The ubiquitous Kronecker product

Journal of Computational and Applied Mathematics

Volumn 123, Issue 1-2, 2000, Pages 85-100

  Loan, Charles F Van ^a  

^a Department of Computer Science and School of Operations Research and Information Engineering   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER PROGRAMMING; FOURIER TRANSFORMS; IMAGE PROCESSING; MATRIX ALGEBRA; SIGNAL PROCESSING; VECTORS;

KRONECKER PRODUCT; LINEAR TRANSFORMS; MATRIX FACTORIZATION; QUANTUM COMPUTING; SEMIDEFINITE PROGRAMMING;

ALGORITHMS;

EID: 0034320029     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00393-9     Document Type: Article

References (74)

1. Andre T.F., Nowak R.D., Van Veen B.D. Low rank estimation of higher order statistics. IEEE Trans. Signal Process. 45:1997;673-685.
2. Andrews H.C., Kane J. Kronecker matrices, computer implementation, and generalized spectra. J. Assoc. Comput. Mach. 17:1970;260-268.
3. Barham R.H., Drane W. An algorithm for least squares estimation of nonlinear parameters when some of the parameters are linear. Technometrics. 14:1972;757-766.
4. Barrlund A. Efficient solution of constrained least squares problems with Kronecker product structure. SIAM J. Matrix Anal. Appl. 19:1998;154-160.
5. Bartels R.H., Stewart G.W. Solution of the equation. AX +XB = C Comm. ACM. 15:1972;820-826.
6. Brewer J.W. Kronecker products and matrix calculus in system theory. IEEE Trans. Circuits Systems. 25:1978;772-781.
7. Calvetti D., Reichel L. Application of ADI iterative methods to the image restoration of noisy images. SIAM J. Matrix Anal. Appl. 17:1996;165-174.
8. Chan T.F. An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Statist. Comput. 9:1988;766-771.
9. Chan T.F., Olkin J.A. Preconditioners for Toeplitz-block matrices. Numer. Algorithms. 6:1993;89-101.
10. Chan R., Jin X.-Q. A family of block preconditioners for block systems. SIAM J. Sci. Statist. Comput. 13:1992;1218-1235.
11. T.F. Coleman, Y. Li, A. Mariano, Separation of pulmonary nodule images using total variation minimization, Technical Report, Cornell Theory Center, Ithaca, New York, 1998.
12. Cullum J., Willoughby R.A. Lanczos Algorithms for Large Sparse Symmetric Eigenvalue Computations, Vol. I (Theory), II (Programs). 1985;Birkhauser, Boston.
13. Datta K. The matrix equation. XA - BX = R and its applications Linear Algebra Appl. 109:1988;91-105.
14. Davio M. Kronecker products and shuffle algebra. IEEE Trans. Comput. c-30:1981;116-125.
15. Davis P., Rabinowitz P. Numerical Integration. 1967;Blaisdell, Waltham, MA.
16. de Boor C. A Practical Guide to Splines. 1978;Springer, New York.
17. de Boor C. Efficient computer manipulation of tensor products. ACM Trans. Math. Software. 5:1979;173-182.
18. Dorr F.W. The direct solution of the discrete poisson equation on a rectangle. SIAM Rev. 12:1970;248-263.
19. Dutt A., Rokhlin V. Fast fourier transforms for nonequispaced data. SIAM J. Sci. Comput. 14:1993;1368-1398.
20. L. Eldin, I. Skoglund, Algorithms for the regularization of Ill-conditioned least squares problems with tensor product structure and applications to space-variant image restoration, Technical Report LiTH-Mat-R-82-48, Department of Mathematics, Linkoping University, Sweden, 1982.
21. Fausett D.W., Fulton C.T. Large least squares problems involving Kronecker products. SIAM J. Matrix Anal. 15:1994;219-227.
22. Fausett D.W., Fulton C.T., Hashish H. Improved parallel QR method for large least squares problems involving Kronecker products. J. Comput. Appl. Math. 1997.
