Simultaneous Real Eigenvalue Embedding in Second Order Systems

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Given matrices

$$\Lambda_1 = \left[ \begin{array}{cccc} \lambda_1 & & & \\ & \lambda_2 & & \\ & & \dots & \\ & & & \lambda_r \end{array} \right] ,$$

$$\Sigma_1 = \left[ \begin{array}{cccc} \mu_1 & & & \\ & \mu_2 & & \\ & & \dots & \\ & & & \mu_r \end{array} \right] ,$$

$$X_1 = \left[ \begin{array}{cccc} x_1 & x_2 & \dots & x_r \end{array} \right]$$

where (compatibility equation)

$$M X_1 \Lambda_1^2 + D X_1 \Lambda + K X_1 = 0 ,$$

we calculate matrices $W$ and $Z$, together with diagonal matrices $E_m$, $E_d$, e $E_k$, such that the matrices of the symmetric model $(\tilde{M},\tilde{D},\tilde{K})$, given by

$$\begin{array}{l} \tilde{M} = M - W E_m W^T \\ \tilde{D} = D + Z E_d W^T + W E_d Z^T \\ \tilde{K} = K - Z E_k Z^T \end{array}$$

satisfy the updating equation

$$\tilde{M}X_1 \Sigma_1^2 + \tilde{D} X_1 \Sigma_1 + \tilde{K} X_1 = 0$$

and the no spill-over equation

$$\tilde{M}X_2 \Sigma_2^2 + \tilde{D} X_2 \Sigma_2 + \tilde{K} X_2 = 0$$

where $(\Sigma_2, X_2)$ stands for any set (generally unknown) of eigenvalues - eigenvectors, which are not included in the updating.

Here, we provide an computer program to compute the set of updating matrix parameters $E_m, E_d , E_k, W, Z$. Once given the input data corresponding to matrices $M, D, K$ , $\Lambda_1$ and $X_1$, the quantity

$$\Vert M X_1 \Lambda_1^2 + D X_1 \Lambda_1 + K X_1 \Vert _F$$

is computed, in order to validate the compatibility equation.

Here we provide a computer program to perform the simultaneous embedding.