J. Vanthournout

On a new type of mixed interpolation

Abstract We approximate every function f by a function f n ( x ) of the form a cos kx + b sin kx + Σ n −2 i =0 c i x i so that f ( jh ) = f n ( jh ) for the n + 1 equidistant points jh , j = 0,…, n . That interpolation function f n ( x ) is proved to be unique and can be written as the sum of the n th-degree interpolation polynomial based on the same points and two correction terms. The error term is also discussed. The results for this mixed type of interpolation reduce to the known results of the polynomial case as the parameter k is tending to 0. This new interpolation theory will be used in the future for the construction of quadrature rules and multistep methods for ordinary differential equations.

On a class of modified Newton-Cotes quadrature formulae based upon mixed-type interpolation

Abstract We present quadrature rules which integrate exactly not only polynomials up to a certain degree, but also the functions sin kx and cos kx (where k is a free parameter). The formulae we obtain are modified Newton—Cotes formulae. They are derived by replacing the integrand by an interpolation function of the form a cos kx + b sin kx + Σn-2j = 0cjxj based on equally spaced nodes. The total truncation error of the modified quadrature formulae is discussed and a rigorous proof of the error term is given for the modified Simpsons 3 8 rule. Numerical examples show the efficiency of the modified rules and the importance of the error term.

A modified Numerov integration method for second order periodic initial-value problems

A two-step P-stable method for the numerical integration of periodic initial value problems is derived. By the analysis of the error term a good estimate for the present period can be made. It is shown that the obtained implicit method can be made explicit by use of an analogous lower order method. In both cases a stability analysis is given. The superiority of these methods over other available methods is illustrated by two examples.

Families of backward differentiation methods based on a new type of mixed interpolation

Abstract Backward differentiation methods based on a new type of mixed interpolation for the first-order initial-value problems whose solutions are known to be periodic are constructed. The angular frequency k is calculated by minimizing the local truncation error within each integration interval. The resulting methods depend on a parameter θ = hk , h the step length, and reduce to classical backward methods if k → 0. Numerical examples are provided to illustrate the algorithms.

Modified backward differentiation methods of the Adams-type based on exponential interpolation

Abstract For the numerical solution of first-order differential equations multistep methods of the Adams-type are established which integrate exactly the functions exp ( kx ) x j , j = 0, 1, … , n , where k is a parameter. These methods have stepsize-dependent coefficients which also depend on k . Within each integration segment one can choose for k the value which minimizes in a certain sense the local truncation error. In this way the new method leads to highly accurate results in comparison with the classical methods of the Adams-type, especially when the solution of the differential equation shows an exponential behaviour. Numerical examples are provided to illustrate this.

On the symmetric representations of SU(5). Matrix elements of the generators in the SO(3) basis

By making use of the results of a previous paper [J. Math. Phys. 29, 1958 (1988)], in which the matrix elements of the SU(5) generators were found in closed form for symmetric representations and in an SO(4) labeled state basis, it is shown how the matrix elements of the generators can also be derived in the physical SO(3) state basis. The main step consists in studying the action of the SU(5) generators upon a particular subset of so‐called intrinsic SU(2)×SU(2) states out of which the physical SO(3) states are projected by means of the Hill–Wheeler technique.

Totally symmetric irreducible representations of the group SO(6) in the principal SO(3) subgroup basis

Explicit matrix elements are found for the generators of the group SO(6) in an arbitrary totally symmetric irreducible representation, using the physical principal SO(3) subgroup in the chain SO(6)⊇SO(5)⊇SO(3). The internal one missing label problem is solved through the definition of intrinsic states associated to the SU(2)×SU(2) subgroup in the chain SO(6)⊇SO(5)⊇SU(2)×SU(2) and out of which is projected a complete set of states in the physical basis by integrations over the physical rotation group manifold. The matrix elements of the SO(6) generators in the SU(2)×SU(2) basis are themselves obtained by the intermediate use of an SU(2)×SU(2)×U(1) basis, the latter group being a subgroup of SO(6) but not of SO(5).

Numerical quadrature based on an exponential type of interpolation

It is shown that a function f(x) can be approximated in a unique way by a function so that f n (x) and f(x) coincide in at least (n + 1) equidistant points. Several equivalent expressions of the interpolation function f n (x) of exponential type are given, and the error term is derived in closed form. With this type of interpolation a set of modified Newton-Cotes quadrature rules of the closed and of the open type are established. The total truncation error for these rules is discussed. Numerical examples show the efficiency of the modified rules and exhibit the particular advantage that can be taken of the explicit form of the error term.

On the symmetric representations of SU(5). Matrix elements of the generators in the subgroup bases SU(2)×SU(2)×U(1) and SU(2)×SU(2)

Matrix elements of the group generators for the totally symmetric irreducible representations of SU(5) are obtained in closed form employing the decomposition chain SU(5)⊇SU(4)×U(1)⊇SO(4)×U(1). The SU(4)≈SU(2)×SU(2) subgroup herein also occurs at the tail of the inclusion chain SU(5)⊇SO(5)⊇SO(4). Therefore, closed form expressions for the matrix elements of the SU(5) generators in the latter basis are established, too.

On the spectrum of a third‐order SO(3) scalar in the enveloping algebra of SO(6)

With the aid of previously derived expressions for the SO(3) reduced matrix elements of the SO(6) generators, which were obtained by considering an intermediate SO(6)↓SU(2)⊗SU(2) reduction, a method is set up to evaluate analytic expressions for the eigenvalues of a third‐order scalar operator belonging to the integrity basis of SO(3) scalar operators in the enveloping algebra of SO(6).