×

On inverses and eigenpairs of periodic tridiagonal Toeplitz matrices with perturbed corners. (English) Zbl 1467.15003

Consider an \(n\times n\) matrix \(A=(a_{ij})\) with main diagonal \((\alpha_1,-3\beta,\dots,-3\beta,\gamma_n)\), upper side diagonal \((2\beta,\dots,2\beta,0)\), lower side diagonal \((0,\beta,\dots,\beta)\), \(a_{1n}=\gamma_1\), \(a_{n1}=\alpha_n\), and all remaining entries zero. Here, \(\alpha_1,\alpha_n,\beta,\gamma_1,\gamma_n\) are complex numbers and \(\beta\ne 0\). The authors present formulas for \(\det{A}\), \(A^{-1}\), and the eigenpairs of \(A\), and give three algorithms for computing them. The transpose \(A^T\) can be studied similarly.

MSC:

15A15 Determinants, permanents, traces, other special matrix functions
15A09 Theory of matrix inversion and generalized inverses
15B05 Toeplitz, Cauchy, and related matrices
15A18 Eigenvalues, singular values, and eigenvectors
65F05 Direct numerical methods for linear systems and matrix inversion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Bender, M. M¨uller, M. Otaduy, T. Matthias and M. Miles,A survey on position-based simulation methods in computer graphics, Comput. Graph. Forum., 2014, 33, 228-251.
[2] R. H. Chan, X. Q. Jin,Circulant and skew-circulant preconditioners for skewHermitian type Toeplitz systems, BIT 31, 1991, 632-646. · Zbl 0743.65026
[3] S. S. Cheng,Partial difference equations, Taylor and Francis, London, 2003. · Zbl 1016.39001
[4] M. Dow,Explicit inverses of Toeplitz and associated matrices, ANZIAM J., 2008, 44, 185-215. · Zbl 1116.15300
[5] M. El-Mikkawy and F. Atlan,A novel algorithm for inverting a generalktridiagonal matrix, Appl. Math. Lett., 2014, 32, 41-47. · Zbl 1311.65029
[6] M. El-Mikkawy and F. Atlan,A new recursive algorithm for inverting general k-tridiagonal matrices, Appl. Math. Lett., 2015, 44, 34-39. · Zbl 1315.65027
[7] M. El-Mikkawy,A new computational algorithm for solving periodic tridiagonal linear systems, Appl. Math. Comput., 2005, 161, 691-696. · Zbl 1061.65024
[8] M. El-Shehawey, G. El-Shreef and A. ShAl-Henawy,Analytical inversion of general periodic tridiagonal matrices, J. Math. Anal. Appl., 2008, 345, 123- 134. · Zbl 1147.15002
[9] Q. H. Feng and F. W. Meng,Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method, Optik, 2016, 127, 7450- 7458.
[10] C. M. da Fonseca and J. Petronilho,Explicit inverse of a tridiagonalk-Toeplitz matrix, Numerische Mathematik, 2005, 100, 457-482. · Zbl 1076.15011
[11] C. M. da Fonseca,On the eigenvalues of some tridiagonal matrices, J. Comput. Appl. Math., 2007, 200, 283-286. · Zbl 1119.15012
[12] G. H. Golub and C. F. Van Loan,Matrix Computations third ed., The John Hopkins University Press, Baltimore, 1996. · Zbl 0865.65009
[13] S. Holmgren, K. Otto,Iterative solution methods and preconditioners for nonsymmetric non-diagonally dominant block-tridiagonaI systems of equations, Dept. of Computer Sci., Uppsala Univ., Sweden, 1989.
[14] R. W. Hockney and C.R. Jesshope,Parallel Computers, Adam Hilger, Bristol, 1981. · Zbl 0523.68004
[15] Y. Huang and W. F. McColl,Analytical inversion of general tridiagonal matrices, J. Phys. A: Math. Gen., 1997, 30, 7919. · Zbl 0927.15003
[16] X. Y. Jiang and K. Hong,Skew cyclic displacements and inversions of two innovative patterned matrices, Appl. Math. Comput., 2017, 308, 174-184. · Zbl 1411.15022
[17] X. Y. Jiang, K. Hong and Z. W. Fu,Skew cyclic displacements and decompositions of inverse matrix for an innovative structure matrix, J. Nonlinear Sci. Appl., 2017, 10, 4058-4070. · Zbl 1412.15006
[18] Z. L. Jiang and D. D. Wang,Explicit group inverse of an innovative patterned matrix, Appl. Math. Comput., 2016, 274, 220-228. · Zbl 1410.15008
[19] Z. L. Jiang, X. T. Chen and J. M. Wang,The explicit inverses of CUPL-Toeplitz and CUPL-Hankel matrices, E. Asian J. Appl. Math., 2017, 7, 38-54. · Zbl 1373.15010
