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Lagrange interpolation and Runge's

exampleTemple H. Fay

a& Porter G. Webster

a

aDepartment of Mathematics, University of Southern

Mississippi, Hattiesburg, Mississippi 394065045, USA

Version of record first published: 09 Jul 2006.

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INT. J. MATH. EDUC. SCI. TECHNOL., 1996, VOL. 27, NO. 6, 785-795

Lagrange interpolation and Runge's exampleby TEM PLE H. FAY andP O R TER G. WEBSTER

Department ofMathematics, University ofSouthern Mississippi,Hattiesburg, Mississippi 39406-5045, USA(Received 7November 1994)

It iswell known th at interpolation by polynomials ofhigh degree isnot wellbehaved. In this paper wegive a theoretical basis for this by examining C.Runge's famous example of a continuous function f(x) over aclosed intervalfor which the Lagrange interpolating polynomials fail to converge. Rungeshowed that there is a convergence curve defined in thecomplex plane whoseinterior contains the (real) closed interval on which the function f(x) isdefined.If /(x) has no (complex) singularities within this convergence curve, then theLagrange interpolating polynomials will converge uniformly tof(x) over theinterval. This discussion uses only elementary topics from advanced calculusand beginning complex variable theory, and thus provides for acapstone seriesof one ortwo lectures toshow the applicability of these courses on anappliedproblem of considerable interest. Theadvent of computer algebra programssuch as Mathematica and M aple V make this discussion all the more tractab le.

1 . In t roduc t ionIt has been almost 95 years since C. Runge first published his famous exampleof a continuous function defined over a closed interval forwhich theLagrangeinterpolating polynomials (taken over poin ts (x,-, f(x)) with these,- equally spaced)fail toconverge tofix). This example isstriking since the function is

Indeed, itgoes against one's intuition th at polynomial interpolation should behaveso poorly.Since numerical approxim ations are so im portant, itis appropriate tha t stu dentshave anappreciation of thecomplexities of even themost simple and intuitiveapproaches. In traditional advanced calculus andcomplex variable classes at theunder-graduate andbeginning graduate level, topics covered typically includeCauchy's Integral Theorem, the Residue Theorem, the Maximum ModulusPrinciple, the Gamma function, and Stirlimgs approximation formulas. But seldomare these ideas put together togive apractical application. The point ofthis noteis to suggest that Runge's example and the accompanying theory canprovidemotivation forthe selection of topics tobe covered insuch classes and provideforan elegant capstone series of one or two lectures. We know of norecent text inwhich this development is readily available.

Besides this pedagogical motive, thedevelopment includes a procedure fordetermining precisely when the Lagrange interpolating polynomials will convergeuniformly to afunction f(x). This isaccomplished bydetermining a convergence0020-739X/96 $1200 1995 Taylor & Francis Ltd.

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786 T. H . Fay and P. G. Webster

Figure 1. Runge's example f{x) = \+\$ xi.curvedefined in the complex plane whose interior con tains the (real) closed intervalon which f(x) is defined. If f(x) has no (complex) singular points within thiscurve, then the Lagrange polynomials will converge uniformly to f(x) over theinterval. This derivation is interesting in that it only uses elementary results toobtain an applied result of considerable usefulness.Furthermore, we use computer algebra software packages Maple V, Mathe-matica, and Derive to investigate this problem. Figures 1 and 2 were producedusing Maple V (with Scientific Workplace as the text editor/front end) and figure3 was produced using the CounterPlot command from Mathematica. Derive wasused for some com putations, although either M aple V or Mathematica could havebeen used as well.The next section is abbreviated from [1] and is included to give motivation forthe lengthier and deeper following section on the derivation of the convergence curve.

2. Runge's exampleIf we choose to interpolate f(x) = 1/(1 + 25x2) by polynomials at the points(xj, f(x{)) for equidistant Xj = j/n, j 0 +1,... + n, then the Lagrange interpolationpolynomials for n = 1, 2, and 3 are given by:25 2^ ~ 2 6 *

1250 AI x 1 3 2 2 5 2/>,(*) = 1 A;22 7 5 42 1 1 6 0 0+

/> 3(*) = 1 -p4{x) = 1 -

2x2 +ill

201937524089

98 3662257 450274

96356 4126 5 625 (2)

96 365, 228 601250 ,x -\ x 3 725137

383 00000 0 , 200000 000x6 +3 725137 3 725137

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Lagrange interpolation and Runge's example 787These polynomials can be calculated easily in Maple V, Mathematica, or Derive.Plotting these polynomials along with f(x), figure 2 clearly shows that as nincreases and for \x \ close to 1, the approximations are getting worse. In fact, wewill demonstrate that

lim ( Max \f(x)-pn{x)\) = ao (3)However, for if \x\ < 0-726, then the p n(x ) converge uniformly to f(x) as n -* oo. Itis our intention to explain this interesting and predictable behaviour.