23. A. Fijany, C.P. Williams, Quantum wavelet transforms: fast algorithms and complete circuits, Technical Report 9809004, Los Alamos National Laboratory, Los Alamos, New Mexico, 1998.
24. T = E ACM Trans. Math. Software. 18:1992;232-238.
25. Golub G.H. Some modified eigenvalue problems. SIAM Rev. 15:1973;318-344.
26. Golub G.H., Luk F., Overton M. A block Lanzcos method for computing the singular values and corresponding singular vectors of a matrix. ACM Trans. Math. Software. 7:1981;149-169.
27. Golub G.H., Nash S., Van Loan C. A Hessenberg-Schur method for the matrix problem. AX + XB = C IEEE Trans. Automat. Control. AC-24:1979;909-913.
28. Golub G.H., Pereya V. The differentiation of pseudoinverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal. 10:1973;413-432.
29. Golub G.H., Van Loan C. Matrix Computations. 3rd Edition:1996;Johns Hopkins University Press, Baltimore, MD.
30. Graham A. Kronecker Products and Matrix Calculus with Applications. 1981;Ellis Horwood, Chichester, England.
31. Granata J., Conner M., Tolimieri R. Recursive fast algorithms and the role of the tensor product. IEEE Trans. Signal Process. 40:1992;2921-2930.
32. Granata J., Conner M., Tolimieri R. The tensor product: a mathematical programming language for FFTs and other fast DSP operations. IEEE SP Mag. January 1992;40-48.
33. Greengard L., Strain J. The fast Gauss transform. SIAM J. Sci. Statist. Comput. 12:1991;79-94.
34. S.R Heap, D.J. Lindler, Block iterative restoration of astronomical images with the massively parallel processor Proceedings of the First Aerospace Symposium on Massively Parallel Scientific Computation, 1986, pp. 99-109.
35. Henderson H.V., Pukelsheim F., Searle S.R. On the history of the Kronecker product. Linear Multilinear Algebra. 14:1983;113-120.
36. Henderson H.V., Searle S.R. The vec-permutation matrix, the vec operator and Kronecker products: a review. Linear Multilinear Algebra. 9:1981;271-288.
37. Horn R.A., Johnson C.A. Topics in Matrix Analysis. 1991;Cambridge University Press, New York.
38. R.M. Hristev, private communication, 1998.
39. Huang C-H., Johnson J.R., Johnson R.W. Multilinear algebra and parallel programming. J. Supercomput. 5:1991;189-217.
40. Jain A.K. Fundamentals of Digital Image Processing. 1989;Prentice-Hall, Englewood Cliffs, NJ.
41. Johnson J., Johnson R.W., Rodriguez D., Tolimieri R. A methodology for designing, modifying, and implementing fourier transform algorithms on various architectures. Circuits Systems Signal Process. 9:1990;449-500.
42. Kagstrom B. Perturbation analysis of the generalized Sylvester equation. (AR-LB,DR-LE) = (C,F) SIAM J. Matrix Anal. Appl. 15:1994;1045-1060.
43. J. Kamm, J.G. Nagy, Kronecker product and SVD approximations in image restoration, Linear Algebra Appli. (1998a), to appear.
44. J. Kamm, J.G. Nagy, Kronecker product and SVD approximations for separable spatially variant blurs, SMU Mathematics Technical Report 98-04, Dallas, TX, 1998b.
45. J. Kamm, J.G. Nagy, Optimal kronecker product approximation of block Toeplitz matrices, SMU Mathematics Technical Report 98-05, Dallas, TX, 1998c.
46. Kaufman L. A variable projection method for solving separable nonlinear least squares problems. BIT. 15:1975;49-57.
47. Klaus B., Horn P. Robot Vision. 1990;MIT Press, Cambridge, MA.
48. B. Kumar, C.H. Huang, J. Johnson, R.W. Johnson, P. Sadayappan, A tensor product formulation of Strassen's matrix multiplication algorithm with memory reduction, Seventh International Parallel Processing Symposium, 1993, pp. 582-588.
