Scilab Function
Last update : 12/4/2006

me_llt - Moreau envelope, LLT algorithm

Calling Sequence

M = me_llt(X,f,S,fusionopt)

Parameters

Description

Compute numerically the discrete Moreau envelope of a set of planar points (X(i),f(i)) at slopes S(j), i.e.

	   				      2
	   M(j) = min f(i) + || s(j) - x(i) ||.
		   i
It reduces computation to computing the Legendre conjugate through the formula
           2                                                      2
M(j) = s(j) - 2 g*(j) with g*(j) = max [ s(j) * x(i) - 1/2 * (x(i) +  f(i)) ]
                                    i
thereby resulting in a theta(n + m) linear-time algorithm with n=length(X)=length(f) and m=length(S).

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));
	M=me_llt(X,Y,S)

See Also

me_llt2d ,   me_direct ,   me_nep ,   me_pe ,  

Author

Yves Lucet, University of British Columbia, BC, Canada

Bibliography

Y. Lucet, 2006, Fast Moreau Envelope Computation I: Numerical Algorithms, Numerical Algorithms, 43 (2006), 235-249

Used Function

The core computation is done by lft_llt .