CYBERNETICS AND CONTROL THEORY

A. S. KRONROD

ON INTEGRATION WITH ACCURACY CONTROL

(Presented by Academician M. V. Keldysh, 27 VII 1963)

1°. For integrating \(f(x)\) on \([-1,+1]\), quadrature formulas of the form

\[ A(f)=C_1 f(x_1)+\cdots+C_n f(x_n)\qquad (x_i\ne x_k \text{ for } i\ne k) \tag{1} \]

are considered. By the accuracy of a quadrature \(A\) we shall mean the greatest natural number \(T(A)\) for which all polynomials of degree \(T(A)\) are integrated exactly by formula (1). The number \(Ц(A)=n\) will be called the cost of \(A\). Among quadratures of cost \(n\), the Gaussian quadrature \(\Gamma_n\) has the maximal accuracy \(2n-1\). The accuracy of other quadratures is \(\leqslant 2n-2\).

2°. Let a pair of quadratures \(A,B\) be given. This means that \(A\) and \(B\) are different quadratures and \(T(A)\leqslant T(B)\). By the cost \(Ц(A,B)\) of the pair \(A,B\) we shall mean the total number of distinct points \(x_i\) occurring in \(A\) and \(B\), and by the accuracy of the pair the number \(T(A,B)=T(A)\).

3°. Lemma. Let \(x_1,\ldots,x_k\) be arbitrary distinct points. There exists a quadrature \(D\)

\[ D(f)=d_1 f(x_1)+\cdots+d_k f(x_k) \tag{2} \]

of accuracy \(\geqslant k-1\). Among quadratures of accuracy \(\geqslant k-1\), it is unique.

Indeed, for \(f(x)=x^s\) \((s=0,1,\ldots,k-1)\), the determinant of the system

\[ D(x^s)=2:(s+1)\qquad (s=0,1,\ldots,k-1) \tag{2a} \]

is a Vandermonde determinant, different from zero.

4°. Theorem 1. Let \(A,B\) be a pair of quadratures. Then

\[ Ц(A,B)\geqslant T(A,B)+2. \]

Indeed, let \(x_s\,[s=1,2,\ldots,N=Ц(A,B)]\) be all the points entering into the formulas \(A\) and \(B\). The quadratures \(A\) and \(B\) have the form (2). By the lemma, they are identical if \(T(A)\geqslant Ц(A,B)-1\).

5°. For integration with accuracy control, with permitted cost
\(N=2n+1\), we shall call optimal* a pair \(A,B\) with maximal accuracy \(T(A,B)\), minimal cost \(Ц(A,B)\leqslant N\) among all such pairs, with minimal \(Ц(A)\) among these, and with maximal \(T(B)\) among these. According to 1° and 4° we have \(A\equiv\Gamma_n\). Therefore \(T(A,B)=2n-1=N-2\). We construct a quadrature \(K_n\) which, together with \(\Gamma_n\), gives an optimal pair. Let \(L\) be the Legendre polynomial of degree \(n\) with leading coefficient 1. Let \(I_p\) be the \((n+p)\)-th moment of \(L\). Let \(t=0\) for even \(n\), and \(t=1\) for odd \(n\). Put

\[ b_0=1;\qquad b_{2k}=-(b_0 I_{2k}+b_2 I_{2k-2}+\cdots+b_{2k-2}I_2):I_0; \tag{3} \]

\[ \varphi(x)=b_0x^{n+1}+b_2x^{n-1}+\cdots+b_{n+t}x^{1-t}. \tag{4} \]

Then \(F(x)=L(x)\cdot\varphi(x)\) is orthogonal to \(x^s\) \((s=0,1,\ldots,n+1-t)\). The function \(F(x)\) has \(2n+1\) distinct roots \(\{\beta_s\}\) on \([-1,+1]\)**; moreover—

* For an even permitted cost, I do not know how to approach this problem reasonably.

** For arbitrary \(n\) I do not know how to prove this, but up to \(n\leqslant 40\) the roots have been computed directly.

whose roots are symmetric with respect to the point \(\beta_{n+1}=0\). Put

\[ K_n(f)=d_1 f(\beta_1)+\ldots+d_{2n+1}f(\beta_{2n+1}). \tag{5} \]

By the lemma we choose \(d_i\) so that \(T(K'_n)\ge 2n\). Such a quadrature formula is unique and \(d_i=d_{2n+2-i}\). Hence it follows that \(T(K_n)\ge 2n+1\).

6°. Theorem 2. \(T(K_n)=3n+1\) for even \(n\), and \(T(K_n)=3n+2\) for odd \(n\).

Indeed, \(F(x)\) is orthogonal to \(x^s\) \((s=0,1,\ldots,n+1-t)\). Further, \(\{\beta_i\}\) are the roots of \(F(x)\). Hence \(K_n\) is exact for certain polynomials of degree \(2n+1+s\) \((s=0,1,\ldots,n+1-t)\) with leading coefficient 1, namely for \(F(x)\cdot x^s\). Consequently, \(K_n\) is exact for all polynomials up to degree \((3n+1+t)\).

7°. In the table, the second column gives the nodes, i.e. the roots of \(F(x)\), for \(2 \leqslant n \leqslant 16\). In the third and fourth columns the weights of the Gaussian and refining quadratures are given. All pertain to the case of integration on \([0,1]\). The nodes and weights, by virtue of their symmetry with respect to \(0.5\), are given only for the left half of the interval \([0,1]\).

In the last column of the table are given the relative errors \(\Gamma_p^{(n)}\) and \(K_p^{(n)}\) of the Gaussian and refining quadratures in computing the integral of \(x^p\) on \([-1,+1]\) for the first even \(p\) for which the quadratures \(\Gamma_n\) and \(K_n\) are no longer exact.

All computations were carried out on the machine of the Institute of Theoretical and Experimental Physics. Programs from the library of long-range [[unclear: continuation on next page]] were used.

of A. V. Uskov’s floating-point library and A. Zhivotovsky and V. Pruss’s integer library.

Remark 1. A pair of quadrature formulas—a Gaussian one and a refining one—is convenient to use in solving integral equations with accuracy control in the metric \(C\), since the even nodes of the refining quadrature formula are the nodes of the Gaussian quadrature formula.

Remark 2. A comparison of the corresponding remainder terms \(\Gamma_p^{(2n+1)}\) and \(K_p^{(n)}\) of the Gaussian and refining quadrature formulas (for example, \(\Gamma_{30}^{(15)}\), \(\Gamma_{32}^{(15)}\), \(\Gamma_{34}^{(15)}\) and \(K_{30}^{(7)}\), \(K_{32}^{(7)}\), \(K_{34}^{(7)}\)) shows that, if the required accuracy of integration is not very high, then the refining quadrature formula is only slightly inferior to the Gaussian quadrature formula of the same cost.

Institute of Theoretical and Experimental Physics
of the State Committee for the Use of Atomic Energy

Received
8 VII 1963