Numerical Quadratures Using the Interpolation Method of Hurwitz-Radon Matrices  

作　　者：Dariusz Jacek Jakóbczak 

机构地区：[1]Department of Computer Science and Management

出　　处：《Advances in Linear Algebra & Matrix Theory》2014年第2期100-108,共9页线性代数与矩阵理论研究进展（英文）

摘　　要：Mathematics and computer sciences need suitable methods for numerical calculations of integrals. Classical methods, based on polynomial interpolation, have many weak sides: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomials considerably. We cannot forget about the Runge’s phenomenon. To deal with numerical interpolation and integration dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical integration. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for vector spaces of dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the curve point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes;interpolation of L points of the curve is connected with the computational cost of rank O(L);MHR interpolation is not a linear interpolation.Mathematics and computer sciences need suitable methods for numerical calculations of integrals. Classical methods, based on polynomial interpolation, have many weak sides: they are useless to interpolate the function that fails to be differentiable at one point or differs from the shape of polynomials considerably. We cannot forget about the Runge’s phenomenon. To deal with numerical interpolation and integration dedicated methods should be constructed. One of them, called by author the method of Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. This novel method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from that matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of function interpolation and numerical integration. Created from the family of N-1 HR matrices and completed with the identical matrix, system of matrices is orthogonal only for vector spaces of dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very significant for stability and high precision of calculations. MHR method is interpolating the curve point by point without using any formula of function. Main features of MHR method are: accuracy of curve reconstruction depending on number of nodes and method of choosing nodes;interpolation of L points of the curve is connected with the computational cost of rank O(L);MHR interpolation is not a linear interpolation.

关 键 词：CURVE INTERPOLATION NUMERICAL Integration Hurwitz-Radon MATRICES MHR Method 

分 类 号：O1[理学—数学]