function v = splinetx(x,y,u)
%SPLINETX  Textbook spline function.
%  v = splinetx(x,y,u) finds the piecewise cubic interpolatory
%  spline S(x), with S(x(j)) = y(j), and returns v(k) = S(u(k)).
%
%  See SPLINE, PCHIPTX.

%   Copyright 2014 Cleve Moler
%   Copyright 2014 The MathWorks, Inc.

%  First derivatives

   h = diff(x);
   delta = diff(y)./h;
   d = splineslopes(h,delta);

%  Piecewise polynomial coefficients

   n = length(x);
   c = (3*delta - 2*d(1:n-1) - d(2:n))./h;
   b = (d(1:n-1) - 2*delta + d(2:n))./h.^2;

%  Find subinterval indices k so that x(k) <= u < x(k+1)

   k = ones(size(u));
   for j = 2:n-1
      k(x(j) <= u) = j;
   end

%  Evaluate spline

   s = u - x(k);
   v = y(k) + s.*(d(k) + s.*(c(k) + s.*b(k)));


% -------------------------------------------------------

function d = splineslopes(h,delta)
%  SPLINESLOPES  Slopes for cubic spline interpolation.
%  splineslopes(h,delta) computes d(k) = S'(x(k)).
%  Uses not-a-knot end conditions.

%  Diagonals of tridiagonal system

   n = length(h)+1;
   a = zeros(size(h)); b = a; c = a; r = a;
   a(1:n-2) = h(2:n-1);
   a(n-1) = h(n-2)+h(n-1);
   b(1) = h(2);
   b(2:n-1) = 2*(h(2:n-1)+h(1:n-2));
   b(n) = h(n-2);
   c(1) = h(1)+h(2);
   c(2:n-1) = h(1:n-2);

%  Right-hand side

   r(1) = ((h(1)+2*c(1))*h(2)*delta(1)+ ...
          h(1)^2*delta(2))/c(1);
   r(2:n-1) = 3*(h(2:n-1).*delta(1:n-2)+ ...
              h(1:n-2).*delta(2:n-1));
   r(n) = (h(n-1)^2*delta(n-2)+ ...
          (2*a(n-1)+h(n-1))*h(n-2)*delta(n-1))/a(n-1);

%  Solve tridiagonal linear system

   d = tridisolve(a,b,c,r);