Gram-Schmidt Orthogonalization on Encrypted Vectors

2011 ◽  
pp. 93-103 ◽  
Author(s):  
Pierluigi Failla ◽  
Mauro Barni
Keyword(s):  
Schmidt Orthogonalization

scholarly journals An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed Al-Smadi ◽  
Nadir Djeddi ◽  
Shaher Momani ◽  
Shrideh Al-Omari ◽  
Serkan Araci
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Reproducing Kernel ◽  
Numerical Approach ◽  
Accurate Solution ◽  
Stability And Convergence ◽  
Type Model ◽  
Schmidt Orthogonalization ◽  
And Performance ◽  
Abel Differential Equation

AbstractOur aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

Gram-Schmidt Orthogonalization by Gauss Elimination

1991 ◽  
Vol 98 (6) ◽  
pp. 544-549 ◽  
Author(s):  
Lyle Pursell ◽  
S. Y. Trimble
Keyword(s):  
Gauss Elimination ◽  
Schmidt Orthogonalization
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