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We describe an effective approach for computing the maximum amplification of the solution norm of linear differential-algebraic systems that arise, in particular, when approximating with respect to space variables the linearized viscous incompressible flow equations for disturbances of laminar flows. In this context the square of the maximum amplification is the largest amplification of the kinetic energy of the disturbances whose knowledge is important in stability investigations and laminar-turbulent transition analysis. First, we reduce such a differential-algebraic system to an ordinary differential one. Then, the maximum amplification is computed as the matrix exponential norm for which a special low-rank approximation is used. To obtain an additional decrease in the computational cost, we use two initial differential-algebraic systems corresponding to coarse and fine grid approximations. The first one is used to compute a rough value of the maximum amplification, and the second one is used to refine the computation. We illustrate the efficiency of this approach with two sample flows of grooved-channel and boundary-layer types. [ABSTRACT FROM AUTHOR]

This work is devoted to the numerical analysis of small flow disturbances, i.e. velocity and pressure deviations from the steady state, in ducts of constant cross sections. The main emphasis is put on the disturbances causing the most kinetic energy density growth, the so-called optimal disturbances, whose knowledge is important in laminar-turbulent transition and robust flow control investigations. Numerically, this amounts to computing the maximum amplification of the 2-norm of a matrix exponential exp{ tS} for a square matrix S at t ≥ 0. To speed up the computations, we propose a new algorithm based on low-rank approximations of the matrix exponential and prove that it computes the desired amplification with a given accuracy. We discuss its implementation and demonstrate its efficiency by means of numerical experiments with a duct of square cross section. [ABSTRACT FROM AUTHOR]

We propose a version of the Newton--Kantorovich method which, given a nondegenerate square $n\backslash times\; n$ matrix and a number

This paper introduces and analyzes the convergence properties of a method that computes an approximation to the invariant subspace associated with a group of eigenvalues of a large not necessarily diagonalizable matrix. The method belongs to the family of projection type methods. At each step, it refines the approximate invariant subspace using a linearized Riccati's equation which turns out to be the block analogue of the correction used in the Jacobi-Davidson method. The analysis conducted in this paper shows that the method converges at a rate quasi-quadratic provided that the approximate invariant subspace is close to the exact one. The implementation of the method based on multigrid techniques is also discussed and numerical experiments are reported. [ABSTRACT FROM AUTHOR]

In this note we propose an algorithm based on theLanczos bidiagonalization to approximate the backward perturbation bound forthe large sparse linear squares problem. The algorithm requires O((m+n)l)operations where m and n are the size of the matrix under consideration andl – min(m,n). The import of the proposed algorithm is illustrated bysome examples coming from the Harwell–Boeing collection of test matrices. [ABSTRACT FROM AUTHOR]

We propose an algorithm that transforms a real symplectic matrix with a stable structure to a block diagonal form composed of three main blocks. The two extreme blocks of the same size are associated respectively with the eigenvalues outside and inside the unit circle. Moreover, these eigenvalues are symmetric with respect to the unit circle. The central block is in turn composed of several diagonal blocks whose eigenvalues are on the unit circle and satisfy a modification of the Krein-Gelfand-Lidskii criterion. The proposed algorithm also gives a qualitative criterion for structural stability. [ABSTRACT FROM AUTHOR]

We propose an algorithm that block-diagonalizes regular matrix pencils using well conditioned transformations. This algorithm is used for approximating the pseudospectra of matrix pencils. Several numerical experiments illustrate the behavior of the proposed algorithm. [ABSTRACT FROM AUTHOR]

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