Abstract
A method for solving least squares problems (A ⊗ Bi)x = b whose coefficient matrices have generalized Kronecker product structure is presented. It is based on the exploitation of the block structure of the Moore-Penrose inverse and the reflexive minimum norm g-inverse of the coefficient matrix, and on the QR method for solving least squares problems. Firstly, the general case where A is a rectangular matrix is considered, and then the special case where A is square is analyzed. This special case is applied to the problem of bivariate polynomial regression, in which the involved matrices are structured matrices (Vandermonde or Bernstein-Vandermonde matrices). In this context, the advantage of using the Bernstein basis instead of the monomial basis is shown. Numerical experiments illustrating the good behavior of the proposed algorithm are included.
Similar content being viewed by others
References
Bapat, R.B.: Linear Algebra and Linear Models, 3rd edn. Springer (2012)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses, Theory and Applications, 2nd ed. Springer (2003)
Björck, A.: Numerical Methods for Least Squares Problems, SIAM Philadelphia (1996)
Björck, A.: Numerical Methods in Matrix Computations, Texts in Applied Mathematics. Springer (2016)
Caliari, M., De Marchi, S., Vianello, M.: Algorithm 886: Padua2D-Lagrange interpolation at Padua points on bivariate domains. ACM Trans. Math. Softw. 35, Article 21 (2008)
Caliari, M., De Marchi, S., Sommariva, A., Vianello, M.: Padua2DM: Fast interpolation and cubature at the Padua points in Matlab/Octave. Numer. Algor. 56, 45–60 (2011)
Delgado, J., Peña, J. M.: Optimal conditioning of Bernstein collocation matrices. SIAM J. Matrix Anal. Appl. 31, 990–996 (2009)
Farouki, R.T., Rajan, V.T.: On the numerical condition of polynomials in Bernstein form. Comput. Aided Geom. Des. 4, 191–216 (1987)
Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Comput. Aided Geom. Des. 5, 1–26 (1988)
Fausett, D.W., Fulton, C.T.: Large least squares problems involving Kronecker products. SIAM J. Matrix Anal. Appl. 15, 219–227 (1994)
Fausett, D.W., Fulton, C.T., Hashish, H.: Improved parallel QR method for large least squares problems involving Kronecker products. J. Comput. Appl. Math. 78, 63–78 (1997)
Franke, R.: A critical comparison of some methods for interpolation of scattered data. Naval Postgraduate School Tech. Rep NPS-53-79-003 (1979)
Gasca, M., Martínez, J.-J.: On the solvability of bivariate Hermite-Birkhoff interpolation problems. J. Comput. Appl. Math. 32, 77–82 (1990)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. The Johns Hopkins University Press, Baltimore (2012)
Grosse, E.: Tensor spline approximations. Linear Algebra Appl. 34, 29–41 (1980)
Hansen, P.C.: Deconvolution and regularization with Toepliz matrices. Numer. Algor. 29, 323–378 (2002)
Koev, P.: http://www.math.sjsu.edu/~koev
Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007)
Marco, A., Martínez, J.-J.: A fast and accurate algorithm for solving Bernstein-Vandermonde linear systems. Linear Algebra Appl. 422, 616–628 (2007)
Marco, A., Martínez, J.-J.: Polynomial least squares fitting in the Bernstein basis. Linear Algebra Appl. 433, 1254–1264 (2010)
Marco, A., Martínez, J.-J.: Accurate computations with totally positive Bernstein-Vandermonde matrices. Electron. J. Linear Algebra 26, 357–380 (2013)
Marco, A., Martínez, J.-J.: Ajuste polinómico por mínimos cuadrados usando la base de Bernstein. La Gaceta de la Real Sociedad Matemática Española 18, 135–153 (2015)
Marco, A., Martínez, J.-J.: Bidiagonal decomposition of rectangular totally positive Said-Ball-Vandermonde matrices: Error analysis, perturbation theory and applications. Linear Algebra Appl. 495, 90–107 (2016)
Marco, A., Martínez, J.-J., Viaña, R.: Accurate polynomial interpolation by using the Bernstein basis. Numer. Algor. 75, 665–674 (2017)
Martínez, J.-J.: A generalized Kronecker product and linear systems. Int. J. Math. Educ. Sci. Technol. 30, 137–141 (1999)
Pisinger, G., Zimmermann, A.: Linear least squares problems with data over incomplete grids. BIT Numer. Math. 47, 809–824 (2007)
Regalia, P.A., Mitra, S.K.: Kronecker products, unitary matrices and signal processing applications. SIAM Rev. 31, 586–613 (1989)
Van Loan, C.F.: The ubiquitous Kronecker product. J. Comput. Appl. Math. 123, 85–100 (2000)
Acknowledgements
We are grateful to the anonymous reviewers for their suggestions, which have contributed to improve our paper.
Funding
This research has been partially supported by Spanish Research Grant MTM2015-65433-P (MINECO/FEDER) from the Spanish Ministerio de Economía y Competitividad. A. Marco, J. J. Martínez and R. Viaña are members of the Research Group asynacs (Ref. ccee2011/r34) of Universidad de Alcalá.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Marco, A., Martínez, JJ. & Viaña, R. Least squares problems involving generalized Kronecker products and application to bivariate polynomial regression. Numer Algor 82, 21–39 (2019). https://doi.org/10.1007/s11075-018-0592-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-018-0592-1
Keywords
- Least squares
- Generalized Kronecker product
- Generalized inverse
- Bernstein-Vandermonde matrices
- Total positivity