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smt: A Matlab toolbox for structured matrices. (English) Zbl 1238.65036

Summary: The full exploitation of the structure of large scale algebraic problems is often crucial for their numerical solution. Matlab is a computational environment which supports sparse matrices, besides full ones, and allows one to add new types of variables (classes) and define the action of arithmetic operators and functions on them. The smt toolbox for Matlab introduces two new classes for circulant and Toeplitz matrices, and implements optimized storage and fast computational routines for them, transparently to the user. The toolbox, available in Netlib, is intended to be easily extensible, and provides a collection of test matrices and a function to compute three circulant preconditioners, to speed up iterative methods for linear systems. Moreover, it incorporates a simple device to add to the toolbox new routines for solving Toeplitz linear systems.

MSC:

65F50 Computational methods for sparse matrices
65F08 Preconditioners for iterative methods
65Y15 Packaged methods for numerical algorithms
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[1] Aricò, A., Rodriguez, G.: toms729gw: a Matlab (Fortran) MEX Gateway for TOMS Algorithm 729, by P. C Hansen and T. Chan. Available at: http://bugs.unica.it/\(\sim\)gppe/soft/ (2008)
[2] Aricò, A., Rodriguez, G.: A fast solver for linear systems with displacement structure. Numer. Algorithms 55(4), 529–556 (2010). doi: 10.1007/s11075-010-9421-x . · Zbl 1203.65062
[3] Arushanian, O.B., Samarin, M.K., Voevodin, V.V., Tyrtyshnikov, E., Garbow, B.S., Boyle, J.M., Cowell, W.R., K.Dritz, W.: The TOEPLITZ package users’ Guide. Technical Report ANL-83-16, Argonne National Laboratory (1983)
[4] Chan, R.H., Jin, X.-Q.: An Introduction to Iterative Toeplitz Solvers. SIAM, Philadelphia (2007) · Zbl 1146.65028
[5] Chan, R.H., Jin, X.-Q., Yeung, M.C.: The circulant operator in the Banach algebra of matrices. Linear Algebra Appl. 149, 41–53 (1991) · Zbl 0717.15017
[6] Chan, T.F.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Statist. Comput. 9(4), 766–771 (1988) · Zbl 0646.65042
[7] Davis, P.J.: Circulant Matrices. Wiley, New York (1979)
[8] Demmel, J., Dongarra, J.: ST-HEC: reliable and scalable software for linear algebra computations on high end computers. Available at: http://www.cs.berkeley.edu/\(\sim\)demmel/Sca-LAPACK-Proposal.pdf (2005)
[9] Eddins, S.: Multithreaded FFT functions in MATLAB R2009a. The MathWorks, Natick. Available at: http://blogs.mathworks.com/steve/2009/04/17/multithreaded-fft-functions-in-matlab-r2009a/ (2009)
[10] Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. IEEE Proc. 93(2), 216–231 (2005). Software available at: http://www.fftw.org/
[11] Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The John Hopkins University Press, Baltimore (1996) · Zbl 0865.65009
[12] Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM Monographs on Mathematical Modeling and Computation. SIAM, Philadelphia (1998)
[13] Hansen, P.C., Chan, T.F.: Fortran subroutines for general Toeplitz systems. ACM Trans. Math. Softw. 18(3), 256–273 (1992) · Zbl 0894.65010
[14] Higham, N.J.: The Test Matrix Toolbox for Matlab (version 3.0). Technical Report 276, Manchester Centre for Computational Mathematics (1995)
[15] Kailath, T., Sayed, A.H. (eds.): Fast Reliable Algorithms for Matrices with Structure. SIAM, Philadelphia (1999)
[16] The MathWorks, Natick: Matlab ver. 7.7 (2008)
[17] Nagy, J.: RestoreTools, an object oriented Matlab package for image restoration. Emory University, Department of Mathematics and Computer Science. Available at: http://www.mathcs.emory.edu/\(\sim\)nagy/RestoreTools/ (2007)
[18] Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math. 74, 171–176 (1986) · Zbl 0621.65025
[19] Tismenetsky, M.: A decomposition of Toeplitz matrices and optimal circulant preconditioning. Linear Algebra Appl. 154/156, 105–121 (1991) · Zbl 0734.65039
[20] Tyrtyshnikov, E.E.: Optimal and superoptimal circulant preconditioners. SIAM J. Matrix Anal. Appl. 13(2), 459–473 (1992) · Zbl 0774.65024
[21] Van Barel, M.: Software produced by members of the MaSe-Team (Matrices having Structure). Katholieke Universiteit Leuven, Department of Computer Science. Available at: http://www.cs.kuleuven.ac.be/\(\sim\)marc/software/ (2008)
[22] van der Mee, C.V.M., Rodriguez, G., Seatzu, S.: Fast computation of two-level circulant preconditioners. Numer. Algorithms 41(3), 275–295 (2006) · Zbl 1094.65046
[23] van der Mee, C.V.M., Seatzu, S.: A method for generating infinite positive self-adjoint test matrices and Riesz bases. SIAM J. Matrix Anal. Appl. 26(4), 1132–1149 (2005) · Zbl 1114.42014
[24] The Working Group on Software (WGS): The control and systems library SLICOT. Available at: http://www.slicot.org/ (1998)
[25] The Working Group on Software (WGS): Basic software tools for structured matrix decompositions and perturbations. Available at: http://www.icm.tu-bs.de/NICONET/NICtask1B.html (2001)
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