New Approach for the Inversion of Structured Matrices via Newton’s Iteration

Affiliation(s)

Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA, USA.

Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA, USA.

ABSTRACT

Newton’s iteration is a fundamental tool for numerical solutions of systems of
equations. The well-known iteration rapidly refines a crude
initial approximation X_{0} to the inverse of a general
nonsingular matrix. In this paper, we will extend and apply this method to ** n**×

KEYWORDS

Newton Iteration, Structured Matrices, Superfast Algorithm, Displacement Operators, Matrix Inverse.

Newton Iteration, Structured Matrices, Superfast Algorithm, Displacement Operators, Matrix Inverse.

Cite this paper

Tabanjeh, M. (2015) New Approach for the Inversion of Structured Matrices via Newton’s Iteration.*Advances in Linear Algebra & Matrix Theory*, **5**, 1-15. doi: 10.4236/alamt.2015.51001.

Tabanjeh, M. (2015) New Approach for the Inversion of Structured Matrices via Newton’s Iteration.

References

[1] Kailath, T. and Sayed, A. (1999) Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics, Philadelphia. http://dx.doi.org/10.1137/1.9781611971354

[2] Pan, V.Y., Branham, S., Rosholt, R. and Zheng, A. (1999) Newton’s Iteration for Structured Matrices and Linear Systems of Equations, SIAM Volume on Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics, Philadelphia.

[3] Pan, V.Y., Zheng, A.L., Huang, X.H. and Dias, O. (1997) Newton’s Iteration for Inversion of Cauchy-Like and Other Structured Matrices. Journal of Complexity, 13, 108-124.

http://dx.doi.org/10.1006/jcom.1997.0431

[4] Bini, D. and Pan, V.Y. (1994) Polynomial and Matrix Computations, Vol. 1 Fundamental Algorithms. Birkhauser, Boston.

[5] Pan, V.Y. (2001) Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhauser, Boston.

[6] Kailath, T., Kung, S.-Y. and Morf, M. (1979) Displacement Ranks of Matrices and Linear Equations. Journal of Mathematical Analysis and Applications, 68, 395-407. http://dx.doi.org/10.1016/0022-247X(79)90124-0

[7] Kailath, T. and Sayed, A.H. (2002) Displacement Structure: Theory and Applications. SIAM Review, 37, 297-386.

http://dx.doi.org/10.1137/1037082

[8] Pan, V.Y. and Rami, Y. (2001) Newton’s Iteration for the Inversion of Structured Matrices. In: Bini, D., Tyrtyshnikov, E. and Yalamov, P., Eds., Structured Matrices: Recent Developments in Theory and Computation, Nova Science Publishers, New York, 79-90.

[9] Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th Edition, John Hopkins University Press, Baltimore.

[10] Heinig, G. (1995) Inversion of Generalized Cauchy Matrices and the Other Classes of Structured Matrices. The IMA Volume in Mathematics and Its Applications, 69, 63-81.

[11] Pan, V.Y. (1993) Decreasing the Displacement Rank of a Matrix. SIAM Journal on Matrix Analysis and Application, 14, 118-121. http://dx.doi.org/10.1137/0614010

[1] Kailath, T. and Sayed, A. (1999) Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics, Philadelphia. http://dx.doi.org/10.1137/1.9781611971354

[2] Pan, V.Y., Branham, S., Rosholt, R. and Zheng, A. (1999) Newton’s Iteration for Structured Matrices and Linear Systems of Equations, SIAM Volume on Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics, Philadelphia.

[3] Pan, V.Y., Zheng, A.L., Huang, X.H. and Dias, O. (1997) Newton’s Iteration for Inversion of Cauchy-Like and Other Structured Matrices. Journal of Complexity, 13, 108-124.

http://dx.doi.org/10.1006/jcom.1997.0431

[4] Bini, D. and Pan, V.Y. (1994) Polynomial and Matrix Computations, Vol. 1 Fundamental Algorithms. Birkhauser, Boston.

[5] Pan, V.Y. (2001) Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhauser, Boston.

[6] Kailath, T., Kung, S.-Y. and Morf, M. (1979) Displacement Ranks of Matrices and Linear Equations. Journal of Mathematical Analysis and Applications, 68, 395-407. http://dx.doi.org/10.1016/0022-247X(79)90124-0

[7] Kailath, T. and Sayed, A.H. (2002) Displacement Structure: Theory and Applications. SIAM Review, 37, 297-386.

http://dx.doi.org/10.1137/1037082

[8] Pan, V.Y. and Rami, Y. (2001) Newton’s Iteration for the Inversion of Structured Matrices. In: Bini, D., Tyrtyshnikov, E. and Yalamov, P., Eds., Structured Matrices: Recent Developments in Theory and Computation, Nova Science Publishers, New York, 79-90.

[9] Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th Edition, John Hopkins University Press, Baltimore.

[10] Heinig, G. (1995) Inversion of Generalized Cauchy Matrices and the Other Classes of Structured Matrices. The IMA Volume in Mathematics and Its Applications, 69, 63-81.

[11] Pan, V.Y. (1993) Decreasing the Displacement Rank of a Matrix. SIAM Journal on Matrix Analysis and Application, 14, 118-121. http://dx.doi.org/10.1137/0614010