-0.5 00-0.5

-1

x 5 f

Figure 2. {a ) / . ,(*) = 1 -f x2.

\

Figure 2.

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788 T. H . Fay and P. G. Webster

Figure 2. (c) p3(x) = 1 - ^ x2 +

Figure 2. (d )

Given a continuous function f(x) defined over a closed interval [a, b~\, firsttranslate and rescale so as to have the domain of f(x) being the interval [ 1, 1] .Consider the problem of whether or not f(x) can be obtained as the uniform limitof Lagrange interpolating polynomials for equidistant points. Runge [2] showedthat by considering f(z) where z is a complex variable, that one can write f(z) =Ln(z) + Rn(z ) where Ln(z) is the Lagrange interpolating polynomial and Rn(z) is aremainder term which can be estimated. He gave conditions which assure that

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Lagrange interpolation and Runge's example 789

Figure 2. (e) Runge's example with approximating polynomials.Ln[z) converges to f(z) and Rn(z ) converges to zero. These conditions have to dowith the singularities of f(z). M ost im portan tly, he derived the equation of a simpleclosed curve A(z) = c , which we call the convergence curve, that encloses theinterval [ 1, 1] when c = 2. He showed that if f(z) has no singularities on theinterior of this curve, then the interpolating polynomials, converge uniformly tof(z). If f(z) has one or more singularities lying within this curve, then one choosesa smaller convergence curve not containing any singular points within its interior,and this curve determines a subinterval of [ 1 , 1] over which the interpolatingpolynomials will converge.The singulari t ies of f(x) are determined by factors of the form

1ax + fix + y (4)

where ax2 + fix + y is an irreducible quadratic; there are no real singularities, sincewe assume f(x) to be well defined an d con tinu ous over the entir e interval [ 1, 1 ].We will look at the examples (1/(1 + x2) and 1/(1 + 25x 2), but the reader will beable to consider any such factor.The equation for the family of convergence curves is:( l - 0 ) ( 1 ~ 2 ) | exp(7t| /()|) = c2, for c(z) = |(1 + 1 (5)

where I(z) denotes the imaginary part of z. This curve is easier to study by lettingz = x + ly and setting r, = [(1 + x)2 + > 2 ] 1 / 2 , r2 = [(1 - x)2 + y 2 ] 1 / 2 , and letting1 1= tan " 1 Cy/(1 4- x)) and a2 = tan -1(}>/(l x)). Then we can rewrite A(z) as

(x, y) = > ex p ( [ - ( a , + a2)y + n\y\]) = c2 (6 )This function of the two real variables x and y now can be investigated rathereasily. Note that the curve is symmetric with respect to both the x and y-axes.The va lue c = 2 gives a closed ellipse-like curve (plotted in figure 3) which

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790 T. H. Fay and P. G . Web ster2

- 2Figure 3. The convergence curve A(x,y) = c2 for c2 = 1-801 43, 4, 9-620 95.

passes through the real points z = 1 and z = 1. It is not difficult to plot thesecurves for various values of c using Ma thematica. Ma thematica makes determiningthe interval of convergence easy.

Examp le 1. Consider g(x) = 1/(1 + x2) for xe [ 1, 1]. The complex valuedfunction g(w) has singular points at z = +/'. Substituting x = 0 and y = 1 intoequation (6), we obtain c2 = 2 exp(7r/2) 9-620 95 > 4 . The convergence curveA(x,y) = 9-620 95 encloses the interval [ 1, 1] (see figure 3), so the LagrangeInterpolation Polynomials converge uniformly to g(x) on the interval [1,1].Indeed, using Mathematica's FindRoot command, it is easy to solve the equationln(^4(r, 0)) = ln(9-620 95) for r, obtain ing uniform convergence over the interval[-1-293 68, 1-293 68].Examp le 2. Runge's example f(x) = 1/(1 + 25x2) has singular points at z =+ 0-2i. Substituting x = 0 and y = 0-2 into equation (6), we obtain c2 = 1-801 4 3.The curve A(x,y) = 1-801 43 crosses the x-axis at x = +0-726 677. Polynomialinterpolation will converge for \x\ < 0726 677, but if 0726677 s$ \x\, f(x) is notrepresented well by L2n +i(z) is these regions.Thus the process is now clear. Given f(x), choose the singularity z = x + \yclosest to the origin. Substitute the values for x and y into equation (6) todetermine the value c2. If c2 ^ 4, then the Lagrange interpolating polynomials

converge uniformly to f(x) over the interval [ 1, 1 ]. If c2 < 4, then use Mathematica(or an equivalent) to plot the convergence curve (if desired), and to solve theequation \n(A(r, 0)) = 2 ln(c) for r to determine the subinterval \_ r,r~] over whichthe convergence is uniform.Perhaps it is worth noting that Mathematica cannot solve the equationA(r, 0) = c2 for r because the equation is still reasonably com plicated. However, bytaking the logarithm of both sides , solving ln(^4(r, 0)) = 2 ln(c) for r is easy

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Lagrange interpolation and Runge's example 791and very quick. One needs a starting point, as FindRoot uses Newton's Method.Thus the initial contour plot of A(x, y) = c2 is sometimes useful in determiningthe initial starting point.

