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2.2 Interpolating
As I have said, a Yorick array often represents the values of a
continuous function at a number of discrete points. In order to find
the values of the function at other points, you need to know how it
varies between (or beyond) the given points. In general, to interpolate
(or extrapolate), you need a detailed understanding of how the function
was discretized in the first place. However, when the list of values
accurately represents the function, linear interpolation between the
known points will suffice. A function which is linear between
successive points is called "piecewise linear".
The interp function is a mechanism for converting a list of
function values at discrete points into a piecewise linear function
which can be evaluated at any point.
 theta = span(0, pi, 100);
x_circle = cos(theta);
y_circle = sin(theta);
x = span(2, 2, 64);
y = interp(y_circle, x_circle, x);

This code fragment produces a y array with the same number of
points as x (64), with the values of the piecewise linear function
defined by the points (x_circle, y_circle). Outside the range
covered by x_circle, the piecewise linear function remains
constant  the simplest possible extrapolation rule.
Regarded as a function of its third argument, interp behaves just
like the sin or cos function  its first two arguments are
really parameters specifying which piecewise linear function
interp will evaluate.
The integ function works just like interp, except that it
returns the integral of the piecewise linear function. The integration
constant is chosen so that integ returns zero at the first point
of the piecewise linear function. (This point will actually have the
maximum value of x if the x array is decreasing.) Thus, the integral
of the piecewise linear approximation to the semicircle and the exact
integral of the semicircle can be computed by:
 yi = integ(y_circle, x_circle, x);
yi_exact = 0.5*(acos(max(min(x,1),1))  x*sqrt(1min(x^2,1)));

Again, the piecewise linear function is assumed to remain constant
beyond the first and last points specified. Hence, integ is a
linear function when extrapolating, and piecewise parabolic when
interpolating.
Use integ only when you need the indefinite integral of a
piecewise linear function. Yorick has more efficient ways to compute
definite integrals. Again, think of integ, like interp, as
a continuous function of its third argument; the first two arguments are
parameters specifying which function.
Neither interp nor integ makes sense unless its second
argument is either increasing or decreasing. There is no way to decide
which branch of a multivalued function should be returned.
Internally, both interp and integ need a lookup function 
that is, a function which finds the index of the point in x_circle just
beyond each of the x values. This lookup function can also be called
directly; its name is digitize.
