## A numerical method for computing the principal square root of a matrix.(English)Zbl 1474.65124

Summary: It is shown how the mid-point iterative method with cubical rate of convergence can be applied for finding the principal matrix square root. Using an identity between matrix sign function and matrix square root, we construct a variant of mid-point method which is asymptotically stable in the neighborhood of the solution. Finally, application of the presented approach is illustrated in solving a matrix differential equation.

### MSC:

 65F60 Numerical computation of matrix exponential and similar matrix functions 15A16 Matrix exponential and similar functions of matrices

mftoolbox
Full Text:

### References:

  Higham, N. J., Functions of Matrices: Theory and Computation (2008), Philadelphia, Pa, USA: Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA · Zbl 1167.15001  Kanzow, C.; Nagel, C., Semidefinite programs: new search directions, smoothing-type methods, and numerical results, SIAM Journal on Optimization, 13, 1, 1-23 (2002) · Zbl 1029.90052  Cross, G. W.; Lancaster, P., Square roots of complex matrices, Linear and Multilinear Algebra, 1, 289-293 (19774) · Zbl 0283.15008  Higham, N. J., Newton’s method for the matrix square root, Mathematics of Computation, 46, 174, 537-549 (1986) · Zbl 0614.65045  Björck, Å.; Hammarling, S., A Schur method for the square root of a matrix, Linear Algebra and its Applications, 52/53, 127-140 (1983) · Zbl 0515.65037  Denman, E. D.; Beavers, J., The matrix sign function and computations in systems, Applied Mathematics and Computation, 2, 1, 63-94 (1976) · Zbl 0398.65023  Gutierrez, J. M.; Hernandez, M. A.; Salanova, M. A., Calculus of $$n$$ th roots and third order iterative methods, Nonlinear Analysis: Theory, Methods & Applications, 47, 4, 2875-2880 (2001) · Zbl 1042.65523  Iannazzo, B., A note on computing the matrix square root, Calcolo, 40, 4, 273-283 (2003) · Zbl 1117.65059  Frontini, M.; Sormani, E., Some variant of Newton’s method with third-order convergence, Applied Mathematics and Computation, 140, 2-3, 419-426 (2003) · Zbl 1037.65051  Lakić, S.; Petković, M. S., On the matrix square root, Zeitschrift für Angewandte Mathematik und Mechanik, 78, 3, 173-182 (1998) · Zbl 0906.65047  Potra, F. A., On $$Q$$-order and $$R$$-order of convergence, Journal of Optimization Theory and Applications, 63, 3, 415-431 (1989) · Zbl 0663.65049  Iannazzo, B., On the Newton method for the matrix $$p$$ th root, SIAM Journal on Matrix Analysis and Applications, 28, 2, 503-523 (2006) · Zbl 1113.65054  Gander, W., On Halley’s iteration method, The American Mathematical Monthly, 92, 2, 131-134 (1985) · Zbl 0574.65041  Guo, C. H., On Newton’s method and Halley’s method for the principal $$p$$ th root of a matrix, Linear Algebra and its Applications, 432, 8, 1905-1922 (2010) · Zbl 1190.65065  Kenney, C. S.; Laub, A. J., Rational iterative methods for the matrix sign function, SIAM Journal on Matrix Analysis and Applications, 12, 2, 273-291 (1991) · Zbl 0725.65048  Haghani, F. K.; Soleymani, F., On a fourth-order matrix method for computing polar decomposition, Computational and Applied Mathematics (2014) · Zbl 1317.65110  Soleymani, F.; Tohidi, E.; Shateyi, S.; Haghani, F. K., Some matrix iterations for computin g matrix sign function, Journal of Applied Mathematics, 2014 (2014) · Zbl 1442.65079  Guo, C. H.; Higham, N. J., A Schur-Newton method for the matrix $$p$$ th root and its inverse, SIAM Journal on Matrix Analysis and Applications, 28, 3, 788-804 (2006) · Zbl 1128.65030  Hoste, J., Mathematica Demystified (2009), New York, NY, USA: McGraw-Hill, New York, NY, USA  Meini, B., The matrix square root from a new functional perspective: theoretical results and computational issues, 1455 (2003), Pisa, Italy: Dipartimento di Matematica, Universit di Pisa, Pisa, Italy
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.