OK, I did have time to go read ppval; indeed, it does presume coefficients are evaluated from each section as the origin, not in absolute coordinates--inside the function is
...
[b,c,l,o]=unmkpp(pp); % unpacks pp-->[bkpt,coefs,pieces,order]
[~,index] = histc(xs,[-inf,b(2:l),inf]); % get which section each input xq goes with
...
% now go to local coordinates ... % subtracts beginning breakpoint x from input
xs = xs-b(index); % vector of x so it thinks each is at origin
...
and goes on to evaluate the functional from there.
To make it work as you've defined the coefficients and coordinates, you have to evaluate over a modified xq vector that counteracts this internal shifting...
[~,ix]=histc(x,[-inf pp.breaks(2:pp.pieces) inf]); % get the equivalent index vector
xq=x+pp.breaks(ix); % "real" x plus breakpoint by section
with that then,
plot(x,y,'*-')
hold on
plot(x,ppval(pp,xq),'r-')
returns the expected result
Alternatively, compute the coefficients for each region based on the breakpoint for the region so each section origin is the breakpoint for the section.
Have to read the details of the documentation very carefully; the examples don't show that detail at all clearly. The description for the coefs though does spell it out; of course I didn't read that until just now after all the above! :)
coefs -- Poynomial coefficients
Polynomial coefficients, specified as an L-by-k matrix with the ith row coefs(i,:) containing
the local coefficients of an order k polynomial on the ith interval, [breaks(i), breaks(i+1)].
In other words, the polynomial is
coefs(i,1)*(X-breaks(i))^(k-1) + coefs(i,2)*(X-breaks(i))^(k-2) + ... + coefs(i,k-1)*(X-breaks(i)) + coefs(i,k).
