# The circle theorem and related theorems for Gauss-type quadrature rules.

Abstract. In 1961, P.J. Davis and P. Rabinowitz established a beautiful "circle theorem" for Gauss and Gauss-Lobatto quadrature rules. They showed that, in the case of Jacobi weight functions, the Gaussian weights, suitably normalized and plotted against the Gaussian nodes, lie asymptotically for large orders on the upper half of the unit circle centered at the origin. Here analogous results are proved for rather more general weight functions--essentially those in the Szego class--, not only for Gauss and Gauss-Lobatto, but also for Gauss-Radau formulae. For much more restricted classes of weight functions, the circle theorem even holds for Gauss-Kronrod rules. In terms of potential theory, the semicircle of the circle theorem can be interpreted as the reciprocal density of the equilibrium measure of the interval [-1, 1]. Analogous theorems hold for weight functions supported on any compact subset [DELTA] of (-1, 1), in which case the (normalized) Gauss points approach the reciprocal density of the equilibrium measure of [DELTA]. Many of the results are illustrated graphically.

Key words. Gauss quadrature formulae, circle theorem, Gauss-Radau, Gauss-Lobatto and Gauss-Kronrod formulae, Christoffel function, potential theory, equilibrium measure

AMS subject classifications. 65D32, 42C05

1. Introduction. One of the gems in the theory of Gaussian quadrature relates to the distribution of the Gaussian weights. In fact, asymptotically for large orders, the weights, when suitably normalized and plotted against the Gaussian nodes, come to lie on a half circle drawn over the support interval of the weight function under consideration. This geometric view of Gauss quadrature rules was first taken by Davis and Rabinowitz [2, [section]II], who established the asymptotic property described--a "circle theorem", as they called it--in the case of Jacobi weight functions w (t) = [(1- t).sup.[alpha]] [(1 + t).sup.[beta]], [alpha] > -1, [beta] > -1, not only for the Gauss formula, but also for the Gauss-Lobatto formula. For the Gauss-Radau formula, they only conjectured it "with meager numerical evidence at hand". It should be mentioned, however, that the underlying asymptotic formula (see eqn (2.2) below) has previously been obtained by Erdos and Turan [4, Theorem IX], and even earlier by Akhiezer [1, p. 81, footnote 9], for weight functions w(t) on [-1,1] such that w(t)[square root of (1-[t.sup.2])] is continuous and w(t)[square root of (1-[t.sup.2])] [greater than or equal to] m > 0 on [-1,1]. This answers, in part, one of the questions raised in [2, last paragraph of [section]IV] regarding weight functions other than those of Jacobi admitting a circle theorem. In [subsection]2-4 we show that the circle theorem, not only for Gaussian quadrature rules, but also for Gauss-Radau and Gauss-Lobatto rules, holds essentially for all weight functions in the Szego class, i.e., weight functions w on [-1,1] for which

(1.1) In w(t)/[square root of (1-[t.sup.2])][member of][L.sub.1](-1,1).

We say "essentially", since an additional, mild condition, viz.

1/w(t)[member of][L.sub.1]([DELTA]),

must also be satisfied, where [DELTA] is any compact subinterval of (-1,1). In [section]5, we show, moreover, that circle theorems, under suitable assumptions, hold also for Gauss-Kronrod formulae. In [section]6 we give a potential-theoretic interpretation of the circle theorem, namely that the semicircle in question is the reciprocal density of the equilibrium measure of the interval [-1,1]. This is true in more general situations, where the support of the given weight function is any compact subset [DELTA] of (-1,1), in which case the (normalized) Gauss points come to lie on the reciprocal density of the equilibrium measure of [DELTA]. This is illustrated in the case of [DELTA] being the union of two disjoint symmetric subintervals of [-1,1]. The equation of the limiting curve can be written down in this case and answers in the affirmative another question raised in [2, last sentence of [section]IV].

