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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(1-min(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 multi-valued 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.
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