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doi: 10.3934/dcdss.2021026

Extended Krylov subspace methods for solving Sylvester and Stein tensor equations

1. 

Laboratory LAMAI, University Cadi Ayyad, Marrakesh, Morocco

2. 

Laboratory LabSIPE, ENSA d'El Jadida, University Chouaib Doukkali, El Jadida, Morocco

* Corresponding author: Smahane El-Halouy

Received  August 2020 Revised  January 2021 Published  March 2021

This paper deals with Sylvester and Stein tensor equations with low rank right hand sides. It proposes extended Krylov-like methods for solving Sylvester and Stein tensor equations. The expressions of residual norms are presented. To show the performance of the proposed approaches, some numerical experiments are given.

Citation: Abdeslem Hafid Bentbib, Smahane El-Halouy, El Mostafa Sadek. Extended Krylov subspace methods for solving Sylvester and Stein tensor equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021026
References:
[1]

J. Ballani and L. Grasedyck, A projection method to solve linear systems in tensor format, Numerical Linear Algebra with Applications, 20 (2013), 27–43. doi: 10.1002/nla.1818.  Google Scholar

[2]

F. P. A. Beik, F. S. Movahed and S. Ahmadi-Asl, On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations, Numerical Linear Algebra with Applications, 23 (2016), 444–466. doi: 10.1002/nla.2033.  Google Scholar

[3]

A. H. Bentbib, K. Jbilou and El M. Sadek, On some extended block Krylov based methods for large scale nonsymmetric Stein matrix equations, Mathematics, 5 (2017), 21. doi: 10.3390/math5020021.  Google Scholar

[4]

A. H. Bentbib, S. El-Halouy and El M. Sadek, Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side, Numerical Algorithms, (2020), 1–20. doi: 10.1007/s11075-020-00874-0.  Google Scholar

[5]

M. Chen and D. Kressner, Recursive blocked algorithms for linear systems with Kronecker product structure, Numerical Algorithms, (2019), 1–18. doi: 10.1007/s11075-019-00797-5.  Google Scholar

[6]

D. Calvetti and L. Reichel, Application of ADI iterative methods to the restoration of noisy images, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 165–186. doi: 10.1137/S0895479894273687.  Google Scholar

[7]

Z. Chen and L. Lu, A projection method and Kronecker product pre-conditioner for solving Sylvester tensor equations, Science China Mathematics, 55 (2012), 1281-1292.  doi: 10.1007/s11425-012-4363-5.  Google Scholar

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Z. Chen and L. Lu, A gradient based iterative solutions for Sylvester tensor equations, Mathematical Problems in Engineering, (2013). doi: 10.1155/2013/819479.  Google Scholar

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M. Heyouni and K. Jbilou, An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation, Electronic Transactions on Numerical Analysis, 33 (2009), 53–62.  Google Scholar

[11]

M. Heyouni, Extended Arnoldi methods for large low-rank Sylvester matrix equations, Applied Numerical Mathematics, 60 (2010), 1171–1182. doi: 10.1016/j.apnum.2010.07.005.  Google Scholar

[12]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455–500. doi: 10.1137/07070111X.  Google Scholar

[13]

D. Kressner and C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 1688–1714. doi: 10.1137/090756843.  Google Scholar

[14]

A. N. Langville and W. J. Stewart, A Kronecker product approximate preconditioner for SANs, Numerical Linear Algebra with Applications, 11 (2004), 723–752. doi: 10.1002/nla.344.  Google Scholar

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N. Lee and A. Cichocki, Fundamental tensor operations for large-scale data analysis using tensor network formats, Multidimensional Systems and Signal Processing, 29 (2018), 921–960. doi: 10.1007/s11045-017-0481-0.  Google Scholar

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T. Li, Q. W. Wang and X. F. Duan, Numerical algorithms for solving discrete Lyapunov tensor equation, Journal of Computational and Applied Mathematics, 370 (2020), 112676. doi: 10.1016/j.cam.2019.112676.  Google Scholar

