In this paper, osculatory rational functions of Thiele-type introduced by Salzer (1962) are extended to the case of vector valued quantities using tile t'ormalism of Graves-Moms (1983). In the computation of the o...In this paper, osculatory rational functions of Thiele-type introduced by Salzer (1962) are extended to the case of vector valued quantities using tile t'ormalism of Graves-Moms (1983). In the computation of the osculatory continued h.actions, the three term recurrence relation is avoided and a new coefficient algorithm is introduced, which is the characteristic of recursive operation. Some examples are given to illustrate its effectiveness. A sutficient condition for cxistence is established. Some interpolating properties including uniqueness are discussed. In the end, all exact interpolating error formula is obtained.展开更多
A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar de...A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar denominator. The generalized inverse introduced by [3]is gen-eralized to rectangular matrix case in this paper. An exact error formula for interpolation is ob-tained, which is an extension in matrix form of bivariate scalar and vector valued rational interpola-tion discussed by Siemaszko[l2] and by Gu Chuangqing [7] respectively. By defining row and col-umn-transformation in the sense of the partial inverted differences for matrices, two type matrix algorithms are established to construct corresponding two different BGIRI, which hold for the vec-tor case and the scalar case.展开更多
This paper studies error formulas for Lagrange projectors determined by Cartesian sets. Cartesian sets are properly subgrids of tensor product grids. Given interpolated functions with all order continuous partial deri...This paper studies error formulas for Lagrange projectors determined by Cartesian sets. Cartesian sets are properly subgrids of tensor product grids. Given interpolated functions with all order continuous partial derivatives, the authors directly construct the good error formulas for Lagrange projectors determined by Cartesian sets. Owing to the special algebraic structure, such a good error formula is useful for error estimate.展开更多
The multiple regression formulas and correlation of ocular components with refractive errors are presented by Gaussian optics. The refractive error changing rate for the cornea and lens power, the axial length, anteri...The multiple regression formulas and correlation of ocular components with refractive errors are presented by Gaussian optics. The refractive error changing rate for the cornea and lens power, the axial length, anterior chamber depth(ACD) and vitreous chamber depth(VCD) are calculated, including nonlinear terms for more accurate rate functions than the linear theory. Our theory, consistent with the empirical data, shows that the Pearson correlation coefficients for spherical equivalent(SE) and ocular components are highest for SE with axial length, ACD and VCD and weakest for corneal power, lens power and lens thickness. Moreover, our regression formulas show the asymmetric feature of the correlation that the axial length, ACD and VCD are more strongly correlated(with higher negative regression constants) with refractive errors in eyes with hyperopia than in eyes with myopia, particularly for severe hyperopia.展开更多
We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the ...We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.展开更多
文摘In this paper, osculatory rational functions of Thiele-type introduced by Salzer (1962) are extended to the case of vector valued quantities using tile t'ormalism of Graves-Moms (1983). In the computation of the osculatory continued h.actions, the three term recurrence relation is avoided and a new coefficient algorithm is introduced, which is the characteristic of recursive operation. Some examples are given to illustrate its effectiveness. A sutficient condition for cxistence is established. Some interpolating properties including uniqueness are discussed. In the end, all exact interpolating error formula is obtained.
文摘A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar denominator. The generalized inverse introduced by [3]is gen-eralized to rectangular matrix case in this paper. An exact error formula for interpolation is ob-tained, which is an extension in matrix form of bivariate scalar and vector valued rational interpola-tion discussed by Siemaszko[l2] and by Gu Chuangqing [7] respectively. By defining row and col-umn-transformation in the sense of the partial inverted differences for matrices, two type matrix algorithms are established to construct corresponding two different BGIRI, which hold for the vec-tor case and the scalar case.
基金supported by Chinese National Natural Science Foundation under Grant Nos.11601039,11671169,11501051the Open Fund Key Laboratory of Symbolic Computation and Knowledge Engineering(Ministry of Education)under Grant No.93K172015K06the Education Department of Jilin Province,“13th Five-Year”Science and Technology Project under Grant No.JJKH20170618KJ
文摘This paper studies error formulas for Lagrange projectors determined by Cartesian sets. Cartesian sets are properly subgrids of tensor product grids. Given interpolated functions with all order continuous partial derivatives, the authors directly construct the good error formulas for Lagrange projectors determined by Cartesian sets. Owing to the special algebraic structure, such a good error formula is useful for error estimate.
基金Supported by an Internal Research of New Vision Inc. and Nobel Eye Institute
文摘The multiple regression formulas and correlation of ocular components with refractive errors are presented by Gaussian optics. The refractive error changing rate for the cornea and lens power, the axial length, anterior chamber depth(ACD) and vitreous chamber depth(VCD) are calculated, including nonlinear terms for more accurate rate functions than the linear theory. Our theory, consistent with the empirical data, shows that the Pearson correlation coefficients for spherical equivalent(SE) and ocular components are highest for SE with axial length, ACD and VCD and weakest for corneal power, lens power and lens thickness. Moreover, our regression formulas show the asymmetric feature of the correlation that the axial length, ACD and VCD are more strongly correlated(with higher negative regression constants) with refractive errors in eyes with hyperopia than in eyes with myopia, particularly for severe hyperopia.
文摘We consider the computation of the. Cauchy principal value mtegral by quadrature formulaeof compound type, which are obtained by replacing f by a piecewise defined function F,[;]. The behaviour of the constants m the estimates where quadrature error) is determined for fixed i and which means that not only the. order, but also the coefficient of the main term of is determined. The behaviour of these error constants is compared -with the corresponding ones obtained for the. method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.