2. Gaussian quadrature. We write the Gaussian quadrature formula for the weight function w in the form

(2.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[tau].sup.G.sub.v] are the Gaussian nodes and [[lambda].sup.G.sub.v] the Gaussian weights; cf., e.g., [7, [section] 1.4.2]. (Their dependence on n is suppressed in our notation.) The remainder satisfies

[R.sup.G.sub.n] (p) = 0 for any p [member of] [P.sub.2n-1]

where [P.sub.2n-1] is the class of polynomials of degree [less than or equal to]2n - 1. Without loss of generality we have assumed that the support of the weight function w is the interval [-1,1]. The circle theorem can then be formulated as follows.

THEOREM 2.1. (Circle theorem) Let w be a weight function in the Szego class (cf. [section]1, (1.1)) satisfying 1/w(t)[member of][L.sub.1] (A) for any compact interval [DELTA] [contains] (-1,1). Then

(2.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all nodes [[tau].sup.G.sub.v] (and corresponding weights) that lie in [DELTA]. (The relation [a.sub.n] - [b.sub.n] here means that [lim.sub.n[right arrow][infinity]] [a.sub.n]/[b.sub.n] = 1.)

As mentioned in [section] 1, this was shown to be true by Davis and Rabinowitz [2] in the case of the Jacobi weight function w (t) = (1 - t) a (1 + t)Q on [-1,1], a > -1, [beta] > -1. We illustrate the theorem in Fig. 2.1 by plotting all quantities on the left of (2.2) for [alpha], [beta] = -0.75 : 0.25 : 1.0,1.5 : 0.5 : 3.0, [beta] [greater than or equal to] [alpha], and for n = 20 : 5 : 40 in the plot on the left, and for n = 60 : 5 : 80 in the plot on the right.

[FIGURE 2.1 OMITTED]

The circle theorem for the more general weight function indicated in Theorem 2.1 has been around implicitly for some time. Indeed, it is contained in an important asymptotic result for Christoffel functions [[lambda].sub.n], (t; w) due to Nevai [12, Theorem 34]. According to this result, one has

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

uniformly for t [member of] [DELTA]. Recalling that [lambda].sub.n], ([[tau].sup.G.sub.v] ; w) = [[lambda].sup.G.sub.v] (cf. [5, Theorem 3.2 and last paragraph of [section]I.3]) yields Theorem 2.1.

COROLLARY TO THEOREM 2.1. If w(t) = [(1 - [t.sup.2]).sup.-1/2] on (-1, 1), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Proof. This follows from the well-known fact that [[lambda].sup.G.sub.v] = [pi]/n, v = 1, 2, ... , n, in this case.

REMARK 2.2. Theorem 2.1 in a weaker form (pointwise convergence almost everywhere) holds also when w is locally in Szego's class, i.e., w has support [-1,1] and satisfies

(2.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [DELTA] is an open subinterval of [-1,1]. Then (2.2) holds for almost all [[tau].sub.v] [member of] [DELTA] ([] 1, Theorem 81).

EXAMPLE 1. The Pollaczek weight function w(t; a, b) on [-1,1], a [greater than or equal to] |b| (cf. [14]). The weight function is given explicitly by (ibid., eqn (3), multiplied by 2)

(2.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [omega] = [omega] (t) = (at + b) [(1 - [t.sup.2]).sup.-1/2. It is not in Szego's class, but is so locally. The recurrence coefficients are known explicitly (ibid., eqn (14)),

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From (2.5) and (2.6), it is straightforward to compute the ratios na [[lambda].sub.G.sub.v]/[pi][w(([[tau].sup.G.sub.v] ; a, b). Their behavior, when n = 380 : 5 : 400, is shown in Fig. 2.2 for a = b = 0 on the left, and for a = 4, b = 1 on the right. The circle theorem obviously holds when a = b = 0 (i.e., w = 1), but also, as expected from the above remark, with possible isolated exceptions, for other values of a and b.