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T. Penzl, et al., A Matlab toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear quadratic optimal control problems, https://www.tu-chemnitz.de/sfb393/lyapack/, (2000). Google Scholar

[18]

T. Stykel and V. Simoncini, Krylov subspace methods for projected Lyapunov equations, Applied Numerical Mathematics, 62 (2012), 35-50.  doi: 10.1016/j.apnum.2011.09.007.  Google Scholar

[19]

X. Xu and W. Q. Wang, Extending BiCG and BiCR methods to solve the Stein tensor equation, Computers & Mathematics with Applications, 77 (2019), 3117–3127. doi: 10.1016/j.camwa.2019.01.024.  Google Scholar

show all references

References:
[1]

J. Ballani and L. Grasedyck, A projection method to solve linear systems in tensor format, Numerical Linear Algebra with Applications, 20 (2013), 27–43. doi: 10.1002/nla.1818.  Google Scholar

[2]

F. P. A. Beik, F. S. Movahed and S. Ahmadi-Asl, On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations, Numerical Linear Algebra with Applications, 23 (2016), 444–466. doi: 10.1002/nla.2033.  Google Scholar

[3]

A. H. Bentbib, K. Jbilou and El M. Sadek, On some extended block Krylov based methods for large scale nonsymmetric Stein matrix equations, Mathematics, 5 (2017), 21. doi: 10.3390/math5020021.  Google Scholar

[4]

A. H. Bentbib, S. El-Halouy and El M. Sadek, Krylov subspace projection method for Sylvester tensor equation with low rank right-hand side, Numerical Algorithms, (2020), 1–20. doi: 10.1007/s11075-020-00874-0.  Google Scholar

[5]

M. Chen and D. Kressner, Recursive blocked algorithms for linear systems with Kronecker product structure, Numerical Algorithms, (2019), 1–18. doi: 10.1007/s11075-019-00797-5.  Google Scholar

[6]

D. Calvetti and L. Reichel, Application of ADI iterative methods to the restoration of noisy images, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 165–186. doi: 10.1137/S0895479894273687.  Google Scholar

[7]

Z. Chen and L. Lu, A projection method and Kronecker product pre-conditioner for solving Sylvester tensor equations, Science China Mathematics, 55 (2012), 1281-1292.  doi: 10.1007/s11425-012-4363-5.  Google Scholar

[8]

Z. Chen and L. Lu, A gradient based iterative solutions for Sylvester tensor equations, Mathematical Problems in Engineering, (2013). doi: 10.1155/2013/819479.  Google Scholar

[9]

F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50 (2005), 1216–1221. doi: 10.1109/TAC.2005.852558.  Google Scholar

[10]

M. Heyouni and K. Jbilou, An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation, Electronic Transactions on Numerical Analysis, 33 (2009), 53–62.  Google Scholar

[11]

M. Heyouni, Extended Arnoldi methods for large low-rank Sylvester matrix equations, Applied Numerical Mathematics, 60 (2010), 1171–1182. doi: 10.1016/j.apnum.2010.07.005.  Google Scholar

[12]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455–500. doi: 10.1137/07070111X.  Google Scholar

[13]

D. Kressner and C. Tobler, Krylov subspace methods for linear systems with tensor product structure, SIAM Journal on Matrix Analysis and Applications, 31 (2010), 1688–1714. doi: 10.1137/090756843.  Google Scholar

[14]

A. N. Langville and W. J. Stewart, A Kronecker product approximate preconditioner for SANs, Numerical Linear Algebra with Applications, 11 (2004), 723–752. doi: 10.1002/nla.344.  Google Scholar

[15]

N. Lee and A. Cichocki, Fundamental tensor operations for large-scale data analysis using tensor network formats, Multidimensional Systems and Signal Processing, 29 (2018), 921–960. doi: 10.1007/s11045-017-0481-0.  Google Scholar

