lft_llt - Legendre-Fenchel conjugate, LLT algorithm

Calling Sequence

Conj = lft_llt(X,Y,S)

Conj = lft_llt(X,Y,S,isConvex)

Conj = lft_llt(X,Y,S,isConvex,fusionopt)

Parameters

• X : column vector. A grid of points on which the function is sampled.
• Y : column vector. The value of the function on the grid X: usually Y(i)=f(X(i)) for some function f.
• S : column vector. The grid on which we want to compute the conjugate: f* is evaluated on S.
• isConvex : Optional. boolean. Whether or not the dataset (X,Y) is known to be convex. If false or omitted, bb is first applied to the data.
• fusionopt : Optional. integer. Select the implementation of the fusion algorithm. fusionopt=1 (or omitted) selects fusionsci, a fast implementation using scilab syntax but with nonlinear complexity. Any over value selects the fusion implementation, a (slower) loop-based implementation that runs in linear-time.
• Conj : column vector. Contains the value of the conjugate f* of the function f evaluated on at the points S(j). In other words: Conj(j) = Max(S(j) * X(i) - Y(i) | over all indexes i)

Description

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

Examples

    X=[-5:0.5:5]';
    Y=X.^2;
    S=(Y(2:size(Y,1))-Y(1:size(Y,1)-1))./(X(2:size(X,1))-X(1:size(X,1)-1));
    Conj=lft_llt(X,Y,S);
    Conj2=lft_llt(X,Y,S,1);
  

See Also

lft_llt_d,  lft_direct,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

Used Function