Scilab Function
Last update : 22/07/2008

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

Calling Sequence

Conj = lft_direct(X,Y,S)

Parameters

Description

Warning: This function is provided only for comparison purposes and unit testing, use the more efficient lft_llt or lft_plt for faster computation.

Compute numerically the discrete Legendre transform of a set of planar points (X(i),Y(i)) at slopes S, i.e.,

                 Conj(j)=max[S(j)*X(i)-Y(i)].
                          i
It uses straight computation for a quadratic-time algorithm theta(n*m) with n=length(X)=length(Y) and m=length(S).

Examples

    S=(-10:1:10)';
    X1=(-2:0.2:2)';f=abs(X1);C1 = lft_direct(X1, f, S);
    X2=(-4:0.4:4)';f=abs(X2);C2 = lft_direct(X2, f, S);
    X3=(-8:0.8:8)';f=abs(X3);C3 = lft_direct(X3, f, S);
    X4=(-160:16:160)';f=abs(X4);C4 = lft_direct(X4, f, S);

    scf(0);clf;
    plot2d([S S S S],[C1 C2 C3 C4],rect=[-3,-0.5,3,10]);
    xtitle("Convergence of the discrete Legendre transform of the absolute value to the indicator of [-1,1]");
  

See Also

lft_direct_d,  lft_llt,  

Author

Yves Lucet, University of British Columbia, BC, Canada