[16]

T. Li, Q. W. Wang and X. F. Duan, Numerical algorithms for solving discrete Lyapunov tensor equation, Journal of Computational and Applied Mathematics, 370 (2020), 112676. doi: 10.1016/j.cam.2019.112676.  Google Scholar

[17]

T. Penzl, et al., A Matlab toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear quadratic optimal control problems, https://www.tu-chemnitz.de/sfb393/lyapack/, (2000). Google Scholar

[18]

T. Stykel and V. Simoncini, Krylov subspace methods for projected Lyapunov equations, Applied Numerical Mathematics, 62 (2012), 35-50.  doi: 10.1016/j.apnum.2011.09.007.  Google Scholar

[19]

X. Xu and W. Q. Wang, Extending BiCG and BiCR methods to solve the Stein tensor equation, Computers & Mathematics with Applications, 77 (2019), 3117–3127. doi: 10.1016/j.camwa.2019.01.024.  Google Scholar

Figure 1.  Residual norm of the global and the extended global Arnoldi algorithms for example $ 3 $ with the first set of matrices and $ n = 225 $
Figure 2.  Residual norm of the global and the extended global Arnoldi algorithms for example $ 3 $ with the second set of matrices and $ n = 225 $
Figure 3.  Residual norm of the extended block and global Arnoldi algorithms for example $ 4 $ with $ n = 300 $
Algorithm 1 Extended Block Arnoldi (EBA)
1: Input: $ n\times n $ matrix $ A $, $ n\times r $ matrix $ B $ and an integer $ m $.
2: Compute the QR decomposition of $ \left[ B,A^{-1}B\right]=U_1\Lambda, \mathbb{U}_0=\left[ \;\right] $ for $ j=1,\ldots ,m $
3: Set $ U_j^{(1)} $: first $ r $ columns of $ U_j $, $ U_j^{(2)} $: second $ r $ columns of $ U_j $
4: $ \mathbb{U}_j=\left[ \mathbb{U}_{j-1},U_j\right] $; $ W=\left[ AU_j^{(1)},A^{-1}U_j^{(2)}\right] $
5: Orthogonalize $ W $ w.r. to $ \mathbb{U}_j $ to get $ U_{j+1} $ i.e.,
6: for $ i=1,\ldots ,j $ $ H_{i,j}=U_i^TW $; $ W=W-U_iH_{i,j} $
7: Compute the QR decomposition of $ W=U_{j+1}H_{j+1,j} $
Algorithm 1 Extended Block Arnoldi (EBA)
1: Input: $ n\times n $ matrix $ A $, $ n\times r $ matrix $ B $ and an integer $ m $.
2: Compute the QR decomposition of $ \left[ B,A^{-1}B\right]=U_1\Lambda, \mathbb{U}_0=\left[ \;\right] $ for $ j=1,\ldots ,m $
3: Set $ U_j^{(1)} $: first $ r $ columns of $ U_j $, $ U_j^{(2)} $: second $ r $ columns of $ U_j $
4: $ \mathbb{U}_j=\left[ \mathbb{U}_{j-1},U_j\right] $; $ W=\left[ AU_j^{(1)},A^{-1}U_j^{(2)}\right] $
