ON THE NUMERICAL EVALUATION OF CERTAIN 2-D SINGULAR INTEGRALS

. The problem of approximating certain two-dimensional Cauchy principal value integrals is here considered, product integration formulas with multiple nodes are presented and the behaviour of the remainder is analyzed. Next a particular class of cubature rules is generated, having the peculiar property that the set of the nodes holds ﬁxed while their multiplicities vary. Some numerical examples of application of the latter rules are also provided.


INTRODUCTION
It is well known that the numerical evaluation of the following two-dimensional Cauchy principal value (CPV) integrals ( 1) is of interest in many applications; the approaches for approximating (1) are of two types: local and global.Local methods based on the use of splines have been considered, for instance in [2]; they are suitable, in particular, when f is not smooth, while, when f is differentiable, global methods are probably to be preferred.In this concern, product formulas based on the use of orthogonal polynomials have been proposed (see, for instance, [8], [10]); it is the case of formulas having the form This work was supported by MURST of Italy.
Here p 1 and p 2 are Jacobi weight functions, the rules are of interpolatory type, {x i } m i=1 , {y j } n j=1 are the zeros of orthogonal polynomials relative to p 1 and p 2 respectively, or of Chebyshev polynomials of the first or second kind.
The question of the convergence of {Q m,n (f ; ξ, η)} for m, n diverging, was also dealt with in the quoted papers.Yet, very few numerical examples for the evaluation of (2) are provided in the literature, while one of the goals of this paper is to provide several numerical tests showing the behaviour of the cubature rules here proposed.
One of the difficulties which occur in handling (2), is due to the fact that an augment of the precision degree in (1) requires to increase the number of the nodes: this implies a twofold disadvantage, because not only all the nodes must be evaluated again, but also numerical cancellation may occur due to the decreasing of min i |x i − ξ| and/or min j |y j − η|, although the case x i = ξ, y j = η, i = 1, ..., m, j = 1, ..., n is always assumed to hold.
Another general approach, which enables one to increase the precision degree of an integration rule, amounts to construct formulas of Turán type, that is having multiple nodes with odd multiplicities, say 2s + 1; in which case a higher precision is attainable by a higher multiciplity of the nodes.The use of Turán quadrature rules for approximating one-dimensional CPV integrals has been developed in [7].However, also in the case of rules based on multiple nodes, a change in the multiplicity generally implies a change of the zeros of the s-orthogonal polynomials involved in the construction; thus, the just mentioned computational problems may occur, as well.
For this reason the introduction of formulas with multiple nodes assumes a particular value in those cases in which the nodes are independent of the multiplicity, so that the computational problems quoted before can be avoided and the corresponding formulas are really effective.
In this context, are crucial some results concerning an invariance property of some classes of s-orthogonal polynomials, which are briefly summarized in Section 2. In Section 3, a general cubature rule for approximating the twodimensional CPV integrals (1) is given and an expression of the remainder is provided.In Section 4, the results of [6] are exploited in order to construct certain formulas such that the cubature sum and the remainder have particularly interesting features, which, among other things, allow one to deduce the asymptotic behaviour of the remainder.Finally, Section 5 is devoted to the development of some examples, which put in evidence the good numerical performances of the integration formulas based on the zeros of s-orthogonal polynomials enjoying a "s-invariance" property.
We recall that, given on the real line, an interval A, finite or infinite, a weight function w, satisfying the condition w(x) ≥ 0, ∀x ∈ A and such that all the moments This minimization leads to the conditions every P ms has m zeros {x s i } m i=1 real and simple in the interior of A. A well known result of Bernstein [3] shows that, when w is the Chebyshev weight function of the first kind, the polynomials of minimal weighted L p norm, for any p ∈ [1, +∞), are the corresponding Chebyshev polynomials this means that the sequence of s-orthogonal polynomials is independent of s.
More recently, some other cases of invariance with respect to s have been considered in [5] and [6].In [5] it was introduced a particular class of weight functions w µ , depending on a real parameter µ > −1 such that the polynomials of second degree, s-orthogonal in [−1, 1] with respect to w µ , are invariant for any s and any µ.In [6], this result was extended to polynomials of any degree n, identifying a wide class W n of weight functions, enjoying an analogous invariance property.
This class, containing in particular the weight functions w µ , is characterized as follows: let w ∞ denote the Chebyshev weight function of the first kind; a weight function w n is said to belong to W n if it fulfills the relation In fact, in [6] it was proved that, for any given n and any w n ∈ W n , the polynomial T n satisfies the following condition min where γ ∈ , γ ≥ 1.Thus, assuming γ = 2s + 2, it turns out that T n is s-orthogonal in [−1, 1] with respect to w n , independent of s.
A subset W n,µ of W n is provided by the functions w n,µ defined by: (3) where is the Chebyshev polynomial of the second kind.
We observe that all the weight functions in W n,µ are generalized smooth Jacobi weights [9], and assuming µ = − 1 2 one finds out the mentioned result of Bernstein.

