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doi: 10.3934/dcdss.2021026
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Extended Krylov subspace methods for solving Sylvester and Stein tensor equations

Abdeslem Hafid Bentbib 1, , Smahane El-Halouy 2,, and  El Mostafa Sadek 2,

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 Early access 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.

Keywords: Sylvester tensor equation, Stein tensor equation, Extended Krylov subspace, Arnoldi process, High dimensional PDEs.
Mathematics Subject Classification: 2020 MSC.
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

[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

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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

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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

[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

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[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 $
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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 $
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Figure 3.  Residual norm of the extended block and global Arnoldi algorithms for example $ 4 $ with $ n = 300 $
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Table201 
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} $
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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} $
Table202 
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) $
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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) $
Table203 
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} $.
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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} $.
Table204 
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} $.
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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 $
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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 $
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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 $
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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 $
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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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