5: Orthogonalize $ W $ w.r. to $ \mathbb{U}_j $ to get $ U_{j+1} $ i.e.,
6: for $ i=1,\ldots ,j $ $ H_{i,j}=U_i^TW $; $ W=W-U_iH_{i,j} $
7: Compute the QR decomposition of $ W=U_{j+1}H_{j+1,j} $
Algorithm 2 Extended Global Arnoldi (EGA)
1: Input: $ n\times n $ matrix $ A $, $ n\times r $ matrix $ B $ and an integer $ m $.
2: Compute the global QR decomposition of $ \left[ B,A^{-1}B\right]=V_1(\Omega\otimes I_r) $; $ \mathbb{V}_0=\left[ \;\right] $ for $ j=1,\ldots ,m $
3: Set $ V_j^{(1)} $: first $ r $ columns of $ V_j $, $ V_j^{(2)} $: second $ r $ columns of $ V_j $
4: $ \mathbb{V}_j=\left[ \mathbb{V}_{j-1},V_j\right] $; $ U=\left[ AV_j^{(1)},A^{-1}V_j^{(2)}\right] $
5: Orthogonalize $ U $ w.r. to $ \mathbb{V}_j $ to get $ V_{j+1} $ i.e.,
6: for $ i=1,\ldots ,j $ $ H_{i,j}=V_i^T\lozenge U $; $ U=U-V_iH_{i,j} $
7:     Compute the QR decomposition of $ U=V_{j+1}(H_{j+1,j}\otimes I_r) $
Algorithm 2 Extended Global Arnoldi (EGA)
1: Input: $ n\times n $ matrix $ A $, $ n\times r $ matrix $ B $ and an integer $ m $.
2: Compute the global QR decomposition of $ \left[ B,A^{-1}B\right]=V_1(\Omega\otimes I_r) $; $ \mathbb{V}_0=\left[ \;\right] $ for $ j=1,\ldots ,m $
3: Set $ V_j^{(1)} $: first $ r $ columns of $ V_j $, $ V_j^{(2)} $: second $ r $ columns of $ V_j $
4: $ \mathbb{V}_j=\left[ \mathbb{V}_{j-1},V_j\right] $; $ U=\left[ AV_j^{(1)},A^{-1}V_j^{(2)}\right] $
5: Orthogonalize $ U $ w.r. to $ \mathbb{V}_j $ to get $ V_{j+1} $ i.e.,
6: for $ i=1,\ldots ,j $ $ H_{i,j}=V_i^T\lozenge U $; $ U=U-V_iH_{i,j} $
7:     Compute the QR decomposition of $ U=V_{j+1}(H_{j+1,j}\otimes I_r) $
Algorithm 3 Sylvester Stein EBA Process
1: Coefficient matrices $ A^{(i)} $, $ i=1,\ldots ,N $, and the right hand side in low rank representation $ \mathcal{B}=\left[B^{(1)},\ldots ,B^{(N)} \right] $.
2: Output: Approximate solutions, $ \mathcal{X}_m $, to the equations (1) / (2).
3: Choose a tolerance $ \epsilon>0 $, a maximum number of iterations $ itermax_i $, a step-size parameter $ k $, set for $ i=1,\ldots ,N $, $ m_i=k $.
4: For $ i=1,\ldots ,N $ do
  For $ m_i=k $, construct the orthonormal basis $ \mathbb{V}_{m_i} $ and the restriction matrices $ \mathbb{T}_{m_i} $ by EBA algorithm (1).
5: Compute $ \mathcal{Y}_m $ / $ \mathcal{Z}_m $ the solution of the low dimensional equation (12) / (13).
6: Compute the residual norms $ r_m=\|\mathcal{R}_m\| $ as in theorem (4.1)/ $ q_m=\|\mathcal{Q}_m\| $ as in theorem (4.2).