[20] J. T. Jia and S. M. Li,Symbolic algorithms for the inverses of generalktridiagonal matrices, Comput. Math. Appl., 2015, 70, 3032-3042. · Zbl 1443.65034
[21] J. T. Jia and S. M. Li,On the inverse and determinant of general bordered tridiagonal matrices, Comput. Math. Appl., 2015, 69, 503-509. · Zbl 1443.15004
[22] J. T. Jia and Q. X. Kong,A symbolic algorithm for periodic tridiagonal systems of equations, J. Math. Chem., 2014, 52, 2222-2233. · Zbl 1300.65014
[23] J. T. Jia, Tomohiro Sogabe and Moawwad El-Mikkawy,Inversion ofktridiagonal matrices with Toeplitz structure, Comput. Math. Appl., 2013, 65, 116-125. · Zbl 1268.15002
[24] J. T. Jia,Tomohiro Sogabe and Moawwad El-Mikkawy, Inversion ofktridiagonal matrices with Toeplitz structure, Comput. Math. Appl., 2013, 65, 116-125. · Zbl 1268.15002
[25] M. Krizek, F. Luca and L. Somer,17 Lectures on Fermat Numbers: From Number Theory to Geometry, Springer Science & Business Media, 2013. · Zbl 1010.11002
[26] M. Myllykoski, R. Glowinski, T. K¨arkk¨ainen and T. Rossi,A GPU-accelerated augmented Lagrangian based L1-mean curvature image denoising algorithm implementation, WSCG 2015 Conference on Computer Graphics, Visualization and Computer Vision, Plzen, Czech Republic, Union Agency, 2015.
[27] M. Myllykoski, T. Rossi and J. Toivanen,On solving separable block tridiagonal linear systems using a GPU implementation of radix-4 PSCR method, J. Parallel Distrib. Comput., 2018, 115, 56-66.
[28] H. J. Nussbaumer,Fast Fourier Transform and convolution algorithms, Springer Science & Business Media, 1981. · Zbl 0476.65097
[29] H. J. Nussbaumer,Digital filtering using complex Mersenne transforms, IBM J. Res. Dev., 1976, 20, 498-504. · Zbl 0339.65068
[30] H. J. Nussbaumer,Digital filtering using pseudo Fermat number transform, IEEE Trans. Acoust. Speech, Signal Processing, 1977, 25, 79-83. · Zbl 0374.94003
[31] C. M. Rader,Discrete convolution via Mersenne transforms, IEEE Trans. Comput C., 1972, 21, 1269-1273. · Zbl 0251.65083
[32] R. M. Robinson,Mersenne and Fermat numbers, P. Am. Math. Soc., 1954, 5, 842-846. · Zbl 0058.27504
[33] K. H. Rosen,Discrete mathematics and its applications, McGraw-Hill, New York, 2011.
[34] J. Shao, Z. W. Zheng and F. W. Meng,Oscillation criteria for fractional differential equations with mixed nonlinearities, Adv. Differ. Equ-ny., 2013, 2013, 323. · Zbl 1391.34069
[35] Y. G. Sun and F. W. Meng,Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput., 2008, 198, 375-381. · Zbl 1141.34317
[36] W. C. Siu and A. G. Constantinides,Fast mersenne number transforms for the computation of discrete fourier transforms, Signal Processing, 1985, 9, 125-131.
[37] H. Tim and K. Emrah,An analytical approach: Explicit inverses of periodic tridiagonal matrices, J. Comput. Appl. Math., 2018, 335, 207-226. · Zbl 1441.15002
[38] R. A. Usmani,Inversion of a tridiagonal Jacobi matrix, Linear Algebra Appl., 1994, 212, 413-414. · Zbl 0813.15001
[39] H. H. Wang,A parallel method for tridiagonal equations, ACM Trans. Math. Softw., 1981, 7, 170-183. · Zbl 0473.65010
[40] R. Xu and F. W. Meng,Some new weakly singular integral inequalities and their applications to fractional differential equations, J. Inequal. Appl., 2016, 2016, 78. · Zbl 1337.26022
[41] W. D. Yang, K. L. Li and K. Q. Li,A parallel solving method for blocktridiagonal equations on CPU-GPU heterogeneous computing systems, J. Supercomput., 2017, 73, 1760-1781.
[42] W. C. Yueh and S. S. Cheng,Explicit eigenvalues and inverses of tridiagonal Toeplitz matrices with four perturbed corners, ANZIAM J., 2008, 49, 361-387. · Zbl 1149.15010
[43] Y. P. Zheng, S. Shon and J. Kim,Cyclic displacements and decompositions of inverse matrices for CUPL Toeplitz matrices, J. Math. Anal. Appl., 2017, 455, 727-741. · Zbl 1370.15030
[44] F. Z. Zhang,The Schur Complement and Its Applications, Springer Science & Business Media, New York, 2006.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.