zbMATH — the first resource for mathematics

A new recursive algorithm for inverting general \(k\)-tridiagonal matrices. (English) Zbl 1315.65027
Summary: In the present article we give a new breakdown-free recursive algorithm for inverting general \(k\)-tridiagonal matrices without imposing any simplifying assumptions. The implementation of the algorithm in Computer Algebra Systems such as Maple, Mathematica and Macsyma is straightforward. Two illustrative examples are given.

65F05 Direct numerical methods for linear systems and matrix inversion
65F50 Computational methods for sparse matrices
65Y10 Numerical algorithms for specific classes of architectures
Full Text: DOI
[1] Aiat Hadj, A. D.; Elouafi, M., A fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix, Appl. Math. Comput., 202, 441-445, (2008) · Zbl 1153.65030
[2] El-Mikkawy, M. E.A., On the inverse of a general tridiagonal matrix, Appl. Math. Comput., 150, 669-679, (2004) · Zbl 1039.65024
[3] El-Mikkawy, M. E.A., A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput. Appl. Math., 166, 581-584, (2004) · Zbl 1051.65062
[4] El-Mikkawy, M. E.A., A. karawia, inversion of general tridiagonal matrices, Appl. Math. Lett., 19, 712-720, (2006) · Zbl 1119.65022
[5] El-Mikkawy, M.; Rahmo, E., A new recursive algorithm for inverting tridiagonal and anti-tridiagonal matrices, Appl. Math. Comput., 204, 368-372, (2008) · Zbl 1157.65338
[6] El-Mikkawy, M.; Atlan, F., Algorithms for solving linear systems of equations of tridiagonal type via transformations, Appl. Math., 5, 413-422, (2014)
[7] Huang, Y.; McColl, W. F., Analytic inversion of general tridiagonal matrices, J. Phys. A, 30, 7919-7933, (1997) · Zbl 0927.15003
[8] Kilic, E., Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions, Appl. Math. Comput., 197, 345-357, (2008) · Zbl 1151.65021
[9] Lewis, J. W., Inversion of the tridiagonal matrices, Numer. Math., 38, 333-345, (1982) · Zbl 0457.15002
[10] Mallik, R. K., The inverse of a tridiagonal matrix, Linear Algebra Appl., 325, 109-139, (2001) · Zbl 0980.15004
[11] Ran, R.-S.; Huang, T.-Z.; Liu, X.-P.; Gu, T.-X., An inversion algorithm for general tridiagonal matrix, Appl. Math. Mech. (English Ed.), 30, 247-253, (2009) · Zbl 1165.65008
[12] Yamamoto, T., Inversion formulas for tridiagonal matrices with applications to boundary value problems, Numer. Funct. Anal. Optim., 22, 357-385, (2001) · Zbl 0996.15006
[13] El-Mikkawy, M. E.A.; Sogabe, T., A new family of k-Fibonacci numbers, Appl. Math. Comput., 215, 4456-4461, (2010) · Zbl 1193.11012
[14] Rimas, J., On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order II, Appl. Math. Comput., 172, 245-251, (2006) · Zbl 1137.65340
[15] Rimas, J., On computing of arbitrary positive integer powers for one type of symmetric pentadiagonal matrices of odd order, Appl. Math. Comput., 204, 120-129, (2008) · Zbl 1157.65360
[16] El-Mikkawy, M.; Atlan, F., A novel algorithm for inverting a general \(k\)-tridiagonal matrix, Appl. Math. Lett., 32, 41-47, (2014) · Zbl 1311.65029
[17] Jia, J.; Sogabe, T.; El-Mikkawy, M., Inversion of \(k\)-tridiagonal matrices with Toeplitz structure, Comput. Math. Appl., 65, 116-125, (2013) · Zbl 1268.15002
[18] Sogabe, T.; El-Mikkawy, M. E.A., Fast block diagonalization of \(k\)-tridiagonal matrices, Appl. Math. Comput., 218, 2740-2743, (2011) · Zbl 06043893
[19] Sogabe, T.; Yilmaz, F., A note on a fast breakdown-free algorithm for computing the determinants and the permanents of \(k\)-tridiagonal matrices, Appl. Math. Comput., 249, 98-102, (2014) · Zbl 1338.65119
[20] El-Mikkawy, M. E.A., A generalized symbolic Thomas algorithm, Appl. Math., 3, 4, 342-345, (2012)
[21] Burden, R. L.; Faires, J. D., Numerical analysis, (2001), Books & Cole Publishing Pacific Grove, CA
[22] Yalciner, A., The LU factorization and determinants of the \(k\)-tridiagonal matrices, Asian-Eur. J. Math., 4, 187-197, (2011) · Zbl 1234.15004
[23] Yilmaz, F.; Sogabe, T., A note on symmetric \(k\)-tridiagonal matrix family and the Fibonacci, Int. J. Pure Appl. Math., 96, 2, 289-298, (2014)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.