# Performance Analyses of a RAIM Algorithm for Kalman Filter with GPS and NavIC Constellations
## 7 Conclusions
This is because the maximum eigenvalue of \(\mathbf{\
abla}_{\text{max}}\) is greater than that of \(\mathbf{\
abla}\) (shown in matlab_program_appendix_figure.m for any \(\mathbf{\
abla}_{\text{sub}}^{i}\)) and for any two symmetric matrices \(\mathbf{J}_{1}\) and \(\mathbf{J}_{2}\), \(\lambda_{\mathbf{J}_{1}}^{\text{min}}~{}+~{}\lambda_{\mathbf{J}_{2}}^{\text{ min}}~{}\leq~{}\lambda_{\mathbf{J}}^{\text{min}}\), where \(\mathbf{J}=\mathbf{J}_{1}~{}+~{}\mathbf{J}_{2}\)[44]. Setting \(\mathbf{J}_{1}\) to \(\mathbf{\
abla}^{-1/2}(c-c_{\text{min}})\mathbf{\
abla}^{-1/2}\) and \(\mathbf{J}_{2}\) to \(\mathbf{\
abla}^{-1/2}(c_{\text{min}}\mathbf{I}~{}+~{}\Delta\mathbf{\
abla})\mathbf{\
abla }^{-1/2}\), the inequality \(\lambda_{\mathbf{\
abla}\mathbf{\
abla}}^{\text{max}}<\lambda_{\mathbf{\
abla},\text{max} \mathbf{\
abla}}^{\text{max}}\) can be proved for any other \(c>c_{\text{min}}\). Finally, the fourth subplot of Figure 17 is generated with Algorithm 17, accounting for uncertainties in \(a_{\ell q}\) for fourteen GPS and seven NavIC satellites. \(\mathbf{D}\) is calculated with the pre-determined \(c_{\text{min}}\) for a given \(M\). The asterisk represents the minimum of all \(\lambda_{\mathbf{D}}^{\text{min}}\) over 50,000 simulations in a batch. The minimum difference between \(\lambda_{\mathbf{D}}^{\text{min}}\) and \(c_{\text{min}}/\lambda_{\mathbf{\
abla},\text{max}}^{\text{max}}\) is also shown for each batch. With the initial \(c\) set as 0.05 for \(M=10\) (corresponding \(c_{\text{min}}=0.03792\)), Figure 3 depicts that \(\lambda_{\mathbf{D}}^{\text{max}}<\lambda_{\mathbf{\Omega},1}^{\text{max}}\) and hence is less than one (see the first subplot of the figure). matlab_program_algorithm_algorithm_1_ts1_figure_3.m can also verify this for changing satellite and user speeds and user heights within limits. ## References

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