## na12

 swMATH ID: 11490 Software Authors: Bini, Dario Andrea; Meini, Beatrice Description: Improved cyclic reduction for solving queueing problems. The cyclic reduction technique (Buzbee et al., 1970), rephrased in functional form (Bini and Meini, 1996), provides a numerically stable, quadratically convergent method for solving the matrix equation X = ∑+ ∞ i=0 Xi Ai, where the Ai’s are nonnegative k × k matrices such that ∑+ ∞ i=0 Ai is column stochastic. In this paper we propose a further improvement of the above method, based on a point-wise evaluation/interpolation at a suitable set of Fourier points, of the functional relations defining each step of cyclic reduction (Bini and Meini,1996). This new technique allows us to devise an algorithm based on FFT having a lower computational cost and a higher numerical stability. Numerical results and comparisons are provided. Homepage: http://www.netlib.org/numeralgo/index.html Dependencies: Netlib Keywords: queueing problems; $$M/G/1$$ type matrices; Toeplitz matrices; numerical examples; cyclic reduction; matrix equation; numerical stability Related Software: SMCSolver; Matlab; cqt-toolbox; Algorithm 432; SQUINT; na10; na31; mftoolbox; OPQ; FISHPAK; mctoolbox; FFTW; SLICOT; TELPACK Cited in: 29 Publications

### Standard Articles

1 Publication describing the Software, including 1 Publication in zbMATH Year
Improved cyclic reduction for solving queueing problems. Zbl 0887.65144
Bini, Dario Andrea; Meini, Beatrice
1997
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### Cited by 31 Authors

 16 Meini, Beatrice 11 Bini, Dario Andrea 4 Latouche, Guy 2 Favati, Paola 2 Gemignani, Luca 2 Li, Rencang 2 Meng, Jie 1 Alfa, Attahiru Sule 1 Chen, Cairong 1 Chesnokov, Andrey A. 1 Gu, Guiding 1 Guo, Pei-Chang 1 Gupta, Umesh Chandra 1 Higham, Nicholas J. 1 Kailath, Thomas 1 Kim, Hyunmin 1 Li, Wang 1 Lucia, Marcello 1 Ma, Changfeng 1 Maggio, Fabio 1 Maity, Arunava 1 Myllykoski, Mirko 1 Ramaswami, Vaidyanathan 1 Rodriguez, Giuseppe 1 Rossi, Tuomo 1 Sayed, Ali H. 1 Sengupta, Bhaskar 1 Seo, Sang-Hyup 1 Takine, Tetsuya 1 Van Barel, Marc 1 Xue, Jungong
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### Cited in 17 Serials

 7 Linear Algebra and its Applications 3 Numerical Algorithms 2 SIAM Journal on Matrix Analysis and Applications 1 IMA Journal of Numerical Analysis 1 Mathematics of Computation 1 BIT 1 Calcolo 1 Journal of Computational and Applied Mathematics 1 Numerical Methods for Partial Differential Equations 1 Mathematical and Computer Modelling 1 Journal of Scientific Computing 1 Numerical Linear Algebra with Applications 1 Journal of Mathematical Study 1 ASA-SIAM Series on Statistics and Applied Probability 1 International Journal of Stochastic Analysis 1 East Asian Journal on Applied Mathematics 1 Numerical Mathematics and Scientific Computation
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### Cited in 8 Fields

 23 Numerical analysis (65-XX) 16 Linear and multilinear algebra; matrix theory (15-XX) 14 Probability theory and stochastic processes (60-XX) 3 Partial differential equations (35-XX) 2 Operations research, mathematical programming (90-XX) 1 Field theory and polynomials (12-XX) 1 Operator theory (47-XX) 1 Systems theory; control (93-XX)