plq_rock — Compute the PLQ Rockafellar function R(A,a(k)) of an operator A.
R = plq_rock(B, k)
matrix. A matrix [a;bm;bp] where a, bm (b-), and bp (b+) are row vectors as defined below.
integer. 1<=k<=m. The index of the a values for which to compute R(A,a(k)). i<=k use bm, i>=k use bp.
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)
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