%% 13. Equispaced points, Runge phenomenon
ATAPformats
%%
% So far in this book, we have considered three good methods for
% approximating functions by polynomials: Chebyshev interpolation,
% Chebyshev projection, and best approximation. Now we shall look at a
% catastrophically bad method!---interpolation in equally spaced points.
% This procedure is so unreliable that for generations, it has tainted
% people's views of the whole subject of polynomial interpolation. The
% mathematical tools we will need to understand what is going on are the
% Hermite integral formula and potential theory, as discussed in the last
% two chapters.
%%
% As mentioned in Chapter 5, polynomial interpolation was an established
% tool by the 19th century. The question of whether or not polynomial
% interpolants would converge to an underlying function as $n\to \infty$
% was not given much attention. Presumably many mathematicians would have
% supposed that if the function was analytic, the answer would be yes. In
% 1884 and 1896, $\hbox{M\'eray}$ published a pair of papers in which he
% identified the fact that certain interpolation schemes do not converge
% $\hbox{[M\'eray}$ 1884 & 1896]. In the first paper he writes,
%%
%
% \em It is rather astonishing that practical applications have not yet
% turned up any cases in which the interpolation is illusory.\footnote{``Il
% est assez \'etonnant que les hasards de la pratique n'aient encore fait
% conna\^itre aucun cas dans lequel l'interpolation soit illusoire.'' By
% illusory, M\'eray means nonconvergent.}
%
%
%
% M\'eray's derivations had the key idea of making use of the
% Hermite integral formula. However, the examples he devised were rather
% contrived, and his idio\-syncratically written papers had little impact. It
% was Runge in 1901 who made the possibility of divergence famous by
% showing that divergence of interpolants occurs in general even for
% equispaced points in an real interval and evaluation points in the
% interior of that interval [Runge 1901].
%
%%
%
% Runge illustrated his discovery with an example that has become known as
% the {\em Runge function}: $1/(1+x^2)$ on $[-5,5]$, or equivalently,
% $1/(1+25x^2)$ on $[-1,1]$:
%
%%
% \vskip -2em
x = chebfun('x'); f = 1./(1+25*x.^2);
%%
% We already know from Chapter 8 that there is nothing wrong with this
% function for polynomial interpolation in Chebyshev points: $f$ is
% analytic, and the polynomial interpolants converge geometrically. Now,
% however, let us follow Runge and look at interpolants in equally spaced
% points, which we can compute using the Chebfun overload of Matlab's
% |interp1| command.
%%
% Here is what we get with 8 points:
%%
% \vskip -2em
s = linspace(-1,1,8); p = interp1(s,f,domain(-1,1));
hold off, plot(f), hold on, plot(p,'r'), grid on
plot(s,p(s),'.r'), axis([-1 1 -1 3]), FS = 'fontsize';
title('Equispaced interpolation of Runge function, 8 points',FS,9)
%%
% \vskip 1pt
%%
% Here is the result for 16 points:
%%
% \vskip -2em
s = linspace(-1,1,16); p = interp1(s,f,domain(-1,1));
hold off, plot(f), hold on, plot(p,'r'), grid on
plot(s,p(s),'.r'), axis([-1 1 -1 3])
title('Equispaced interpolation of Runge function, 16 points',FS,9)
%%
% \vskip 1pt
%%
% Is this going to converge as $n\to\infty$? Things look pretty good in the
% middle, but not so good at the edges. Here is the result for 20 points:
%%
% \vskip -2em
s = linspace(-1,1,20); p = interp1(s,f,domain(-1,1));
hold off, plot(f), hold on, plot(p,'r'), grid on
plot(s,p(s),'.r'), axis([-1 1 -1 3])
title('Equispaced interpolation of Runge function, 20 points',FS,9)
%%
% \vskip 1pt
%%
% What is happening is exponential convergence in the middle of the
% interval but exponential divergence near the ends. The next figure shows
% the maximum error over $[-1,1]$ as a function of the number of
% points. We get a hint of convergence at first, but after $n=10$,
% things just get worse. Note the log scale.