THE TUR ÁN TYPE INTEGRATION RULES FOR 2-D CPV INTEGRALS
Let us recall that the general quadrature rule of Turán type is given by are the zeros of the m-th degree polynomials P ms (p; x), monic, s-orthogonal with respect to p.
Moreover, we use the notation below for the one-dimensional CPV integral: (5) subtracting out the singularity one has: x−ζ dx and applying (4) to the first integral in the right hand side, yields [7] (6) where In particular, it has been proved in [7], Now, considering the two-dimensional CPV integral (1) where W (x, y) = p 1 (x)p 2 (y), and applying a product of quadrature rules (6), we have the general integration rule for approximating (1): hi B (2) + C 1 (ξ) where are the zeros respectively of the s-orthogonal polynomials P ms (p 1 ; x) and P ns (p 2 ; y); A (1) •i and A (2) •j are the coefficients of the Turán type quadrature formulas related to the weights p 1 and p 2 respectively.
The error term E s (f ) in (8), vanishes if f is a polynomial of degree M in x and N in y, where In the previous formulas, the existence of the derivatives of f up to the 2s-th order is required at least at the nodes.However, the presence of these derivatives in many practical situations does not constitute a computational problem, since in several cases recurrence relations can be established between any two successive derivatives of the given function [1].
In order to give an explicit expression for the error term in (8), we introduce the notation below: and We assume that the quantities m −1,j (ς) exist finite for j = 1, 2.
Proof.We recall that an interesting result in [12] allows for giving evaluations of the remainder of some approximation formulas in two variables; to be more precise, let T 1 , T 2 denote two linear approximation operators and set R denote the remainders in the approximation of T 1 , T 2 and T respectively, then the following relation holds: Taking into account that in the case being examinated the linear operators T 1 , T 2 reduce to the integral operators I (p 1 f ; ξ) , I (p 2 f ; η) defined in (5) and the corresponding error terms have the form (7), we can write Subtracting out the singularities in the above CPV integrals and by the hypothesis on f , the relation ( 10) follows.

ON THE EFFICIENCY OF A CLASS OF INTEGRATION RULES
The evaluation of both the cubature sum and the remainder term in (8) becomes particularly simple when p 1 and p 2 belong to the class of the weight functions recalled in Section 2; indeed, assuming for instance p 1 ∈ W m and p 2 ∈ W n not only one has for any s, but also the coefficients A hi in (4) can be given explicitly, as shown in [6].Furthermore, in this case, it is possible to state a convergence result of the integration rules (8), for s diverging.
Let DT (J) be the set of Dini type functions, defined on any interval J of length l (J), by where ω(g; •) denotes the modulus of continuity of the function g; and, for some δ > 0, let us consider the subinterval We denote by t i,j the singularities of p j , i = 1, 2, ..., q j , j = 1, 2 belonging to (−1, 1) and with U j the set where t 0,j = −1, t q j +1,j = 1.
Moreover, it is also interesting to point out that if p 1 and p 2 belong to the subset W n,µ of W n , then the terms (9) involved in the expression (10) of the remainder, can be often evaluated in closed form.For instance, when m = 2, there results In order to give an idea of the magnitude of the terms H m in (9), we report in the Tables 4 3.

NUMERICAL RESULTS
The integration rules presented in Section 3 have been tested for several functions and for different choices of the weight functions of type (3).We emphasize the very good performances of such cubatures, when the weight functions are p 1 ∈ W m,µ 1 , p 2 ∈ W n,µ 2 as the following Tables illustrate.
• Table 4 shows the results for the function f (x, y) = sin (2x + 3y), when   We point out that also in the case when the singularity is close to the boundary of the integration interval, the rules here proposed have a good performance.