A note on fast computation of effective independence through QR downdating for sensor placement

A simple and fast computational algorithm of the Effective Independence for sensor placement is presented in the paper. It is based on the relationship between the Effective Independence and Modal Kinetic Energy method, and on a downdating algorithm of the QR decomposition for a reduced modal matrix. Traditional computations of the Effective Independence, which tries to render the target modes as linearly independent at possible and is significant and influential in the field of sensor placement, require the eigenvalue decomposition or inversion of the Fisher Information matrix, and both are costly. By virtue of the connection between the Modal Kinetic Energy and Effective Independence derived by the authors, the norms of the orthonormal rows after the QR decomposition of the modal matrix can be employed to compute the Effective Independence. Moreover, new downdating steps of the QR decomposition based on the combination of the modified Gram–Schmidt and Householder transformations are implemented, which further improved its efficiency by eliminating unnecessary expensive QR decompositions in the iterations. The computational burden for traditional methods and the proposed ones are investigated in terms of the floating point operations required. Finally, a numerical example shows that the new algorithms run much more quickly with fewer computation steps, has the advantages of simplicity and clarity in physical significance, and is stable.