3 . Derivation of the convergence curveIn this section, we outline the derivation of the convergence curve A(z) = c2

used in the above discussion. Indeed, it is quite interesting to see that through theuse of complex variables, we can write a function as a sum of a Lagrangeinterpolation polynomial and a remainder term, the latter which can be estimatedand from which the convergence curve arises quite naturally. The derivation makesuse of such standard results as Cauchy's Integral Theorem, Cauchy's ResidueTheorem, Maximum Modulus Principle, some simple properties of the Gammafunction, and Stirling's Approximation Formulas. All of these results are commonfare in standard advanced calculus and beginning complex analysis courses but areperhaps not put together in such a way as to have a real application.

To begin investigating the real valued function f(x) denned over the closedinterval [a, b~\, as above, we first translate and rescale the independent variable sothat we study f(x) over the interval [1, 1], as a matter of convenience. Moreover,as above, we now consider f(zv) where the real variable x has been replaced by thecomplex variable w and we assume f{tv) is analytic in a domain ( =openconnectedset) D which contains the interval [1, 1]. We recall the Cauchy Integral Theoremwhich states that

| d* (7)Z W

where C is a simple closed curve in D within which f(z) is analytic and w is insideC.

We first consider a general interpolation problem. Let Wj, w2,..., wn be distinctvalues of w which lie inside C, and letw2)---{w wn) (8)

The fundamental functionFn(z)-Fn(zv) (9)z w

turns out to be a polynomial of degree n 1 when considered as a function of w(the numerator is divisible by the denominator). Since Fn(z) is a constant whentreating w as a variable, then when we divide function (9) by Fn(z), we still havea polynomial in of degree n 1 in w. We define

Fn(z ) - Fn(co)q>,,(w, z) =Fn(z)(z - w)1 Fn(w) 1 (10)z w Fn(z) z w

and consequently, =

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79 2 T. H. Fay and P. G. WebsterSubsti tuting equation (11) into equation (7), we obtain

/ ( ; ) = Ln(w) + Rn(w ) (12)where

L B ( ) = ^ r < f f(z)

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Lagrange interpolation and Runge's example 793ou t l / ( 2 " + 1 ) and use the basic functional equation of the Gamma functionr(x + 1) = xT(x) to obtain:

Thus we now simply have to estimate values of the Gamma function. This isaccomplished by using the well-known (technical) formulaz) sin(nz)which can be found derived in DiLillo, Example 9.12 [3] or in Vvedensky [4],Exercise 23, p 251. Using this we w rite

T(nz n) = (22)F(l + n nz)sin(nnz)

and thus , , ( -1 )" (1 + n + nz)(\ + n- nz)T{nz + n)T(n - nz)sin(nnz)

Since F2n +i( z) = ( l)2"+1 i?2+i(2r)> w e may assume z to be in the right halfplane. Then Stirling's approximation formula for the Gamma function (seeWhittaker and Watson [5], p 253) yields1) ^ (-1)"2(1 - z2)1'2 exp( -2) ( l + * ) (" + n z )(l - ^) ("-"2 )sin(w^) (24)

From Stirling's formulae x p ( - 2 ) S ]Z ' , * 2 " + 1 / 2 (25)

N / ^ 2 2 " + 1equation (24) becomes

F7n.-i(z) = = sin(n7T2r). (26 )J n n 2" + l / 2 L 4 J

Now we set"L r 1 J s i I

and rewrite equation (18) as

fii,tl(n)g(1 ~ w ) j (b I y ^ + i - / 1 ^ d^Thus, if we can show that | ' / '2+i(w)/ l/'2n+i( ;3 :)l becomes small as n becomes large,we will have shown that \F 2n+i( lw)/F2n+ iz)\ becomes small as n becomes large,

1 For a pair of sequences of complex numbers {z n} and {zvn}, the notation zn = w nindicates that the two sequences have the property that wn # 0 and (zjw n) -*1 as n -* co.

