A Seventh-Order Scheme for Computing the Generalized Drazin Inverse
Abstract
:1. Introduction
- ,
- ,
- .
2. Derivation of an Efficient Formulation
3. Seventh Rate of Convergence
- (i)
- if and only if ,
- (ii)
- if and only if .
4. Computational Tests
N = 5000; no = 25; ParallelTable[ A[j] = SparseArray[ {Band[{-100, 1100}] -> RandomReal[20], Band[{1, 1}] -> 2., Band[{1000, -50}, {N - 20, N - 25}] -> {2.8, RandomReal[] + I}, Band[{600, 150}, {N - 100, N - 400}] -> {-RandomReal[3], 3. + 3 I} }, {N, N}, 0.], {j, no} ];
For[j = 1, j <= number, j++, { X = A[j]/(Norm[A[j], "Frobenius"]^2); k = 1; X1 = 20 X; Time[j] = Part[ While[k <= 75 && N[Norm[X - X1, 1]] >= 10^(-6), X1 = SparseArray[X]; XX = Id - A[j].X1; X2 = XX.XX; X = Chop@ SparseArray[ X1.(Id + (XX + X2).(Id - XX + X2).(Id + XX + X2))]; k++]; // AbsoluteTiming, 1]; }];
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Ahmed, D.; Hama, M.; Jwamer, K.H.F.; Shateyi, S. A Seventh-Order Scheme for Computing the Generalized Drazin Inverse. Mathematics 2019, 7, 622. https://doi.org/10.3390/math7070622
Ahmed D, Hama M, Jwamer KHF, Shateyi S. A Seventh-Order Scheme for Computing the Generalized Drazin Inverse. Mathematics. 2019; 7(7):622. https://doi.org/10.3390/math7070622
Chicago/Turabian StyleAhmed, Dilan, Mudhafar Hama, Karwan Hama Faraj Jwamer, and Stanford Shateyi. 2019. "A Seventh-Order Scheme for Computing the Generalized Drazin Inverse" Mathematics 7, no. 7: 622. https://doi.org/10.3390/math7070622