Cubature formulae and orthogonal polynomials

CUBATURE FOKMULAE AND ORTHOGONAL POLYNOMIALS* I.P. MYSOVSKIKH Leningrad fhceiued

3 January

19691

IN Ill the proposition (Theorem 2’ ) was stated that kZ common zeros of two orthogonal polynomials of degree k of a plane domain and with a positive weight function, can be taken as the nodes of a cubature formula exact for polynomials of degree 2Iz - 1. Here this statement is proved in a somewhat stronger form: the common zeros of the orthogonal polynomials are not assumed to be different. Lemma

Let the algebraic curves f = 0, g = 0 of order k intersect distinct points Jfi

=

(xi,

Vi),

i= 1,2,..., ka-

r

(0 g

2r <

at exactly h2 - r

kz),

the first r points M,, . . . , Mr being points of tangency of the first curves. There exist kZ + k points Mi, i = k2 - r + 1, , . , , Ni such that the points M,, . . . , M, _r do not lie on a curve of order at the points M,, . . . , M, contactlof the first order with the curve

(1)

order of these Ni = 2k2 + k, 2k - 1, possessing

r,

f = 0.

A point of tangency of the first order can be regarded as two infinitely close points of intersection, so that in the lemnla we have kZ points of intersection, some of which are infinitely close together. Proof. The points (11 do not lie on an algebraic curve of order less than k, tangential to the curve f at the points M,, . . . , Mr. We assume the opposite: such a curve h = 0 exists. If h = 0 is irreducible, it is a component of each of the curves f = 0, g = 0, which contradicts the hypothesis of the lemma. If h = 0 is reducible, there exists a component of it which is a component of both f = 0 and g = 0, which is again excluded.

* Zh.

vGhsi1.

Mat. mat.

Fizi, 10, 2, 444-447,

1970.

219

1. P. Mysovskikh

220

We also notice that because of the hypotheses of the lemma the points (1) are not singular points of the curves f 0 and g = 0. We choose k distinct straight lines not passing through any of the points of (1). On I, we take k + 1 distinct points. On ii we take k + i distinct points chosen on t$,. . . , li_* (i = 2, 3,. . . , k). The number of points chosen on the straight lines equals k2 + i/,k(k 4 1). We denote them by Mj, j = k2 - r + 1,. . , where Nz = 2k2+ I/zk(k 4 1). Nz--r, The points Mj, j = 1,2,..., Nz- r,do not lie on a curve of order 2k - 1, possessing at the points M,, . . . , A$ contact of the first order with the curve f = 0. Otherwise this curve contains the straight lines I,, . . . , I, as components. But then the points (1) lie on a curve of order k - 1, having contact with I = 0 at the points M,, . . . , IV,, which is impossible. Wedenoteby qj(x, y),j=$ the following order:

2,...,

monomials of x: and y, enumerated in

1, x, y, x2, xy, y2, x3,

xzy,.

I

*

.

We also introduce the notation

The statement about the dis~bution of the points to the statement that the N, x N,-matrix

Xj, i = 1. 2,. . . , NZ - 7.

is equivalent

has rank equal to Iv,. The first k* columns of the matrix (21, corresponding to the points of intersection (1) of the curves f = 0 and g = 0, are linearly independent, as is implied by the following theorem ([21, p. 381). Theorem from 121 If two curves of order n, and n, intersect

only at ordinary points, the points

Cubature

formulae

and orthogonal polynomials

of intersection, distinct or infinitely close, impose independent conditions every curve thr order of which is greater than or equal to n, + n2 - 2.

221

on

This also implies the lemma, since a non-zero minor of the matrix (2) can be obtained by deleting the N, - N, columns with ordinal numbers greater than k* corresponding to the points on the straight lines 1,) . . . , 1,. It is obvious from the proof that there exist 2k2 - k - r points, which include the points (1) and which do not lie on an algebraic curve of order 2k - 2, possessing contact of the first order with f = 0 at the points M,, . . . , M,. Theorem

Let f and g be orthogonal polynomials of degree k of the domain !J and of weight p (x, y) > 0 such that the algebraic curves f = 0, g = 0 intersect at exactly k* - r finite and distinct points (l), the points M,, . . . , M, being points of contact of the first order of the curves f = 0 and g = 0. Then the points (1) can be taken as the nodes of the cubature formula

exact for polynomials of degree not greater than 2k - 1. Proof.

