lft_llt2d - 2D Legendre-Fenchel conjugate, LLT algorithm

Calling Sequence

Conj = lft_llt2d (Xr, Xc, f, Sr, Sc)

Conj = lft_llt2d (Xr, Xc, f, Sr, Sc, isConvex)

Parameters

• Xr : column vector of length n.
• Xc : column vector of length m.
• f : 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.
• Sr : column vector of size m1.
• Sc : column vector of size m2. The Conjugate is computed on a grid SrxSc.
• isConvex : Boolean, optional. Whether or not the given function is known to be true. Defaults to false. It is passed to lft_llt.
• Conj : matrix of size m1xm2 containing the Conjugate of the function f.

Description

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)

Examples

    Xr=(-2:0.1:2)'; 
    Xc=(-2:0.1:2)'; 
    Sr=(-2:0.1:2)';
    Sc=(-2:0.1:2)';
    deff('[z]=f(Xr,Xc)',['z= Xr^2 + Xc^2']);
    z=eval3d(f,Xr,Xc);
    result_llt = lft_llt2d (Xr, Xc, z, Sr, Sc, 0); //llt method
    result_direct = lft_direct2d (Xr, Xc, z, Sr, Sc, 1); //direct method
  

See Also

lft_llt,  lft_direct2d,  

Author

Mike Trienis, University of British Columbia, BC, Canada

Used Function