Tatsuo Torii

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.

A high-order iterative formula for simultaneous determination of zeros of a polynomial

Abstract We propose a hybrid method to determine all the zeros of a polynomial simultaneously, by combining the single-root method and the simultaneous one. The present method has a high convergence order even for multiple roots by using the Pade approximation.

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 iterative method for algebraic equation by Pade´ approximation

In this paper, we consider iterative formulae with high order of convergence to solve a polynomial equation,f(z)=0. First, we derive the numerator of the Padé approximant forf(z)/f′(z) by combining Viscovatovs and Euclidean algorithms, and then calculate the zeros of the numerator so as to apply one of the zeros for the next approximation. Regardless of whether the root is simple or multiple, the convergence order of this iterative formula is always attained for arbitrary positive integerm with the Taylor polynomial of degreem for a given polynomialf(z). Since it is easy to systematically obtain formulae of different order, we can choose formulae of suitable order according to the required accuracy.ZusammenfassungIn dieser Arbeit betrachten wir Iterationsverfahren höherer Ordnung zur Lösung einer Polynomgleichungf(z)=0. Durch Anwendung der Verfahren von Viscovatov und Euclid erhalten wir eine Approximation für den Zähler der Padé-Approximierenden vonf(z)/f′(z), und verwenden eine der Wurzeln des Zählerpolynoms für die nächste Approximation. Dieses Verfahren hat die Ordnungm sowohl für einfache als auch mehrfache Wurzeln bei Verwendung des Taylor Polynomsm-ter Ordnung. Da es leicht ist, Verfahren verschiedener Ordnung zu erhalten, können wir gemäß der geforderten Genauigkeit eine passende Ordnung des Verfahrens wählen.

A methodology of parsing mathematical notation for mathematical computation

Mathematical notation has not been parsed effectively up to the present. In order to use mathematical notation on computer directly, and thus to develop many potential applications and to determine if common mathematical notation can be parsed by computer, we inquire into the omission mechanism of parentheses and operators, study the extension mechanism of mathematical notation, and establish a methodology to translate a two-dimensional mathematical expression into a textual functional meaning representation. This methodology is a combination of different formalization methods denoted by a defined box language, various context-sensitive grammars written in a defined metalanguage, and knowledge-based parsers.