Applied numerical linear algebra
Applied numerical linear algebra
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
The Quadratic Eigenvalue Problem
SIAM Review
Efficient svm training using low-rank kernel representations
The Journal of Machine Learning Research
Optimization of the solution of the parameter-dependent Sylvester equation and applications
Journal of Computational and Applied Mathematics
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We consider a second-order damped-vibration equation Mx¨+D(ε)x˙+Kx=0, where M, D(ε), K are real, symmetric matrices of order n. The damping matrix D(ε) is defined by D(ε)=Cu+C(ε), where Cu presents internal damping and rank(C(ε)) = r, where ε is dampers' viscosity. We present an algorithm which derives a formula for the trace of the solution X of the Lyapunov equation ATX + XA = -B, as a function ε → Tr(ZX(ε)), where A = A(ε) is a 2n × 2n matrix (obtained from M, D(ε),K) such that the eigenvalue problem Ay = λy is equivalent with the quadratic eigenvalue problem (λ2M+λD(ε)+K)x=0 (B and Z are suitably chosen positive-semidefinite matrices). Moreover, our algorithm provides the first and the second derivative of the function ε → Tr(ZX(ε)) almost for free. The optimal dampers' viscosity is derived as εopt = argmin Tr(ZX(ε)). If r is small, our algorithm allows a sensibly more efficient optimization, than standard methods based on the Bartels-Stewart's Lyapunov solver.