Numerical Quadratures Using the Interpolation Method of Hurwitz-Radon Matrices
Author(s)
Dariusz Jacek Jakóbczak*
ABSTRACT
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.
Cite this paper
Jacek Jakóbczak, D. (2014) Numerical Quadratures Using the Interpolation Method of Hurwitz-Radon Matrices. Advances in Linear Algebra & Matrix Theory, 4, 100-108. doi: 10.4236/alamt.2014.42008.
Jacek Jakóbczak, D. (2014) Numerical Quadratures Using the Interpolation Method of Hurwitz-Radon Matrices. Advances in Linear Algebra & Matrix Theory, 4, 100-108. doi: 10.4236/alamt.2014.42008.
References
[1] Dahlquist, G. and Bjoerck, A. (1974) Numerical Methods. Englewood Cliffs N. J., Prentice-Hall. [2] Jankowska, J. and Jankowski, M. (1981) Survey of Numerical Methods and Algorithms (Part I). Wydawnictwa Naukowo-Techniczne, Warsaw. (in Polish) [3] Ralston, A. (1965) A First Course in Numerical Analysis. McGraw-Hill Book Company, Boston. [4] Kozera, R. (2004) Curve Modeling via Interpolation Based on Multidimensional Reduced Data. Silesian University of Technology Press, Gliwice. [5] Jakóbczak, D. (2009) Curve Interpolation Using Hurwitz-Radon Matrices. Polish Journal of Environmental Studies, 3B, 126-130. [6] Citko, W., Jakóbczak, D. and Sieńko, W. (2005) On Hurwitz-Radon Matrices Based Signal Processing. Signal Processing’2005, Poznań, 19-24. [7] Sieńko, W., Citko, W. and Jakóbczak, D. (2004) Learning and System Modeling via Hamiltonian Neural Networks. Lecture Notes on Artificial Intelligence, 3070, 266-271. [8] Jakóbczak, D. (2007) 2D and 3D Image Modeling Using Hurwitz-Radon Matrices. Polish Journal of Environmental Studies, 16, 104-107. [9] Jakóbczak, D. and Kosiński, W. (2007) Hurwitz-Radon Operator in Monochromatic Medical Image Reconstruction. Journal of Medical Informatics & Technologies, 11, 69-78. [10] Eckmann, B. (1999) Topology, Algebra, Analysis—Relations and Missing Links. Notices of AMS, 46, 520-527. [11] Lang, S. (1970) Algebra. Addison-Wesley Publishing Company, Boston. [12] Jakóbczak, D. (2010) Object Modeling Using Method of Hurwitz-Radon Matrices of Rank k. In: Wolski, W. and Borawski, M., Eds., Computer Graphics: Selected Issues, University of Szczecin Press, Szczecin, 79-90. [13] Jakóbczak, D. (2010) Shape Representation and Shape Coefficients via Method of Hurwitz-Radon Matrices. Lecture Notes in Computer Science, 6374, 411-419. [14] Jakóbczak, D. (2011) Object Recognition via Contour Points Reconstruction Using Hurwitz-Radon Matrices. In: Józefczyk, J. and Orski, D., eds., Knowledge-Based Intelligent System Advancements: Systemic and Cybernetic Approaches, IGI Global, Hershey, 87-107. [15] Jakóbczak, D. (2010) Application of Hurwitz-Radon Matrices in Shape Representation. In: Banaszak, Z. and swic, A. Eds., Applied Computer Science: Modelling of Production Processes, 1, Lublin University of Technology Press, Lublin, 63-74. [16] Jakóbczak, D. (2010) Implementation of Hurwitz-Radon Matrices in Shape Representation. In: Choras, R.S., Ed., Advances in Intelligent and Soft Computing 84, Image Processing and Communications: Challenges 2, Springer-Verlag, Berlin Heidelberg, 39-50. [17] Jakóbczak, D. (2011) Curve Parameterization and Curvature via Method of Hurwitz-Radon Matrices. Image Processing & Communications—An International Journal, 1-2, 49-56. [18] Jakóbczak, D. (2011) Data Extrapolation and Decision Making via Method of Hurwitz-Radon Matrices. Lecture Notes in Computer Science/LNAI, 6922, 173-182.
[1] Dahlquist, G. and Bjoerck, A. (1974) Numerical Methods. Englewood Cliffs N. J., Prentice-Hall. [2] Jankowska, J. and Jankowski, M. (1981) Survey of Numerical Methods and Algorithms (Part I). Wydawnictwa Naukowo-Techniczne, Warsaw. (in Polish) [3] Ralston, A. (1965) A First Course in Numerical Analysis. McGraw-Hill Book Company, Boston. [4] Kozera, R. (2004) Curve Modeling via Interpolation Based on Multidimensional Reduced Data. Silesian University of Technology Press, Gliwice. [5] Jakóbczak, D. (2009) Curve Interpolation Using Hurwitz-Radon Matrices. Polish Journal of Environmental Studies, 3B, 126-130. [6] Citko, W., Jakóbczak, D. and Sieńko, W. (2005) On Hurwitz-Radon Matrices Based Signal Processing. Signal Processing’2005, Poznań, 19-24. [7] Sieńko, W., Citko, W. and Jakóbczak, D. (2004) Learning and System Modeling via Hamiltonian Neural Networks. Lecture Notes on Artificial Intelligence, 3070, 266-271. [8] Jakóbczak, D. (2007) 2D and 3D Image Modeling Using Hurwitz-Radon Matrices. Polish Journal of Environmental Studies, 16, 104-107. [9] Jakóbczak, D. and Kosiński, W. (2007) Hurwitz-Radon Operator in Monochromatic Medical Image Reconstruction. Journal of Medical Informatics & Technologies, 11, 69-78. [10] Eckmann, B. (1999) Topology, Algebra, Analysis—Relations and Missing Links. Notices of AMS, 46, 520-527. [11] Lang, S. (1970) Algebra. Addison-Wesley Publishing Company, Boston. [12] Jakóbczak, D. (2010) Object Modeling Using Method of Hurwitz-Radon Matrices of Rank k. In: Wolski, W. and Borawski, M., Eds., Computer Graphics: Selected Issues, University of Szczecin Press, Szczecin, 79-90. [13] Jakóbczak, D. (2010) Shape Representation and Shape Coefficients via Method of Hurwitz-Radon Matrices. Lecture Notes in Computer Science, 6374, 411-419. [14] Jakóbczak, D. (2011) Object Recognition via Contour Points Reconstruction Using Hurwitz-Radon Matrices. In: Józefczyk, J. and Orski, D., eds., Knowledge-Based Intelligent System Advancements: Systemic and Cybernetic Approaches, IGI Global, Hershey, 87-107. [15] Jakóbczak, D. (2010) Application of Hurwitz-Radon Matrices in Shape Representation. In: Banaszak, Z. and swic, A. Eds., Applied Computer Science: Modelling of Production Processes, 1, Lublin University of Technology Press, Lublin, 63-74. [16] Jakóbczak, D. (2010) Implementation of Hurwitz-Radon Matrices in Shape Representation. In: Choras, R.S., Ed., Advances in Intelligent and Soft Computing 84, Image Processing and Communications: Challenges 2, Springer-Verlag, Berlin Heidelberg, 39-50. [17] Jakóbczak, D. (2011) Curve Parameterization and Curvature via Method of Hurwitz-Radon Matrices. Image Processing & Communications—An International Journal, 1-2, 49-56. [18] Jakóbczak, D. (2011) Data Extrapolation and Decision Making via Method of Hurwitz-Radon Matrices. Lecture Notes in Computer Science/LNAI, 6922, 173-182.