Scilab Function
Last update : 23/07/2008

op_fitzinf - Evaluate the Fitzpatrick function of infinite order of an operator on a grid using Rockafellar functions.

Calling Sequence

F = op_fitzinf(B, x, xstar)

Parameters

Description

Evaluates the Fitzpatrick function of infinite order of an operator A on a grid (x,x*) using Rockafellar functions, where B is defined as:

    B : x -> conv(A*x):
          m
        union ( {a(i)} cross [bm(i),bp(i)] )
         i=1

    with a(1) < a(2) < ... < a(m),
         bm(1) <= bp(1) <= bm(2) <= bp(2) <= ... <= bm(m) <= bp(m),
         a(0) = bm(0) = bp(0) = -%inf,
         a(m+1) = bm(m+1) = bp(m+1) = %inf,
         m = size(B,2).

This function computes the Rockafellar functions R(A,a(k)) for all k=1..m, evaluates the functions at (x,x*), then uses LLT1d algorithm to find the maximum result. It runs in O(m^2 + m*Nx + N) time, or O(m^2) when m==Nx==Nxstar. See also plq_rock, which returns PLQ Rockafellar functions, and plq_fitzinf0, which returns PLQ Fitzpatrick functions.

    F(A, %inf, x, xstar) =    sup     [ a*xstar + R(A,a)(x) ]
                           a in Dom A

    R(A, k)(x) = { (x-a(i))*bm(i) + sum(j=i+1:k, (a(j-1)-a(j))*bm(j)) , if a(i-1) < x <= a(i) <= a(k)
                 { (x-a(i))*bp(i) + sum(j=k:i-1, (a(j+1)-a(j))*bp(j)) , if a(k) <= a(i) <= x <= a(i+1)

    m = size(A, 2);
    Nx = size(x, 1);
    Nxstar = size(xstar, 1);
    N = Nx * Nxstar;

Examples

a = -4:4;
bm = [-15,-13,-10,-7,-6,-4,0.5,1,1];
bp = [-13,-11,- 8,-6,-5, 0,  1,1,2];
B = [a;bm;bp];
x = -10:10;
xstar = -10:10;
F = op_fitzinf(B, x, xstar),

clf(); alpha=30; theta=60;
plot3d(x, xstar, F, alpha=alpha, theta=theta);
  

See Also

op_fitz_brute,  op_fitz_direct,  op_fitz,  plq_fitzinf0,  plq_fitzinf0_direct,  plq_rock,  

Author

Bryan Gardiner, University of British Columbia, BC, Canada