783 research outputs found
Bivariate hierarchical Hermite spline quasi--interpolation
Spline quasi-interpolation (QI) is a general and powerful approach for the
construction of low cost and accurate approximations of a given function. In
order to provide an efficient adaptive approximation scheme in the bivariate
setting, we consider quasi-interpolation in hierarchical spline spaces. In
particular, we study and experiment the features of the hierarchical extension
of the tensor-product formulation of the Hermite BS quasi-interpolation scheme.
The convergence properties of this hierarchical operator, suitably defined in
terms of truncated hierarchical B-spline bases, are analyzed. A selection of
numerical examples is presented to compare the performances of the hierarchical
and tensor-product versions of the scheme
Numerical Aspects of the Coefficient Computation for LMMs
The numerical solution of Boundary Value Problems usually requires the use of an adaptive mesh selection strategy. For this reason, when a Linear Multistep Method is considered, a dynamic computation of its coefficients is necessary. This leads to solve linear systems which can be expressed in different forms, depending on the polynomial basis used to impose the order conditions. In this paper, we compare the accuracy of the numerically computed coefficients for three different formulations. For all the considered cases Vandermonde systems on general abscissae are involved and they are always solved by the Bj \u308rk-Pereyra algorithm. An adaptation of the forward error analysis given in [8, 9] is proposed whose significance is confirmed by the numerical results
Solution of a quadratic quaternion equation with mixed coefficients
A comprehensive analysis of the morphology of the solution space for a
special type of quadratic quaternion equation is presented. This equation,
which arises in a surface construction problem, incorporates linear terms in a
quaternion variable and its conjugate with right and left quaternion
coefficients, while the quadratic term has a quaternion coefficient placed
between the variable and its conjugate. It is proved that, for generic
coefficients, the equation has two, one, or no solutions, but in certain
special instances the solution set may comprise a circle or a 3-sphere in the
quaternion space . The analysis yields solutions for each case, and
intuitive interpretations of them in terms of the four-dimensional geometry of
the quaternion space .Comment: 19 pages, to appear in the Journal of Symbolic Computatio
Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
AbstractA “double” Pythagorean-hodograph (DPH) curve r(t) is characterized by the property that |r′(t)| and |r′(t)×r″(t)| are both polynomials in the curve parameter t. Such curves possess rational Frenet frames and curvature/torsion functions, and encompass all helical polynomial curves as special cases. As noted by Beltran and Monterde, the Hopf map representation of spatial PH curves appears better suited to the analysis of DPH curves than the quaternion form. A categorization of all DPH curve types up to degree 7 is developed using the Hopf map form, together with algorithms for their construction, and a selection of computed examples of (both helical and non-helical) DPH curves is included, to highlight their attractive features. For helical curves, a separate constructive approach proposed by Monterde, based upon the inverse stereographic projection of rational line/circle descriptions in the complex plane, is used to classify all types up to degree 7. Criteria to distinguish between the helical and non-helical DPH curves, in the context of the general construction procedures, are also discussed
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