Obtaining $\lambda_{min}$ is often impossible and so this choice for $\mu$ is not applicable. The parameter $\mu$ depends on the precision floating point arithmetic of the machine.
%For example for big number of nodes and small shape parameter $\epsilon$
%is better to choose $\mu=\sqrt{\varepsilon},$ where $\varepsilon$
%is the smallest number of the machine.
Obviously, by choosing an appropriate value for $\mu$, we can construct matrix $\mathbf{P}$ well--conditioned but well--conditioning of $\mathbf{\Psi}$ depends on matrix $\mathbf{\Phi}.$\\
Accordingly the user can check the well--conditioning of matrix $\mathbf{P}$ but this choice for matrix $\mathbf{\Psi}$ does not exist. For large number of nodes and small shape parameter $\epsilon$ the matrix $\mathbf{\Psi}$ will be ill--conditioned, …show more content…
In this case the computational condition number of $\mathbf{\Phi}$ (not exact) comes close to the biggest number of the machine and then it is better to choose $\mu=\sqrt{\varepsilon},$ where $\varepsilon$ is the smallest number of the machine. Therefore the condition numbers of matrices $\mathbf{P}$ and $\mathbf{\Psi}$ are almost equal to the square root of the biggest number of the machine. Consequently decreasing the computational errors of solving the systems $\mathbf{P}\mathbf{a}=\mathbf{a_1}$ and $\mathbf{\Psi}\mathbf{a_1}=\mathbf{f}$ by applying the SPD system solvers is
%For example for big number of nodes and small shape parameter $\epsilon$
%is better to choose $\mu=\sqrt{\varepsilon},$ where $\varepsilon$
%is the smallest number of the machine.
Obviously, by choosing an appropriate value for $\mu$, we can construct matrix $\mathbf{P}$ well--conditioned but well--conditioning of $\mathbf{\Psi}$ depends on matrix $\mathbf{\Phi}.$\\
Accordingly the user can check the well--conditioning of matrix $\mathbf{P}$ but this choice for matrix $\mathbf{\Psi}$ does not exist. For large number of nodes and small shape parameter $\epsilon$ the matrix $\mathbf{\Psi}$ will be ill--conditioned, …show more content…
In this case the computational condition number of $\mathbf{\Phi}$ (not exact) comes close to the biggest number of the machine and then it is better to choose $\mu=\sqrt{\varepsilon},$ where $\varepsilon$ is the smallest number of the machine. Therefore the condition numbers of matrices $\mathbf{P}$ and $\mathbf{\Psi}$ are almost equal to the square root of the biggest number of the machine. Consequently decreasing the computational errors of solving the systems $\mathbf{P}\mathbf{a}=\mathbf{a_1}$ and $\mathbf{\Psi}\mathbf{a_1}=\mathbf{f}$ by applying the SPD system solvers is