Note that the output of poly follows the pattern for MATLAB's standard poly command: it is a row vector, and the high-order coefficients come first. Chebfun versions stored functions via their values at Chebyshev points, whereas version 5 switched to Chebyshev coefficients, but this hardly matters to the user, and both representations are exploited for various purposes internally in the system. Conversely, if f is a chebfun, then chebcoeffs f is the vector of its Chebyshev coefficients. Before version 5, the command for this was chebpoly. Unike poly , chebcoeffs returns a column vector with the low-order coefficients first.
If we apply chebcoeffs to a function that is not "really" a polynomial, we will usually get a vector whose last entry i. This reflects the adaptive nature of the Chebfun constructor, which always seeks to use a minimal number of points. By using poly we can print the coefficients of such a chebfun in the monomial basis. The numbers disagree in the last four digits, even though the functions represented in the two columns agree almost to full precision. This reflects the ill-conditioning of the monomial basis for representing functions on an interval.
The two columns represent monomial coefficients of two functions that agree to 15 or 16 digits. The fact that the numbers differ in the final four decimal places reflects the ill-conditioning of monomials as a basis for representing functions on an interval. We can zoom in on the overshoot region by resetting the axes:. What is the amplitude of the Gibbs overshoot for Chebyshev interpolation of a step function?
We can find out by using max :. After the step function, a function with "one more derivative" of smoothness would be the absolute value. For example:. Such plots look good to the eye, but they do not achieve machine precision. We can confirm this by using splitting on to compute a true absolute value and then measuring some norms. Here is an example of a smoother function, one that is in fact analytic. This time the convergence is equally clean but quite different in nature.
Now the straight line appears on the semilogy axes rather than the loglog axes, revealing the geometric convergence. The mathematics of Chebfun can be captured in five theorems about interpolants in Chebyshev points. For analytic functions the convergence is geometric. The essence of the following theorem is due to Bernstein in , though it is not clear where an explicit statement first appeared in print.
For a startling illustration of the implications of this theory, consider these two functions from the Chebfun gallery. Theorem 3 can be used to explain why their lengths are so different. The next theorem asserts that Chebyshev interpolants can be computed by the barycentric formula [Salzer ]. The final theorem asserts that the barycentric formula has no difficulty with rounding errors. Our "theorem" is really just an advertisement; see [Higham ] for a precise statement and proof.
The barycentric formula of Theorem 4 is numerically stable. Nevertheless it is provably stable. For practical computations, it is rarely worth the trouble to compute a best minimax approximation rather than simply a Chebyshev interpolant. Nevertheless best approximations are a beautiful and well-established idea, and it is certainly interesting to be able to compute them. Chebfun makes this possible with the command minimax.
Note that a second output argument from minimax returns the error as well as the polynomial. Chebfun's minimax command can compute rational best approximants too, and it is probably the most robust code in existence for such approximations. CF approximation often comes close to optimal for non-smooth functions too, provided you specify a fourth argument to tell the system which Chebyshev grid to use:. For a further indication of the power of this approach, see the Chebfun example "Digital filters via CF approximation".
Chebfun is based on polynomial interpolants in Chebyshev points, not equispaced points.go
It has been known for over a century that the latter choice is disastrous, even for interpolation of smooth functions [Runge ]. One should never use equispaced polynomial interpolants for practical work unless you will only need the result near the center of the interval of interpolation , but like best approximations, they are certainly interesting. In Chebfun, we can compute them with the interp1 command.
For example, here is an analytic function and its equispaced interpolant of degree Note the vertical scale. Approximation experts will know that one of the tools used in analyzing effects like this is the Lebesgue function associated with a given set of interpolation points. Chebfun has a command lebesgue for computing these functions. Equispaced points are perfect for trigonometric interpolation of periodic functions, of course, accessible in Chebfun with the trig flag, as described in Chapter This is the so-called "Clenshaw-Lord" Chebyshev-Pade approximation; if the flag maehly is specified the code alternatively computes the linearized variation known as the "Maehly" approximation.
Chebyshev-Pade approximation is the analogue for functions defined on an interval of Pade approximation for functions defined in a neighborhood of a point. The results are similar:. The third and fourth options for rational approximation, as mentioned in Section 4. As mentioned in Section 4. We explore the same function yet again, and this time obtain an equioscillating error curve:.
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