[FIGURE 2.2 OMITTED]

3. Gauss-Radau formula. Our analysis of the Gauss-Radau formula (and also the Gauss-Lobatto formula in [section]4) seeks to conclude from the validity of the circle theorem for the Gauss formula (2.1) the same for the corresponding Gauss-Radau formula,

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [R.sup.R.sub.n] ([P.sub.2n]) = 0. (Here, as in (2.1), the nodes and weights depend on n.)

THEOREM 3.1. Let the weight function w satisfy the conditions of Theorem 2.1. Then not only the Gaussian quadrature rule (2.1) for w, but also the Gauss-Radau rule (3.1) for w admits a circle theorem.

Proof. It is known that [[tau].sup.R.sub.v] are the zeros of [[pi].sub.n] (; w-1), the polynomial of degree n orthogonal with respect to the weight function [w.sub.-1] (t) = (t + 1)w(t) (cf. [7, [section]1.4.2, p. 25]).

Let

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

be the elementary Lagrange interpolation polynomials for the nodes [[tau].sup.R.sub.1], [[tau].sup.R.sub.2], ... , [[tau].sup.R.sub.n]. Since the Gauss-Radau formula is interpolatory, there holds

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

If [[lambda]*.sub.v] are the n Gaussian weights for the weight function [w.sub.-1], we have, again by the interpolatory nature of the Gaussian quadrature formula, and by (3.3),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By assumption, the Gauss formula for the weight function w, and hence also the one for the weight function [w.sup.-1] (which satisfies the same conditions as those imposed on w) admits a circle theorem. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

EXAMPLE 2. The logarithmic weight function w(t) = [t.sup.[alpha] ln(1/t) on [0,1], [alpha] > -1. Here, Gauss-Radau quadrature is over the interval [0,1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

A linear transformation of variables, mapping [0,1] onto [-1,1], yields the Gauss-Radau quadrature formula over [-1,1], to which Theorem 3.1 is applicable. The circle theorem, therefore, by a simple computation, now assumes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This is illustrated in Fig. 3.1, on the left for n = 20 : 5 : 40, on the right for n = 60 : 5 80, and [alpha] = -0.75 : 0.25 : 1.0, 1.5 : 0.5 : 3 in both cases.

[FIGURE 3.1 OMITTED]

4. Gauss-Lobatto formula. The argumentation, in this case, is quite similar to the one in [section]3 for Gauss-Radau formulae. We recall that the Gauss-Lobatto formula for the weight function w is

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [R.sup.L.sub.n] ([P.sub.2n + 1]) = 0 and [[tau].sup.L.sub.v] are the zeros of [[pi].sub.n] (.; [[w.sub.[+ or -] 1]), the polynomial of degree n orthogonal with respect to the weight function [[w.sub.[+ or -]1] (t) = (1- [t.sup.2])w(t) (cf. [7, [section] 1.4.2, p. 26]).

THEOREM 4.1. Let the weight function w satisfy the conditions of Theorem 2.1. Then the Gauss-Lobatto rule (4.1) for w admits a circle theorem. Proof. In analogy to (3.2), we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and denote by [[lambda]*.sub.v] the n Gaussian weights for the weight function [w.sub.[+ or -]1],. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

while, on the other hand,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

by Theorem 2.1 and the fact that wf1 satisfies the same conditions as those imposed on w.

5. Gauss-Kronrod formula. While the quadrature rules discussed so far are products of the 19th century, the rules to be considered now are brainchilds of the 20th century ([10]). The idea (1) is to expand the Gaussian n-point quadrature formula (2.1) into a (2n + 1)-point formula by inserting n + 1 additional nodes and redefining all weights in such a manner as to achieve maximum degree of exactness. It turns out, as one expects, that this optimal degree of exactness is 3n + 1; it comes at an expenditure of only n + 1 new function evaluations, but at the expense of possibly having to confront complex-valued nodes and weights. The quadrature formula described, called Gauss-Kronrod formula, thus has the form

(5.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[tau].sub.G.sub.v] are the Gaussian nodes for the weight function w and

(5.2) [R.sup.K.sub.n] (p) = 0 for all p [member of] [P.sub.3n + 1].