7: If $ r_m\geq\epsilon $ / $ q_m\geq\epsilon $, set for $ i=1,\ldots ,N $, $ m_i=mi+k $ and go to step 2.
8: The approximate solutions are given by $ \mathcal{X}_m=\mathcal{Y}_m\times_1\mathbb{V}_{m_1}\ldots\times_N\mathbb{V}_{m_N} $ / $ \mathcal{X}_m=\mathcal{Z}_m\times_1\mathbb{V}_{m_1}\ldots\times_N\mathbb{V}_{m_N} $.
Algorithm 3 Sylvester Stein EBA Process
1: Coefficient matrices $ A^{(i)} $, $ i=1,\ldots ,N $, and the right hand side in low rank representation $ \mathcal{B}=\left[B^{(1)},\ldots ,B^{(N)} \right] $.
2: Output: Approximate solutions, $ \mathcal{X}_m $, to the equations (1) / (2).
3: Choose a tolerance $ \epsilon>0 $, a maximum number of iterations $ itermax_i $, a step-size parameter $ k $, set for $ i=1,\ldots ,N $, $ m_i=k $.
4: For $ i=1,\ldots ,N $ do
  For $ m_i=k $, construct the orthonormal basis $ \mathbb{V}_{m_i} $ and the restriction matrices $ \mathbb{T}_{m_i} $ by EBA algorithm (1).
5: Compute $ \mathcal{Y}_m $ / $ \mathcal{Z}_m $ the solution of the low dimensional equation (12) / (13).
6: Compute the residual norms $ r_m=\|\mathcal{R}_m\| $ as in theorem (4.1)/ $ q_m=\|\mathcal{Q}_m\| $ as in theorem (4.2).
7: If $ r_m\geq\epsilon $ / $ q_m\geq\epsilon $, set for $ i=1,\ldots ,N $, $ m_i=mi+k $ and go to step 2.
8: The approximate solutions are given by $ \mathcal{X}_m=\mathcal{Y}_m\times_1\mathbb{V}_{m_1}\ldots\times_N\mathbb{V}_{m_N} $ / $ \mathcal{X}_m=\mathcal{Z}_m\times_1\mathbb{V}_{m_1}\ldots\times_N\mathbb{V}_{m_N} $.
Algorithm 4 Sylvester / Stein EGA Process
1: Input: Coefficient matrices $ A^{(i)} $, $ i=1,\ldots ,N $, and the right hand side in low rank representation $ \mathcal{B}=\left[B^{(1)},\ldots ,B^{(N)} \right] $.
2: Output: Approximate solutions, $ \mathcal{X}_m $, to the equations (1) / (2).
3: Choose a tolerance $ \epsilon>0 $, a maximum number of iterations $ itermax_i $, a step-size parameter $ k $, set for $ i=1,\ldots ,N $, $ m_i=k $.
4: For $ i=1,\ldots ,N $ do
  For $ m_i=k $, construct the orthonormal basis $ \mathbb{W}_{m_i} $ and the restriction matrices $ T_{m_i} $ by EGA algorithm (2).
5: Compute $ \mathcal{Y}_m $ / $ \mathcal{Z}_m $ the solutions of the low dimensional equations (16) / (19).