%%
% \vskip -2em
ee = []; nn = 2:2:50;
for np = nn
s = linspace(-1,1,np); p = interp1(s,f,domain(-1,1));
ee = [ee norm(f-p,inf)];
end
hold off, semilogy(nn,ee,'.-'), grid on, axis([0 50 5e-2 2e6])
xlabel n+1, title('Divergence as n+1 -> \infty',FS,9)
%%
% \vskip 1pt
%%
% By now the reader may have suspected that what is going wrong here can be
% understood by looking at potentials, as in the last two chapters. Here is
% an adaptation of a code segment from Chapter 11 to plot equipotential
% curves for $n+1 = 8$ and $20$.
%%
% \vspace{-2em}
d = domain(-1.5,1.5);
xgrid = -1.5:.02:1.5; ygrid = -1:.02:1;
[xx,yy] = meshgrid(xgrid,ygrid); zz = xx+1i*yy;
for np = [8 20]
xj = linspace(-1,1,np);
ell = poly(xj,d);
hold off, plot(xj,ell(xj),'.k','markersize',8)
hold on, ylim([-1.2 1.2]), axis equal
ellzz = ell(zz);
levels = ((1.25:.25:3)/exp(1)).^np;
contour(xx,yy,abs(ellzz),levels,'k')
title(['Level curves of |l(x)| for '...
int2str(np) ' equispaced points'],FS,9)
snapnow
end
%%
% \vspace{1pt}
%%
% What we see here is that $[-1,1]$ is very far from being a level curve
% for equispaced interpolation points. From the last two chapters, we
% expect serious consequences of this irregularity. In the second plot
% just shown, for example, it is the fourth curve (from inside out) that
% approximately touches the endpoints $\pm 1$. For Theorem 11.1 to be of
% any use in such a landscape, $f$ will have to be analytic in a region
% larger than the ``football'' enclosed by that curve. Analyticity on
% $[-1,1]$ is not enough for convergence; we will need analyticity in a
% much bigger region of the complex plane. This is what Runge discovered in
% 1901.
%%
% Following the method of the last chapter, we now consider the limit
% $n\to\infty$, where the distribution of interpolation points approaches
% the constant measure (12.9),
% $$ \mu(\tau) = {1\over 2} . \eqno (13.1) $$
% The potential (12.7) associated with this measure is
% $$ u(s) = -1 + {1\over 2}\hbox{Re}
% \left[(s+1)\log(s+1)-(s-1)\log(s-1)\right]. \eqno (13.2) $$
% Here is a code that plots just one level curve of this potential, the one
% passing through $\pm 1$, where the value of the potential is $-1 + \log
% 2$.
%%
% \vskip -2em
x1 = -1.65:.02:1.65; y1 = -0.7:.02:0.7;
[xx,yy] = meshgrid(x1,y1); ss = xx+1i*yy;
u = @(s) -1 + 0.5*real((s+1).*log(s+1)-(s-1).*log(s-1));
hold off
contour(x1,y1,u(ss),(-1+log(2))*[1 1],'k','linewidth',1.4)
set(gca,'xtick',-2:.5:2,'ytick',-.5:.5:.5), grid on
ylim([-.9 .9]), axis equal
hold on, plot(.5255i,'.k')
text(0.05,.63,'0.52552491457i')
title(['Runge region for equispaced interpolation ' ...
'in the limit n -> \infty'],FS,9)
%%
% \vskip 1pt
%%
% For the interpolants to a function $f$ in equispaced nodes to converge as
% $n\to\infty$ for all $x\in[-1,1]$, $f$ must be analytic, not just on
% $[-1,1]$, but throughout this _Runge region,_ which crosses the real axis
% at $\pm 1$ and the imaginary axis at $\pm 0.52552491457\dots i.$ Runge
% reports this number correctly to 4 digits, and writes
%%
%
% \em The curve has somewhat the shape of an ellipse. At the ends of the
% long axis, however, our curve is more pointed than an
% ellipse.\footnote{``Die Kurve$\dots$ hat etwa die Gestalt einer
% Ellipse$\dots.$ An den Enden der grossen Achse ist unsere Kurve aber
% spitzer als eine Ellipse.''}
%%
% Here are the values of (13.2) at the endpoints and the middle:
% $$ u(\pm 1) = -1 + \log 2, \quad u(0) = -1, $$
% and thus
% $$ \exp( u(\pm 1))= {2\over e}, \quad \exp(u(0)) = {1\over e}. $$
% These numbers indicate that in the limit $n\to\infty$, the endpoints of
% an equispaced grid in $[-1,1]$ lie at an average distance $2/e$ from the
% other grid points, in the geometric mean sense, whereas the midpoint lies
% at an average distance of just $1/e$. As emphasized in the last chapter,
% notably in equation (12.6), the effect of such a discrepancy grows
% exponentially with $n$.