49. Lancaster P. Explicit solution of linear matrix equations. SIAM Rev. 12:1970;544-566.
50. Moler C.B., Stewart G.W. An algorithm for generalized matrix eigenvalue problems. SIAM J. Numer. Anal. 10:1973;241-256.
51. J.G. Nagy, Decomposition of block Toeplitz matrices into a sum of Kronecker products with applications in image processing, SMU Mathematics Technical Report 96-01, Dallas, TX, 1996.
52. Nagy J.G., O'Leary D.P. Restoring images degraded by spatially-variant blur. SIAM J. Sci. Comput. 19:1998;1063-1082.
53. Nowak R.D. Penalized least squares estimation of higher order statistics. IEEE Trans. Signal Process. 46:1998;419-428.
54. Nowak R.D., Van Veen B. Tensor product basis approximations for Volterra filters. IEEE Trans Signal Process. 44:1996;36-50.
55. Pereyra V., Scherer G. Efficient computer manipulation of tensor products with applications to multidimensional approximation. Math. Comp. 27:1973;595-604.
56. N.P. Pitsianis, The Kronecker product in approximation, fast transform generation, Ph.D. Thesis, Department of Computer Science, Cornell University, 1997.
57. Pollard J.H. On the use of the direct matrix product in analyzing certain stochastic population models. Biometrika. 53:1966;397-415.
58. Rauhala U.A. Introduction to array algebra. Photogrammetric Eng. Remote Sensing. 46(2):1980;177-182.
59. Rauhala U.A., Davis D., Baker K. Automated DTM validation and progressive sampling algorithm of finite element array relaxation. Photogrammetric Eng. Remote Sensing. 55:1989;449-465.
60. Regalia P.A., Mitra S. Kronecker products, unitary matrices, and signal processing applications. SIAM Rev. 31:1989;586-613.
61. Seberry J., Zhang X-M. Some orthogonal matrices constructed by strong Kronecker product multiplication. Austral. J. Combin. 7:1993;213-224.
62. Snay R.A. Applicability of array algebra. Rev. Geophys. Space Phys. 16:1978;459-464.
63. Steeb W-H. Matrix Calculus and Kronecker Product with Applications and C++ Programs. 1997;World Scientific Publishing, Singapore.
64. Stenger F. Kronecker product extensions of linear operators. SIAM J. Numer. Anal. 5:1968;422-435.
65. L. Stiller, Multilinear algebra and chess endgames, in: Computational Games, Vol. 29, MSRI Publications.
66. Strohmer T. Numerical algorithms for discrete gabor expansions. Feichtinger H.G., Strohmer T. Gabor Analysis and Algorithms. 1998;267-294 Birkhauser, Basel.
67. Swami A., Mendel J. Time and lag recursive computation of cumulants from a state-space model. IEEE Trans. Automat. Control. 35:1990;4-17.
68. S. Thirumalai, High performance algorithms to solve Toeplitz and block Toeplitz matrices, Ph.D. Thesis, University of Illinois, Urbana, IL, 1996.
69. M.J. Todd, On search directions in interior point methods for semidefinite programming, Optim. Methods Software, (1998) , to appear.
70. Tolimieri R., An M., Lu C. Algorithms for Discrete Fourier Transform and Convolution. 1989;Springer, New York.
71. Vandenberghe L., Boyd S. Semidefinite Programming. SIAM Rev. 38:1996;27-48.
72. Van Loan C. Computational Frameworks for the Fast Fourier Transform. 1992;SIAM Publications, Philadelphia, PA.
73. Van Loan C., Pitsianis N.P. Approximation with Kronecker products. Moonen M.S., Golub G.H. Linear Algebra for Large Scale and Real Time Applications. 1992;293-314 Kluwer Publications, Dordrecht.
74. Vetter W.J. Vector structures, solutions of linear matrix equations. Linear Algebra Appl. 10:1975;181-188.