In accordance

with the lemma we supplement the points (1) up to

N 1 - r points in such a way that the points M,, Mz,...,

MN,-,

(4)

do not lie on an algebraic curve of order 2k - 1, possessing at the points M,, . . . , Mr contact of the first order with f = 0. The points (4) can be taken as the nodes of the cubat,ure formula

JJ

pFdxdyr*

P

N,_-l

c j-1

cfFWj)+

(5) j-i

exact for polynomials of degree not greater than 2k - 1. Indeed, noting that (5) is exact for F = ‘pi, i = 1, 2,. . . , Ni, we obtain a linear algebraic system for the unknown coefficients of the formula. The determinant of the system is the same as the non-zero minor of the matrix (2) discussed in the proof of the lemma.

1. P. Mysovskikh

222

We prove that Cj 5 0 in (5) for j > k2 - r. Let 1 be an integer satisfying the inequality k2 - P + 1 G 1 G IV1 - r. We construct a polynomial P, (x, y) such that the curve P, Cx, y) = 0 passes through all the nodes of formula (5) except M,, touches the curve f = 0 at the nodes M,, . . . , M, and P, (Ml) = 1. Such a polynomial exists, since for the discovery of its coefficients we obtain a system with the same determinant as figured in the determination of the coefficients of formula 6). Since the curve P, (x, y) = 0 passes through all the points of intersection the curves f = 0 and g = 0 and has contact of the fist order with them at the points M,, . . . , M,, by Noether’s theorem (see [21, p. 244) the representation

of

(6)

Pl lx, Y) = Af + Bg,

holds, where A and B are polynomials of degree h - 1. It is also easy to give a direct proof of the representation (6). We write that formula (5) is exact when F = P,. We obtain

c, =

ss

pP1 ax au=

P

ss

p(Af+Bg)dsdy

=

0.

a

The integral equals zero, since f and g are orthogonal polynomials. has been proved.

The theorem

It is only for simplicity of notation that we speak of contact of the first order at all the r points. It would also be possible to speak of contact of any order for each point individually. As we have seen, cubature formulas containing values of the partial derivatives of the integrand occur naturally when we take as nodes the common zeros of orthogonal polynomials. Similar quadrature formulae with nodes at the roots of orthogonal polynomials do not exist, since in the one-dimensional case the roots of orthogonal polynomials are simple. We also notice that the nodes and coefficients of (3) are real, if the common zeros of the orthogonal polynomials f and g are real. ExampZe. Let Q be the square --~Gx, degree orthogonal polynomials

YGI,P(S,

u)=I.

Thefourth

Cubature

223

formulae and orthogonal polynomials

where

are such that the curves f = 0 and g = 0 intersect at the 12 points i&a, 01, (0, *a); (fb, ztb); (*cl zkc), and at the points (M. O), (0, AU) tangency of the first order occurs. (We do not give the numerical values of a, b, c here.) By the theorem there exists a eubature formula i 1 F(*, #Ids: dY=A TF(a,oj+ F(- a, o)+ F(0, a)+ F(0, - a)] + ss -i -i +B~F(fb,=tb)+C~F(~cc,=tc)+~(F,(a,O)+F,(-aa,O)+F,(O,a)+~x(O,-u)],

which is exact for all polynomials of the 7-th degree. The inequality of the coefficients for groups of nodes is a consequence of symmetry. From the fact that the formula is exact for F = x, we obtain D = 0, so that the formula does not contain terms with partial derivatives. We obtain a well-known formula G31, p.

458). Translated by J. Berry REFERENCES

1.

MYSOVSKIKH, N. P. Cubature formulae and orthogonal polynomials. Mat. mat. Fiz., 9, 2, 419-425, 1969.

2.

COOLIDGE, J. L. A Treatise 1931.

3.

KRYLOV, V. I. Approximate Evaluation integralovf ‘Nauka”, Moscow, 1967,

on Algebraic

Plane Curves.

of integrals

Zh. u?chisZ.

Clarendon Press, Oxford,

(Priblizhennoe vychislenie