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794 T. H. Fay and P. G.Websterand thus f(w) = L2n+i(w). To dothis wenote first that thesine term in equation(27) can beestimated by

\sin(nnz)\^^exp(nn\I(z)\ (29)where I(z) denotes the imaginary part of thecomplex number z. Hence it sufficesto investigate thebehaviourof

(30)as n becomes large. Thus we set

A{z) = |(1 + z)+*\\ - z)~z)\ exp(7:|/(*)|) (31)Letting z = x + iy andsetting rx = [(1 + x)2 + ^ 2 ) ] 1 / 2 , r2 = [(1 x)2 + y2~]1/2, andletting at = tan~'(y/(l + x))and a2 = tan~*(y/(l x)), werewrite A{z)as

^( 2 ) = ri1+*>f-*> exp[-(a: + a2)y + n|y|] (32)This function of the tworeal variables x andy can be investigated rather easilyusing Mathematica bysetting A{z) = c2 for c ^ 1.

The value c = 2 gives aclosed ellipse-like curve which passes through therealpoints z = 1 andz = 1. Lanczos [5] tabulated values for this curve. Of course,now we can usemuch more sophisticated methods togenerate values employingcomputer algebra software packages such asMathematica. In figure 3, we showplots of equation (32) for various values ofc.

Now assume that thedomain D contains thecurve A(z) = 4 in itsinteriorandthat f(w) has nosingular points within D(i.e. /(w) is regular onD) andconsidertwo other v4-curves, A(z) = c2 andA(z) = c2, which enclose the curve A(z) = 4.The curve A(z) = c2 encloses thecurve A(z) = c\ if c2 > c\. If z lies on thecurveA(z) = c\ andw lies on thecurve A(z) = c2, then

(33)For sufficiently large values of n, |'/'2+i(w)/t/'2n+i(;2r)l ' s arbitrarily small, and

consequently, -R2n+i(w) becomes arbitrarily small if the curve C is taken to beA{z) = c\.This means, for large enough n, f(w) can beapproximated with arbitraryaccuracy byL2n+i(w) if w belongs to thecurve A(z) = c\. Since themaximum ofI/(w) - 2+i(ro)l taken in thewhole domain enclosed by thecurve A(zv)= c\ isno greater than on thecurve itself (theMaximum Modulus Principle, see [7], p132), thefunction /(w) isapproximated byL2n+\(w) in theentire domain enclosedby A(w)= c\ with arbitrary accuracy. These observations remain valid no matterhow large the two ^4-curves are taken provided that they do not contain anysingularities of f(w).It follows that the region of uniform convergence extends over the interiorof that ^4-curve which passes through at least one of thesingularities of /(w) butcontains noother singularity in its interior. If f(zv) has nosingularities, then theregion of convergence extends over theentire complex plane.

On the other hand, if f{w) has singular points in the interior of the curveA(w) = 4, then it is necessary tomove inside the curve to smaller ^4-curves.Let

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Lagrange interpolation and Runge's example 795A(w) = c\ be an ^4-curve which lies entirely within the interior of the region wheref{w) is regular. This ^4-curve will intersect the interval [1, 1] in two points rand r. By slightly deforming the ^4-curve around the points 1 and 1, andcomput ing R2n+i(w) by integrating over this deformed countour, we see that if wlies with in A(w) = c\, the n i^n+iC^O becom es arb itrarily small for sufficiently largevalues of n. This is because A(z) is greater on the part of the deformed contouroutside of the curve A(zv) = c\ than on the curve itself, while A(w) must be smallerthan A(z) because w lies within the ^4-curve. As long as the ^4-curve encloses nosingular points of /(w), i t can be replaced by a large ^4-curve. Thus the region ofconvergence fills up the interior of that ^4-curve which passes through at least onesingular point of f(w) but includes no singular points within i ts interior.In summary, if the singular points of f(w) l ie outside the curve A(w) = 4, thenthe Lagrange polynomial interpolants L2n+i(x) converge to f(x) as n > oo for allx e [ 1, 1 ]. If s ingular points are located inside the yi-curv e A(w) = 4, then theLagrange polynomial in terpolants L2n +\{x) converge to f(x) as -> co for only apart of the interval [ 1 , 1 ] .

References[1 ] FAY, T. H., and WEBSTER, P. G., 1995, On Runge's example for polynomial inter-polation, Primus, V, 73-79.[2 ] RUNGE, C , 1901, Uber empirische Functionen und die Interpolation zwischenaquidistanten Ordinaten, Zeitschrift fur Mathematik und Physik, 46, 224-243.[3 ] DELILLO, N. J., 1982, Advanced Calculus with Ap plications (New York: Macmillan).[4] VVEDENSKY, D., 1993, Partial D ifferential Equations with Ma thematica (Reading, MA:Addison-Wesley).[5 ] WHITTAKER, E. T., and WATSON, G. N., 1940, A Course of Modern Analysis (London:Cambridge University Press.).[6] LANCZOS, C , 1956, App lied A nalysis (Englewood Cliffs, NJ: Prentice Hall).[7] BOAS, R. P., 1987, Invitation to Complex Analysis (New York: Random House).

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