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An automatic integration routine for the computation of highly oscillating functions is presented. The algorithm is based on the use of Gaussian quadrature formulas with suitable weight function.

Quadrature and demodulation phase errors in a quadrature PSK system cause crosstalk between channels. It is shown that the tracking loop signal-to-noise ratio must be at least 5 dB greater than for PSK to give the same performance.

Modifications of Romberg’s integration method are applicable to functions with endpoint singularities. Several authors have reached different conclusions on the usefulness of these schemes due to the potential difficulties in calculating certain exponents in the asymptotic error expansion. In this paper, we consider a method based on the ϵ \epsilon -algorithm which does not require the user to supply these parameters. Tests show this luxury produces a method substantially better than the unmodified Romberg’s method, but not as good as the modified procedure which assumes all the exponents are known. It is possible to design a scheme which incorporates the best of both methods and allows the user to decide how much information he wishes to provide. The algorithm is stable numerically in quadrature applications and converges for functions with endpoint singularities of an algebraic or logarithmic nature.

A Chebyshev quadrature is of the form \[ ∫ − 1 1 w ( x ) f ( x ) d x ≃ c ∑ k = 1 n f ( x k ) \int _{ - 1}^1 {w(x)f(x)dx \simeq } c\sum \limits _{k = 1}^n {f({x_k})} \] It is usually desirable that the nodes x k {x_k} be in the interval of integration and that the quadrature be exact for as many monomials as possible (i.e., the first n + 1 n + 1 monomials). For n = 1 , ⋅ ⋅ ⋅ , 7 n = 1, \cdot \cdot \cdot ,7 and 9 9 , such a choice of nodes is possible, but for n = 8 n = 8 and n > 9 n > 9 , the nodes are complex. In this note, the idea used is that the l 2 {l^2} -norm of the deviations of the first n + 1 n + 1 monomials from their moments be a minimum. Numerical calculations are carried out for n = 8 , 10 n = 8,10 , and 11 11 and one interesting feature of the numerical results is that a “multiple” node at the origin is required. The above idea is then generalized to a minimization of the l 2 {l^2} -norm of the deviations of the first k k monomials, k ≧ n + 1 k \geqq n + 1 , including k = ∞ k = \infty , and corresponding numerical results are presented.

Let { Q n } n = 1 ∞ \{ {Q_n}\} _{n = 1}^\infty denote a sequence of quadrature formulas, Q n ( f ) ≡ ∑ i = 1 k n w i ( n ) f ( x i ( n ) ) {Q_n}(f) \equiv \sum _{i = 1}^{{k_n}}w_i^{(n)}f(x_i^{(n)}) , such that Q n ( f ) → ∫ 0 1 f ( x ) d x {Q_n}(f) \to \int _0^1 f (x)dx for all f ∈ C [ 0 , 1 ] f \in C[0,1] . Let 0 > ε > 1 4 0 > \varepsilon > \frac {1}{4} and a sequence { a n } n = 1 ∞ \{ {a_n}\} _{n = 1}^\infty be given, where a 1 ≧ a 2 ≧ a 3 ≧ . . . {a_1} \geqq {a_2} \geqq {a_3} \geqq ... , and where a n → 0 {a_n} \to 0 as n → ∞ n \to \infty . Then there exists a function f ∈ C [ 0 , 1 ] f \in C[0,1] and a sequence { n k } k = 1 ∞ \{ {n_k}\} _{k = 1}^\infty such that | f ( x ) | ≦ 2 a 1 / | ( 1 − 4 ) | |f(x)| \leqq 2{a_1}/|(1 - 4)| , and such that ∫ 0 1 f ( x ) d x − Q n k ( f ) = a k , k = 1 , 2 , 3 , . . . \int _0^1 f (x)dx - {Q_{{n_k}}}(f) = {a_{k,}}k = 1,2,3,... .

Equal-weight Chebyshev quadrature is not generally used because the nodes become complex for large n. However, interest in these schemes remains because of recent work on minimal norm quadrature as well as schemes for doing real integrals of analytic functions by complex methods. This note presents some properties of these Chebyshev quadratures that may be of interest to other researchers in this area. Proofs are sketched to save space. Equal-weight Chebyshev quadrature is not generally used because the nodes x(" l become complex for n 2 10. However, interest in these schemes remains because of recent work on minimal norm quadrature [1], [2], and [3] as well as schemes for doing real integrals of analytic functions by complex methods [5]. This note presents some properties of these Chebyshev quadratures that may be of interest to other researchers in this area. Proofs are sketched to save space. The nodes for Chebyshev quadrature are defined as the unique solution set of the system n r 2 , [x ']]i = J xi dx, j = 1, ... , Let P,(x) = I-1 (x ) THEoREM 1. If n = 2m, Pz(x) has at least two real zeros in (ts, sn) where t. is the zero of largest magnitude of the nth Legendre polynomial. The proof is immediate by using a Gauss quadrature formula on P"(x). A little known result of Kuzmin [6] is THEOREM 2. P"(x) has O(og n) real zeros. Using this, we can prove COROLLARY. Again, with n = 2m, Theorem 1 is true with the smaller interval (e{m, em). For n = 2m < 100, computation gives exactly two real zeros of P2mf(x). Hence, using the known symmetry of P2m, we get COROLLARY. The positive real zero of P2m(X) lies in the interval (em, tm+,), 2m < 100. The zeros of P"(z) are given for n < 47 in the microfiche section of this issue. THEOREM 3. Let f(z) be analytic in a closed domain including the curve r (defined below) in its interior. Let In be given by Received December 10, 1970, revised March 10, 1971. AMS 1969 subject classifications. Primary 6555.

This report compares the performance of five different adaptive quadrature schemes, based on Newton-Cotes (2 N + 1) point rules ( N = 1, 2, 3, 4, 5), in approximating the set of definite integrals ∫ 1 -1 ( x 2 + p 2 ) -1 dx with relative accuracy ε.