6: Compute the upper bound for the residual norms $ r_m=\left( {\sum_{i=1}^N\|\mathcal{Y}_m\times_iT_{m_i+1,m_i}E_{m_i}^T\|^2}\right)^{1/2} $ as in theorem (4.4)/ $ q_m=\left( \beta_1^2+\sum_{i=1}^N{\beta_2^{(i)}}^2+\sum_{i=1}^N{\beta_3^{(i)}}^2\right)^{1/2} $ as in theorem (4.5).
7: If $ r_m\geq\epsilon $ / $ q_m\geq\epsilon $, set for $ i=1,\ldots ,N $, $ m_i=mi+k $ and go to step 2.
8: The approximate solutions are given by $ \mathcal{X}_m=(\mathcal{Y}_m\otimes\mathcal{I}_R)\times_1\mathbb{W}_{m_1}\ldots\times_N\mathbb{W}_{m_N} $ / $ \mathcal{X}_m=(\mathcal{Z}_m\otimes\mathcal{I}_R)\times_1\mathbb{W}_{m_1}\ldots\times_N\mathbb{W}_{m_N} $.
Algorithm 4 Sylvester / Stein EGA Process
1: Input: Coefficient matrices $ A^{(i)} $, $ i=1,\ldots ,N $, and the right hand side in low rank representation $ \mathcal{B}=\left[B^{(1)},\ldots ,B^{(N)} \right] $.
2: Output: Approximate solutions, $ \mathcal{X}_m $, to the equations (1) / (2).
3: Choose a tolerance $ \epsilon>0 $, a maximum number of iterations $ itermax_i $, a step-size parameter $ k $, set for $ i=1,\ldots ,N $, $ m_i=k $.
4: For $ i=1,\ldots ,N $ do
  For $ m_i=k $, construct the orthonormal basis $ \mathbb{W}_{m_i} $ and the restriction matrices $ T_{m_i} $ by EGA algorithm (2).
5: Compute $ \mathcal{Y}_m $ / $ \mathcal{Z}_m $ the solutions of the low dimensional equations (16) / (19).
6: Compute the upper bound for the residual norms $ r_m=\left( {\sum_{i=1}^N\|\mathcal{Y}_m\times_iT_{m_i+1,m_i}E_{m_i}^T\|^2}\right)^{1/2} $ as in theorem (4.4)/ $ q_m=\left( \beta_1^2+\sum_{i=1}^N{\beta_2^{(i)}}^2+\sum_{i=1}^N{\beta_3^{(i)}}^2\right)^{1/2} $ as in theorem (4.5).
7: If $ r_m\geq\epsilon $ / $ q_m\geq\epsilon $, set for $ i=1,\ldots ,N $, $ m_i=mi+k $ and go to step 2.
8: The approximate solutions are given by $ \mathcal{X}_m=(\mathcal{Y}_m\otimes\mathcal{I}_R)\times_1\mathbb{W}_{m_1}\ldots\times_N\mathbb{W}_{m_N} $ / $ \mathcal{X}_m=(\mathcal{Z}_m\otimes\mathcal{I}_R)\times_1\mathbb{W}_{m_1}\ldots\times_N\mathbb{W}_{m_N} $.
Table 1.  Example $ 1 $
Algorithm $ R $ $ n $ Cycles $ \mathcal{R}_m $ CPU Times (Second)
Algorithm 3 $ 3 $ $ 500 $ $ 5 $ $ 9.34\cdot 10^{-7} $ $ 16.1 $
$ 5 $ $ 300 $ $ 4 $ $ 4.03\cdot 10^{-7} $ $ 31.38 $
Algorithm 4 $ 3 $ $ 500 $ $ 9 $ $ 6.69\cdot 10^{-7} $ $ 4.57 $
$ 5 $ $ 300 $ $ 8 $ $ 4.34\cdot 10^{-7} $ $ 3.76 $
Algorithm $ R $ $ n $ Cycles $ \mathcal{R}_m $ CPU Times (Second)
Algorithm 3 $ 3 $ $ 500 $ $ 5 $ $ 9.34\cdot 10^{-7} $ $ 16.1 $
$ 5 $ $ 300 $ $ 4 $ $ 4.03\cdot 10^{-7} $ $ 31.38 $
