The problem to be examined here is that of determining an algebraic or trigonometric polynomial of minimal degree exactly satisfying given constraints. The constraints to be considered are of the following forms: (A) Ordinates are specified at given distinct abscissas, (B) Slopes are specified at some or all of the points where the ordinates are constrained, (C) pth derivatives are specified at some points for which all lower-order derivatives, including the zeroth (i.e. the Ordinate), are also specified. The polynomials which we will construct to satisfy such constraints will be of the form jP"(x) = ]£j CrXr or

A floating-point error analysis is given for the evaluation of a real polynomial at a real argument by Horner's scheme. A computable error bound is derived. It is observed that when a polynomial has coefficients of constant sign or of strictly alternating sign, one cannot expect better accuracy by reformulating the problem in terms of Chebyshev polynomials. Given that the real polynomial P{x) = 22 A*' is to be evaluated at a real argument a under conditions of normalized floating-point arithmetic, we wish to study the bounds for accumulated round-off effects. Two algorithms for evaluation will be compared: (i) the standard Homer scheme and (ii) Clenshaw's algorithm [1]

The problem to be examined here is that of determining an algebraic or trigonometric polynomial of minimal degree exactly satisfying given constraints. The constraints to be considered are of the following forms: (A) Ordinates are specified at given distinct abscissas, (B) Slopes are specified at some or all of the points where the ordinates are constrained, (C) pth derivatives are specified at some points for which all lower-order derivatives, including the zeroth (i.e. the Ordinate), are also specified. The polynomials which we will construct to satisfy such constraints will be of the form jP"(x) = ]£j CrXr or

No abstract

Abstract. Some algorithms are introduced, whereby a function defined on an arbitrarily spaced set of abscissas may be interpolated or approximated by trigonometric or hyperbolic polynomials. The interpolation may be ordinary or osculatory. Least squares approximation is included; the approximant may be a pure sine series or a cosine series or a balanced trigonometric or hyperbolic polynomial. An application to a periodicity-search is described.An extensive set of algorithms is available for functional approximation and interpolation in terms of polynomials. The present article develops some corresponding algorithms for nonpolynomial approximants. The classes of approximant (interpolant) considered are sine polynomial, cosine polynomial, balanced trigonometric polynomial and their analogs in terms of hyperbolic functions. The classes of approximation considered are interpolation on ordinates, osculatory and hyperosculatory interpolation, weighted least-squares approximation, weighted least-squares approximation subject to some ordinate and derivative constraints. Let /' be the range of 0 induced by the requirement x£/ and let n,,(0) be a function of degree (j, j) such that n,,(0,) = 0 for / g 2j; moreover, these 2j zeros are the only zeros of IT,,-in /'. For consecutive j we can now construct the functions IT,-,-, which are unique to within a normalization factor. Starting with noo(0) = 1, we may define for y = 0,1 • ■ • ní+1,,+i = gX©)!!,-,, where g,(0) = a¡ sin 0 + /3, cos 0 -y¡.

Abstract. The m-point Gauss-Legendre formula gives an exact expression for the integral of an algebraic polynomial of maximum degree 2m -1 in terms of m ordinates. It is shown that analogous formulas can be derived for exponential and trigonometric polynomials. | The Gauss-Legendre quadrature formulas approximate the integral of a function by a weighted sum of function-values. When m function-values are used, the formula is exact for functions belonging to a specific 2m-dimensional space, namely the polynomials of degree zero through 2m -1. This finite-dimensional space is in no sense 'representative' of the infinite space over which the formula gives exact results. The latter space includes all Riemann-integrable functions which are odd with reference to an origin located at the mid-point of the domain of integration ; it also includes an infinite class of even functions, and this class contains an infinite subclass of polynomials. Similar remarks also apply to the Newton-Cotes formulas. In where qm is the mth degree polynomial belonging to a set which is orthogonal with respect to the inner product iqrix), q,ix)) = /* qrix)qsix)wOx)dx, and the x¡ are zeros of qm.Using these results as a basis, we now consider the problem of constructing an m-point quadrature formula which gives exact results whenever the integrand fOx) belongs to a 2m-dimensional space distinct from those considered in [1]. A simple example is provided by the case where fix) is a linear combination of exponentials erx, where r takes consecutive (possibly negative) integer values. In this case by writing ex = z we obtain /* fix)dx = Jfa zagOz)dz, where s is an integer and g a polynomial. The latter integral is in the form (1), so one can find an m-point formula which is valid for up to 2m terms in the linear combination. Specifically, if r takes integer values from p through p 4-2m -1, then s = p -1 ; the polynomial qm and numbers ß, are defined as in (1), (2) with wOz) = zs and integration limits ea, eh. It then follows that

Abstract. A floating-point error analysis is given for the standard recursive method of evaluating trigonometric polynomials. It is shown that, by introducing a phase-shift, one can hold the error growth down to an essentially linear function of the degree. Explicit computable error bounds are derived and numerically verified. , and the principal conclusion was that the cumulative effect of rounding errors could become very severe whenever 0 was small modulo zr. By using the phase-shift ) = F(w/2 -), and one can therefore always avoid evaluating a Fourier series at a small argument. If we arrange to perform the phase-shift whenever 6 is in the range (-ir/4, zr/4) modulo ir, we can guarantee that all evaluations occur with arguments in the range [ir/4, 3 v/A] modulo zr. Under these conditions, the behavior of the Clenshaw algorithm will be shown to be quite good. The transformation consists in determining the coefficients C'" S'r of Gi

Abstract. The m-point Gauss-Legendre formula gives an exact expression for the integral of an algebraic polynomial of maximum degree 2m -1 in terms of m ordinates. It is shown that analogous formulas can be derived for exponential and trigonometric polynomials. | The Gauss-Legendre quadrature formulas approximate the integral of a function by a weighted sum of function-values. When m function-values are used, the formula is exact for functions belonging to a specific 2m-dimensional space, namely the polynomials of degree zero through 2m -1. This finite-dimensional space is in no sense 'representative' of the infinite space over which the formula gives exact results. The latter space includes all Riemann-integrable functions which are odd with reference to an origin located at the mid-point of the domain of integration ; it also includes an infinite class of even functions, and this class contains an infinite subclass of polynomials. Similar remarks also apply to the Newton-Cotes formulas. In where qm is the mth degree polynomial belonging to a set which is orthogonal with respect to the inner product iqrix), q,ix)) = /* qrix)qsix)wOx)dx, and the x¡ are zeros of qm.Using these results as a basis, we now consider the problem of constructing an m-point quadrature formula which gives exact results whenever the integrand fOx) belongs to a 2m-dimensional space distinct from those considered in [1]. A simple example is provided by the case where fix) is a linear combination of exponentials erx, where r takes consecutive (possibly negative) integer values. In this case by writing ex = z we obtain /* fix)dx = Jfa zagOz)dz, where s is an integer and g a polynomial. The latter integral is in the form (1), so one can find an m-point formula which is valid for up to 2m terms in the linear combination. Specifically, if r takes integer values from p through p 4-2m -1, then s = p -1 ; the polynomial qm and numbers ß, are defined as in (1), (2) with wOz) = zs and integration limits ea, eh. It then follows that