lft_llt.
matrix of size m1xm2 containing the Conjugate of the function f.
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 LLT1d 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), this function calls LLT1d n times in one dimension and then n times in the other dimension (2*n^2), giving a linear 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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