%%
% Here are some examples. Equispaced interpolation will converge throughout
% $[-1,1]$ for $f(x) = \exp(-x^2)$, which is analytic everywhere, and for
% $f(x) = (1+x^2)^{-1}$, which has poles at $\pm i$. On the other hand it
% will not converge for $f(x) = (1+16x^2)^{-1}$, which has poles at $\pm
% i/4$. It will converge slowly for $f(x) = (1+(x/0.53)^2)^{-1}$, and
% diverge slowly for $f(x) = (1+(x/0.52)^2)^{-1}$ (Exercise 13.3).
%%
% The worst-case rate of divergence is $2^n$, and this rate will always
% appear if $f$ is not analytic on $[-1,1]$. To be precise, for such a
% function the errors will be of size $O(2^n)$ as $n\to\infty$ but not of
% size $O(C^n)$ for any $C<2$. Here, for example, we take $f$ to be a hat
% function, with just one derivative of bounded variation. The dots show
% errors in Chebyshev interpolation, converging at the rate $O(n^{-1})$ in
% keeping with Theorem 7.2, and the crosses show errors in equispaced
% interpolation, with a dashed line parallel to $2^n$ for comparison.
%%
% \vskip -2em
f = max(0,1-2*abs(x));
eequi = []; echeb = []; nn = 2:2:60;
for n = nn
s = linspace(-1,1,n+1);
pequi = interp1(s,f,domain(-1,1)); eequi = [eequi norm(f-pequi,inf)];
pcheb = chebfun(f,n+1); echeb = [echeb norm(f-pcheb,inf)];
end
hold off, semilogy(nn,2.^(nn-12),'--r')
hold on, axis([0 60 1e-4 1e14]), grid on
semilogy(nn,eequi,'x-r','markersize',8), semilogy(nn,echeb,'.-b')
text(47,3e6,'equispaced','color','r')
text(41,0.8,'Chebyshev','color','b')
text(32,4e8,'C 2^n','fontsize',12,'color','r')
xlabel np, ylabel Error, title('Chebyshev vs. equispaced points',FS,9)
%%
% \vskip 1pt
%%
% All of the above remarks about equispaced interpolation concern the ideal
% mathematics of the problem. On a computer in floating point arithmetic,
% however, a further complication arises: even if convergence ought to take
% place in theory, rounding errors will be amplified by $O(2^n)$, causing
% divergence in practice. Here, for example, are the errors in equispaced
% and Chebyshev interpolation of the entire function $\exp(x)$:
%%
% \vskip -2em
f = exp(x);
eequi = []; echeb = []; nn = 2:2:80;
for n = nn
s = linspace(-1,1,n+1);
pequi = interp1(s,f,domain(-1,1)); eequi = [eequi norm(f-pequi,inf)];
pcheb = chebfun(f,n+1); echeb = [echeb norm(f-pcheb,inf)];
end
hold off, semilogy(nn,2.^(nn-50),'--r')
hold on, axis([0 80 1e-17 1e4]), grid on
semilogy(nn,eequi,'x-r','markersize',8), semilogy(nn,echeb,'.-b')
text(22,6e-6,'C 2^n','fontsize',12,'color','r')
text(42,3e-7,'equispaced','color','r')
text(51,6e-14,'Chebyshev','color','b')
xlabel np, ylabel Error, title('The effect of rounding errors',FS,9)
%%
% \vskip 1pt
%%
% In exact arithmetic we would see convergence for both sets of points, but
% on the computer the divergence for equispaced points sets in early. The
% rate is cleanly $O(2^n)$ until we reach $O(1)$. Notice that the line of
% crosses, if extended backward to $n=0$, meets the $y$ axis at
% approximately $10^{-18}$, i.e., a digit or two below machine precision.
%%
% The $2^n$ divergence of equispaced polynomial interpolants is a
% fascinating subject, and we must remind ourselves that one should only go
% into so much detail in analyzing a method that should never be used! But
% perhaps we should qualify that ``never'' in one respect. As Runge himself
% emphasized, though interpolants in equispaced points do not converge on
% the whole interval of interpolation, they may still do very well near the
% middle. In the numerical solution of differential equations, for
% example, higher order centered difference formulas are successfully used
% based on 5 or 7 equally spaced grid points. A less happy example would
% be Newton--Cotes quadrature formulas, based on polynomial interpolation
% in equally spaced points, where the $O(2^n)$ effect is unavoidable and
% causes serious problems for larger $n$ and divergence as $n\to\infty$, as
% first proved by $\hbox{P\'olya}$ [1933]. We shall discuss quadrature in
% Chapter 19.
