Code covered by the BSD License

# Interpolation Utilities

### Joe Henning (view profile)

22 May 2012 (Updated )

A variety of interpolation utilities

trigint(x, y, xi, c)
```function [yi, ypi] = trigint(x, y, xi, c)

% hRIGINT 1-D piecewise trigonometric interpolation
%    TRIGINT(X,Y,XI,C) interpolates to find YI, the values of the
%    underlying function Y at the points in the array XI, using
%    piecewise trigonometric interpolation.  X and Y must be vectors
%    of length N.
%
%    C specifies the number of data points to use in the
%    interpolation.  The default is to use all points.
%
%    [YI,YPI] = TRIGINT() also returns the interpolated derivative
%    of the underlying function Y at points XI.

% Joe Henning - Fall 2012

% On the Interpolation Trigonometric Polynomial with an Arbitrary Even Number of Nodes
% Ernest Scheiber
% 2011 13th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
% 978-0-7695-4630-8/11 2011 IEEE
% DOI 10.1109/SYNASC.2011.13

if (nargin < 4)
c = 0;
end

n = length(x);

if (n < c)
fprintf('??? Bad c input to trigint ==> c <= length(x)\n');
yi = [];
ypi = [];
return
end

for i = 1:length(xi)
% Find the right place in the table by means of a bisection.
klo = 1;
khi = n;
while (khi-klo > 1)
k = fix((khi+klo)/2.0);
if (x(k) > xi(i))
khi = k;
else
klo = k;
end
end

h = x(khi) - x(klo);
if (h == 0.0)
fprintf('??? Bad x input to trigint ==> x values must be distinct\n');
yi(i) = NaN;
ypi(i) = NaN;
continue;
end

% Evaluate lagrange polynomial
yi(i) = 0;
ypi(i) = 0;
if (c == 0)
if (mod(n,2) == 0)   % even
for k = 1:n
sumx = 0;
for m = 1:n
sumx = sumx + x(m);
end
term = y(k)*(cos(0.5*(xi(i)-x(k))) + sin(0.5*(xi(i)-x(k)))*cot(0.5*sumx));
termp = 0;
termp2 = y(k)*0.5*(-sin(0.5*(xi(i)-x(k))) + cos(0.5*(xi(i)-x(k)))*cot(0.5*sumx));
for m = 1:n
if (k ~= m)
term = term*sin(0.5*(xi(i)-x(m)))/sin(0.5*(x(k)-x(m)));
prod = 0.5*cos(0.5*(xi(i)-x(m)));
for j = 1:n
if ((k ~= j) && (m ~= j))
prod = prod*sin(0.5*(xi(i)-x(j)))/sin(0.5*(x(k)-x(j)));
end
end
termp = termp + y(k)*(cos(0.5*(xi(i)-x(k))) + sin(0.5*(xi(i)-x(k)))*cot(0.5*sumx))*prod/sin(0.5*(x(k)-x(m)));
termp2 = termp2*sin(0.5*(xi(i)-x(m)))/sin(0.5*(x(k)-x(m)));
end
end
yi(i) = yi(i) + term;
ypi(i) = ypi(i) + termp + termp2;
end
else   % odd
for k = 1:n
term = y(k);
termp = 0;
for m = 1:n
if (k ~= m)
term = term*sin(0.5*(xi(i)-x(m)))/sin(0.5*(x(k)-x(m)));
prod = 0.5*cos(0.5*(xi(i)-x(m)));
for j = 1:n
if ((k ~= j) && (m ~= j))
prod = prod*sin(0.5*(xi(i)-x(j)))/sin(0.5*(x(k)-x(j)));
end
end
termp = termp + y(k)*prod/sin(0.5*(x(k)-x(m)));
end
end
yi(i) = yi(i) + term;
ypi(i) = ypi(i) + termp;
end
end
else
if (mod(c,2) == 0)   % even
c2 = c/2;
if (klo < c2)
klo = c2;
end
if (klo > n-c2)
klo = n-c2;
end
khi = klo + 1;
for k = klo-(c2-1):klo+c2
sumx = 0;
for m = klo-(c2-1):klo+c2
sumx = sumx + x(m);
end
term = y(k)*(cos(0.5*(xi(i)-x(k))) + sin(0.5*(xi(i)-x(k)))*cot(0.5*sumx));
termp = 0;
termp2 = y(k)*0.5*(-sin(0.5*(xi(i)-x(k))) + cos(0.5*(xi(i)-x(k)))*cot(0.5*sumx));
for m = klo-(c2-1):klo+c2
if (k ~= m)
term = term*sin(0.5*(xi(i)-x(m)))/sin(0.5*(x(k)-x(m)));
prod = 0.5*cos(0.5*(xi(i)-x(m)));
for j = 1:n
if ((k ~= j) && (m ~= j))
prod = prod*sin(0.5*(xi(i)-x(j)))/sin(0.5*(x(k)-x(j)));
end
end
termp = termp + y(k)*(cos(0.5*(xi(i)-x(k))) + sin(0.5*(xi(i)-x(k)))*cot(0.5*sumx))*prod/sin(0.5*(x(k)-x(m)));
termp2 = termp2*sin(0.5*(xi(i)-x(m)))/sin(0.5*(x(k)-x(m)));
end
end
yi(i) = yi(i) + term;
ypi(i) = ypi(i) + termp + termp2;
end
else   % odd
c2 = floor(c/2);
if (klo < c2+1)
klo = c2+1;
end
if (klo > n-c2)
klo = n-c2;
end
khi = klo + 1;
for k = klo-c2:klo+c2
term = y(k);
termp = 0;
for m = klo-c2:klo+c2
if (k ~= m)
term = term*sin(0.5*(xi(i)-x(m)))/sin(0.5*(x(k)-x(m)));
prod = 0.5*cos(0.5*(xi(i)-x(m)));
for j = 1:n
if ((k ~= j) && (m ~= j))
prod = prod*sin(0.5*(xi(i)-x(j)))/sin(0.5*(x(k)-x(j)));
end
end
termp = termp + y(k)*prod/sin(0.5*(x(k)-x(m)));
end
end
yi(i) = yi(i) + term;
ypi(i) = ypi(i) + termp;
end
end
end
end
```