The formula (5.1) is uniquely determined by the requirement (5.2); indeed (cf. [7, [section]3.1.2]), the inserted nodes [[tau].sup.K.sub.[mu]]--the Kronrod nodes--must be the zeros of the polynomial [[pi].sup.K.sub.n + 1] of degree n+ 1 orthogonal to all lower-degree polynomials with respect to the "weight function" [[pi].sub.n] (t)w(t), where [[pi].sub.n], is the orthogonal polynomial of degree n relative to the weight function w,

(5.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The weights in (5.1) are then determined "by interpolation".

Interestingly, in the simplest case w(t) = 1, the polynomial [[pi].sup.K.sub.n + 1] has already been considered by Stieltjes in 1894, though not in the context of quadrature. It is nowadays, for arbitrary w, called the Stieltjes polynomial for the weight function w.

Orthogonality in the sense (5.3) is problematic for two reasons: the "weight function" [w.sup.K.sub.n] = [[pi].sub.n + 1] w is oscillatory and sign-varying on the interval [-1,1], and it depends on n. The zeros of [[pi].sup.K.sub.n + 1], therefore, are not necessarily contained in (-1,1), or even real, although in special cases they are. A circle theorem for Gauss-Kronrod formulae is therefore meaningful only if all Kronrod nodes are real, distinct, contained in (-1,1), and different from any Gaussian node. If that is the case, and moreover, w is a weight function of the type considered in Theorem 2.1, there is a chance that a circle theorem will hold. The best we can prove is the following theorem.

THEOREM 5.1. Assume that the Gauss-Kronrod formula (5.1) exists with [[tau].sup.K.sub.[mu] distinct nodes in (-1,1) and [[tau].sup.K.sub.[mu]] [not equal to] [[tau].sup.G.sub.v] for all [mu] and v. Assume, moreover, that

(i) the Gauss quadrature formula for the weight function w admits a circle theorem;

(ii) the (n + 1)-point Gaussian quadrature formula for [w.sup.K] (t) = [[pi].sub.n] (t) w (t), with Gaussian weights [[lambda]*.sub.[mu], admits a circle theorem in the sense

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all [mu] such that [[tau].sup.K.sub.[mu]] [member of] [DELTA], where [DELTA] is any compact subinterval of (-1,1); (iii) [[lambda].sup.K.sub.v] 1/2 [[lambda].sup.G.sub.v] as n [right arrow] [infinity] for all v such that [[tau].sup.G.sub.v] [member of] [DELTA].

Then the Gauss-Kronrod formula (5.1) admits a circle theorem in the sense

(5.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all v, [mu] as defined in assumptions (ii) and (iii).

Proof. The first relation in (5.4) is an easy consequence of assumptions (i) and (iii):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

To prove the second relation in (5.4), we first note that the n + 1 Gaussian nodes for [w.sup.K] = 7r,,[[pi].sub.n]w are precisely the Kronrod nodes [[tau].sup.K.sub.[mu]. By assumption (ii),

(5.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since the Gauss formula for wK is certainly interpolatory, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

denoting the elementary Lagrange interpolation polynomials for the nodes [[tau].sup.K.sub.1], [[tau].sup.K.sub.s], ... , [[tau].sup.K.sup.K.sub.n + 1]. On the other hand, by the interpolatory nature of (5.1), we have similarly

(5.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By (5.5) and (5.6), therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

EXAMPLE 3. Jacobi weight function w(t) = [(1 - t).sup.[alpha] [(1 + t).sup.[beta], [alpha], [beta] [member of] [0,5/2).

For these weight functions, (5.4) has been proved by Peherstorfer and Petras [13, Theorem 2], from which assumptions (ii) and (iii) can be recovered by "inverse implication". Assumption (i), of course, is satisfied for these weight functions by virtue of Theorem 2.1.

The circle theorem, in this case, is illustrated in Fig. 5.1, on the left for n = 20 : 5 : 40, on the right for n = 60 : 5 : 80, with [alpha], [beta] = 0 : 0.4 : 2, [beta] > [alpha], in both cases.

