plq_rock - Compute the PLQ Rockafellar function R(A,a(k)) of an operator A.
Compute the PLQ Rockafellar function R(A,a(k)) of an operator A, where B is defined below. This function runs in linear time, computing a PLQ piece for each i=1:m+1 where a(i-1) < x <= a(i).
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). R(A, 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)
a = 1:5; bm = [-10,-4, 1, 2, 5]; bp = [- 5, 0, 2, 4, 7]; B = [a;bm;bp]; plq_rock(B, 1), plq_rock(B, 5),
op_fitz_brute, op_fitz_direct, op_fitz, op_fitzinf, plq_fitzinf0, plq_fitzinf0_direct,
Bryan Gardiner, University of British Columbia, BC, Canada