%%
% We close this chapter with an observation that highlights the fundamental
% nature of the Runge phenomenon and its associated mathematics. Suppose
% you want to persuade somebody that it is important to know something about
% complex variables, even for dealing with real functions. I still
% remember the argument my calculus teacher explained to me: to understand
% why the Taylor series for $1/(1+x^2)$ only converges for $-1 < x < 1$,
% you need to know that Taylor series converge within disks in the complex
% plane, and this function has poles at $\pm i$.
%%
% Runge's observation is precisely a generalization of this fact to
% interpolation points equispaced in an interval rather than all at $x=0$.
% The convergence or divergence of polynomial interpolants to a function
% $f$ again depends on whether $f$ is analytic in a certain region; the
% change is that the region is now not a disk, but elongated. Even the
% phenomenon of divergence in floating-point arithmetic for functions whose
% interpolants ``ought'' to converge is a generalization of familiar facts
% from real arithmetic. Just try to evaluate $\exp(x)$ on a computer for
% $x=-20$ using the Taylor series!
%%
%
% \begin{displaymath}
% \framebox[4.7in][c]{\parbox{4.5in}{\vspace{2pt}\sl
% {\sc Summary of Chapter 13.}
% Polynomial interpolation in equispaced points is exponentially
% ill-conditioned: the interpolant $p_n$ may have oscillations near the
% edge of the interval nearly $\kern .7pt 2^n$ times larger than the function $f$
% being interpolated, even if $f$ is analytic. In particular, even if $f$
% is analytic and the interpolant is computed exactly without rounding
% errors, $p_n$ need not converge to $f$ as $n\to\infty$.\vspace{2pt}}}
% \end{displaymath}
%
%%
% \smallskip \parskip=2pt\small
% {\bf Exercise 13.1. Three examples.}
% Draw plots comparing accuracy of equispaced
% and Chebyshev interpolants as functions of $n$ for
% $\exp(x^2)$, $\exp(-x^2)$, $\exp(-1/x^2)$.
% \par
% {\bf Exercise 13.2. Computing geometric means in Chebfun.}
% (a) What output is produced by the program below? (b) Why?
% \par\verb|x = chebfun('x',[0 1]);|
% \hfill\break\verb|f = chebfun(@(y) prod(abs(x-1i*y)),[0.1 1],'vectorize');|
% \hfill\break\verb|roots(f-2/exp(1))|
% \par
% {\bf Exercise 13.3. Borderline convergence.}
% The claim was made in the text that equi\-spaced
% polynomial interpolants on $[-1,1]$ converge
% for $f(x) = (1+(x/0.53)^2)^{-1}$ and diverge for
% $f(x) = (1+(x/0.52)^2)^{-1}$. Can you observe this difference
% numerically? Run appropriate experiments and discuss the results.
% \par
% {\bf Exercise 13.4. Approaching the sinc function.}
% The sinc function is defined for all $x$ by $S(x) = \sin(\pi x)/(\pi x)$
% (and $S(0) = 1$).
% For any $n\ge 1$, an approximation to $S$ is given by
% $$ S_n = \prod_{k=1}^n (1-x^2/k^2). $$
% Construct $S_{20}$ in Chebfun on the interval $[-20,20]$ and
% compare it with $S$. On what interval around $x=0$ do you
% find $|S_{20}(x)-S(x)| < 0.1$? (It can be shown that for
% every $x$, $\lim_{n\to \infty} S_n(x) = S(x)$.)
% \par
% {\bf Exercise 13.5. Barycentric weights and ill-conditioning.}
% (a) Suppose a function is interpolated by a polynomial in $n+1$
% equispaced points in $[-1,1]$, with $n$ even. From the result of Exercise 5.6,
% derive a formula for the ratio of the barycentric weights at
% the midpoint $x=0$ and the endpoint $x=1$.
% (b) With reference to the barycentric formula (5.11), explain what
% this implies about sensitivity of these polynomial interpolants
% to perturbations in the data at $x=0$.
% \par