[FIGURE 5.1 OMITTED]

We remark that asymptotic results of Ehrich [3, Corollary 3] imply the circle theorem also for negative values of [alpha] = [beta] > 1/2 .

6. Potential-theoretic interpretation and extension of the circle theorem. There is a deep connection between Christoffel functions (and hence Gaussian weights) and equilibrium measures in potential theory. For the necessary potential-theoretic concepts, see [15]. Thus, for example, the density of the equilibrium measure [[omega].sub.[-1,1] of the interval [-1,1] is [[omega]'.sub.[-1,1](t) = 1/([pi][square root of 91- [t.sup.2])], showing that (2.3) can be interpreted by saying that as n [right arrow] [infinity] the ratio n[[lambda].sub.n] (t; w) /w (t) converges to the reciprocal of the density of the equilibrium measure of [-1,1]. Here we consider a weight function w that is compactly supported on a (regular) set E [subset] R and [DELTA] [subset] E an interval on which w satisfies the Szeg6 condition (2.4). Then, for almost all v,

(6.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[omega]'.sub.E] is the density of the equilibrium measure of E (cf. [17, Theorem 1]).

EXAMPLE 4. A weight function supported on two intervals,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where 0 < [xi] <1,p> -1, q > - 1 and [gamma] [member of] R.

The recursion coefficients for the weight function w are explicitly known if [gamma] = [+ or -] and p = q = [+ or -] 1/2 (see [6, [section]5]). The quantities n[[lambda].sup.G.sub.v]/([pi]w([[tau].sup.G.sub.v])) in these cases are therefore easily computable; plotting them for [xi] = 1/2 and n = 60 : 5 : 80, yields the graph in Fig. 6.1.

[FIGURE 6.1 OMITTED]

The limiting curve for general [xi] must be related to the reciprocal density [[omega]'.sub.-1,-[xi]][union][xi],1] of the two support intervals. We can find its equation by using the known fact [6, [section]6] that for y = 1 and p = q = -1/2, when n is even, the Gauss weights [[lambda].sup.g.sub.v] are all equal to [pi]/n. Consequently, for these n, and [[tau].sup.G.sub.v] [member of] [[xi],1],

(6.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

so that the right branch of the limiting curve, and by symmetry the curve itself, has the equation y = [phi](t), where

[phi](t) = [|t|.sup.-1] [([t.sup.2] - [[xi].sup.2]).sup.1/2] [(1 - [t.sup.2]).sup.1/2].

The extrema of [phi] are attained at [t.sub.0] = [+ or -] [square root ([xi])] and have the value [[phi].sub.0] = 1 - [xi]. For = [xi], 1/2 therefore, [t.sub.0] = [+ or -][square root of (1/2)] = 0.7071 ... , [[phi].sub.0] = 1/2.

We conclude from (6.1) and (6.2) that

(6.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Actually, the equilibrium measure is known for any set E whose support consists of several intervals and is an inverse polynomial image of [-1,1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [[tau].sub.N] is a polynomial of degree N. Then indeed [9, p. 577],

(6.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In the case at hand, E = [-1, [xi]] [union] [[xi],1], 0 < [xi] < 1, we have

[[tau].sub.2] (t) = 2[t.sup.2] - [[xi].sup.2] - 1/1-[[xi].sup.2],

and (6.4) becomes (6.3).

Acknowledgment. The author gratefully acknowledges helpful discussions with V. Totik. He is also indebted to a referee for the references [1], [4] and for the remark in the last paragraph of the paper.

(1) In a germinal form, the idea can already be found in work of Skutsch [16]; see [8].

* Received December 20, 2004. Accepted for publication November 8, 2005. Recommended by D. Lubinsky.

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Dedicated to Ed Saff on the occasion of his 60th birthday

WALTER GAUTSCHI ([dagger])

([dagger]) Department of Computer Sciences, Purdue University, West Lafayette, Indiana 47907-2066 (wxg@cs.purdue.edu).
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