Algorithm 4 $ 3 $ $ 500 $ $ 9 $ $ 6.69\cdot 10^{-7} $ $ 4.57 $
$ 5 $ $ 300 $ $ 8 $ $ 4.34\cdot 10^{-7} $ $ 3.76 $
Table 2.  Example 2
Algorithm $ R $ $ n $ Cycles $ \|\mathcal{R}_m\| $ $ \|\mathcal{X}^{\ast}-\mathcal{X}_m\| $ CPU Times (Second)
Algorithm 3 (case $ 1 $) $ 3 $ $ 300 $ $ 2 $ $ 4.08\cdot 10^{-9} $ $ 3.85\cdot 10^{-6} $ $ 1.58 $
(case $ 2 $) $ 3 $ $ 300 $ $ 2 $ $ 1.26\cdot 10^{-8} $ $ 2.82\cdot 10^{-7} $ $ 1.53 $
Algorithm 4 (case $ 1 $) $ 3 $ $ 300 $ $ 6 $ $ 5.82\cdot 10^{-8} $ $ 4.21\cdot 10^{-6} $ $ 1.46 $
(case $ 2 $) $ 3 $ $ 300 $ $ 7 $ $ 5.39\cdot 10^{-8} $ $ 4.17\cdot 10^{-8} $ $ 2.22 $
Algorithm 4 $ 5 $ $ 500 $ $ 10 $ $ 4.54\cdot 10^{-7} $ $ - $ $ 26.01 $
Algorithm $ R $ $ n $ Cycles $ \|\mathcal{R}_m\| $ $ \|\mathcal{X}^{\ast}-\mathcal{X}_m\| $ CPU Times (Second)
Algorithm 3 (case $ 1 $) $ 3 $ $ 300 $ $ 2 $ $ 4.08\cdot 10^{-9} $ $ 3.85\cdot 10^{-6} $ $ 1.58 $
(case $ 2 $) $ 3 $ $ 300 $ $ 2 $ $ 1.26\cdot 10^{-8} $ $ 2.82\cdot 10^{-7} $ $ 1.53 $
Algorithm 4 (case $ 1 $) $ 3 $ $ 300 $ $ 6 $ $ 5.82\cdot 10^{-8} $ $ 4.21\cdot 10^{-6} $ $ 1.46 $
(case $ 2 $) $ 3 $ $ 300 $ $ 7 $ $ 5.39\cdot 10^{-8} $ $ 4.17\cdot 10^{-8} $ $ 2.22 $
Algorithm 4 $ 5 $ $ 500 $ $ 10 $ $ 4.54\cdot 10^{-7} $ $ - $ $ 26.01 $
Table 3.  Example 3, $ N = 3 $ and $ R = 5 $
Matrices Algorithm $ \|\mathcal{R}_m\| $ CPU Times (Second)
fdm Algorithm $ 2 $ [4] $ 1.45\cdot 10^{-6} $ $ 56.0 $
Algorithm 4 $ 3.6\cdot 10^{-9} $ $ 0.74 $
poisson Algorithm $ 2 $ [4] $ 2.82\cdot 10^{-6} $ $ 14.5 $
Algorithm 4 $ 2.24\cdot 10^{-9} $ $ 0.67 $
Matrices Algorithm $ \|\mathcal{R}_m\| $ CPU Times (Second)
fdm Algorithm $ 2 $ [4] $ 1.45\cdot 10^{-6} $ $ 56.0 $
Algorithm 4 $ 3.6\cdot 10^{-9} $ $ 0.74 $
poisson Algorithm $ 2 $ [4] $ 2.82\cdot 10^{-6} $ $ 14.5 $
Algorithm 4 $ 2.24\cdot 10^{-9} $ $ 0.67 $
Table 4.  Example $ 4 $
Algorithm $ R $ $ n $ Cycles $ \mathcal{R}_m $ CPU Times (Second)
Algorithm 3 $ 3 $ $ 300 $ $ 6 $ $ 1.58\cdot 10^{-6} $ $ 27.49 $
Algorithm 4 $ 3 $ $ 300 $ $ 4 $ $ 1.56\cdot 10^{-7} $ $ 0.83 $
$ 5 $ $ 300 $ $ 4 $ $ 8.77\cdot 10^{-7} $ $ 0.85 $
Algorithm $ R $ $ n $ Cycles $ \mathcal{R}_m $ CPU Times (Second)
Algorithm 3 $ 3 $ $ 300 $ $ 6 $ $ 1.58\cdot 10^{-6} $ $ 27.49 $
Algorithm 4 $ 3 $ $ 300 $ $ 4 $ $ 1.56\cdot 10^{-7} $ $ 0.83 $
$ 5 $ $ 300 $ $ 4 $ $ 8.77\cdot 10^{-7} $ $ 0.85 $
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