Takemitsu Hasegawa

An automatic quadrature for Cauchy principal value integrals

An automatic quadrature is presented for computing Cauchy principal value integrals Q(f; c) = Faf(t)/(t c) dt, a < c < b, for smooth functions f(t) . After subtracting out the singularity, we approximate the function f(t) by a sum of Chebyshev polynomials whose coefficients are computed using the FTT. The evaluations of Q(f; c) for a set of values of c in (a, b) are efficiently accomplished with the same number of function evaluations. Numerical examples are also given.

An algorithm based on the FFT for a generalized Chebyshev interpolation

An algorithm for a generalized Chebyshev interpolation procedure, increasing the number of sample points more moderately than doubling, is pre- sented. The FFT for a real sequence is incorporated into the algorithm to enhance its efficiency. Numerical comparison with other existing algorithms is given.

Generalized Chebyshev interpolation and its application to automatic quadrature

A generalized Chebyshev interpolation procedure increasing a fixed number of sample points at a time is developed and analvzed. It is incorporated into an efficient automatic quadrature scheme of Clenshaw-Curtis tvpe. Numerical examples indicate that the present method is efficient not onlv for well-behaved functions but for those w^ith discontinuous low order derivatives by virtue of adequate error estimation as wvell as saving of sample points.

Indefinite integration of oscillatory functions by the Chebyshev series expansion

Abstract An automatic quadrature scheme is presented for evaluating the indefinite integral of oscillatory function ∫ x 0 ƒ(t)e iωt dt, 0⩽x⩽1 , of a given function ƒ( t ), which is usually assumed to be smooth. The function ƒ( t ) is expanded in the Chebyshev series to make an efficient evaluation of the indefinite integral. Combining the automatic quadrature method obtained and Sidis extrapolation method makes an effective quadrature scheme for oscillatory infinite integral ∫ ∞ a ƒ( x ) cos ω x d x for which numerical examples are also presented.

Hilbert and Hadamard transforms by generalized Chebyshev expansion

Abstract An automatic quadrature is presented for approximating Hadamard finite-part (fp) integrals of a smooth function, with a double pole singularity within the range of integration. The quadrature rule is derived from the differentiation of an approximation to a Cauchy principal value integral or the Hilbert transform. The approximation to the fp integral is represented as a function of the value of pole by using Chebyshev polynomials of the second kind. Since the error can be estimated independently of the value of pole, a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance, with the same number of function evaluations. Numerical examples are also included to illustrate the performance of the methods.

Application of a modified FFT to product type integration

Abstract An automatic integration scheme is proposed for evaluating the so-called product type (indefinite) integral Q(K, ƒ) = ∫ y x >K(t)ƒ(t) dt, −1 ⩽ x, y where ƒ(t) is assumed to be a smooth function and K(t) are some singular or badly-behaved functions. Typical examples for K(t) are 1n|t − c|, |t − cα, α > − 1, Cauchy principal value 1/(t − c) and eiωt, |ω| ≫ 1. The function ƒ(t) is approximated by a truncated Chebyshev series pN(t) of degree N, whose coefficients are efficiently computed using the FFT. The approximation QN (K, ƒ) to the integral Q(K, ƒ) is given by Q(K, pN. The sequence {pN(t)} is recursively generated until the required tolerance for the integral is satisfied. To enhance the efficiency of the automatic quadrature, the degree N is increased more slowly than doubling, which is usually the case. The evaluations of QN(K, ƒ)=Q(K, pN) for a set of {(x, y, c)} can be efficiently made by using recurrence relations for the singular kernels K(t) above. Numerical examples for the algebraic singular kernel K(t)=|t − c|α, α > − 1, are included.

An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=eixg(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals ∫a∞K(ωt)f(t) and ∫a∞L(ωt)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=Jν(x) andL(x)=Yν(x), whereJν(x) andYν(x) are the Bessel functions of order ν. The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.

Numerical integration of functions with poles near the interval of integration

Abstract An automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole (or poles) of order 2, i.e., a double pole on the real axis or a complex conjugate pair of double poles, near the interval of integration. The present scheme is based on product integration rules of interpolatory type, using function values of the abscissae only in the interval of integration. The integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the Chebyshev polynomials. The fast Fourier transform (FFT) technique is used to generate efficiently the sequence of the finite Chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error. Numerical examples are included to illustrate the performance of the method.

An algorithm for nondominant solutions of linear second-order inhomogeneous difference equations

An algorithm is given for computing a weighted sum of a nondominant solution of a linear second-order inhomogeneous difference equation to a prescribed accuracy by estimating the truncation error. The present method is an extension of both the stable numerical method due to Olver and Sookne and a summation technique due to Deuflhard for computing minimal solutions of a homogeneous difference equation. The method is illustrated by numerical examples.

An extended doubly-adaptive quadrature method based on the combination of the Ninomiya and the FLR schemes

An improvement is made to an automatic quadrature due to Ninomiya (J. Inf. Process. 3:162–170, 1980) of adaptive type based on the Newton–Cotes rule by incorporating a doubly-adaptive algorithm due to Favati, Lotti and Romani (ACM Trans. Math. Softw. 17:207–217, 1991; ACM Trans. Math. Softw. 17:218–232, 1991). We compare the present method in performance with some others by using various test problems including Kahaner’s ones (Computation of numerical quadrature formulas. In: Rice, J.R. (ed.) Mathematical Software, 229–259. Academic, Orlando, FL, 1971).