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CCA (Computational Convex Analysis) >> CCA (Computational Convex Analysis) > pl_lft_direct_2d

# pl_lft_direct_2d

[For comparison only] 2D Legendre-Fenchel conjugate, direct computation

### Calling Sequence

    Conj = pl_lft_direct_2d (Xr, Xc, f, Sr, Sc)

### 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.

Conj

  matrix of size m1xm2 containing the Conjugate of the function f.

### Description

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)

### 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 = pl_lft_llt_2d (Xr, Xc, z, Sr, Sc, 0); //llt method
    result_direct = pl_lft_direct_2d (Xr, Xc, z, Sr, Sc, 1); //direct method

### See Also

- pl_lft_llt_2d
- pl_lft_direct

### Authors

`Mike Trienis`, University of British Columbia, BC, Canada

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