This article describes the subroutine library SLICOT that provides Fortran 77 implementations of numerical algorithms for computations in systems and control theory. Around a nucleus of basic numerical linear algebra subroutines, this library builds methods for the design and analysis of linear control systems. A brief history of the library is given together with a description of the current version of the library and the on-going activities to complete and improve the library in several aspects. 1 Introduction Systems and control theory are disciplines widely used to describe, control, and optimize industrial and economical processes. There is now a huge amount of theoretical results available which has lead to a variety of methods and algorithms used throughout industry and academia. Although based on theoretical results, these methods often fail when applied to real-life problems, which often tend to be ill-posed or of high dimensions. This failing is frequently due to the lack of...

A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order p ", where " is the machine precision, the new method computes the eigenvalues to full possible accuracy. Keywords. eigenvalue problem, Hamiltonian pencil (matrix), symplectic pencil (matrix), skew-Hamiltonian matrix AMS subject classification. 65F15 1 Introduction The eigenproblem for Hamiltonian and symplectic matrices has received a lot of attention in the last 25 years, since the landmark papers of Laub [13] and Paige/Van Loan [20]. The reason for this is the importance of this problem in many applications in c...

We investigate the numerical solution of continuous-time algebraic Riccati equations via Newton's method on serial and parallel computers with distributed memory. We apply and extend the available theory for Newton's method endowed with exact line search to accelerate convergence. We also discuss a new stopping criterion based on recent observations regarding condition and error estimates. In each iteration step of Newton's method a stable Lyapunov equation has too be solved. We propose to solve these Lyapunov equations using iterative schemes for computing the matrix sign function. This approach can be efficiently implemented on parallel computers using ScaLAPACK. Numerical experiments on an ibm sp2 multicomputer report the accuracy, scalability, and speed-up of the implemented algorithms.

A new backward stable, structure preserving method of complexity O(n 3 ) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuous-time algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix H and the extended matrix 0 H H 0 and makes use of the symplectic URV-like decomposition that was recently introduced by the authors. Keywords. Eigenvalue problem, Hamiltonian matrix, algebraic Riccati equation, sign function, invariant subspace. AMS subject classification. 65F15, 93B40, 93B36, 93C60. 1 Introduction It is a well accepted fact in numerical analysis that a numerical algorithm should reflect as many of the structural properties of the physical problem or the resulting mathematical model. For the solution of eigenvalue problems this means that use of the symmetry structures of the matrix or the spectrum is made. While for symme...

An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute eigenvalues, eigenvectors, and invariant subspaces of large and sparse Hamiltonian matrices and low-rank approximations to the solution of continuous-time algebraic Riccati equations with large and sparse coefficient matrices.

A collection of benchmark examples is presented for the numerical solution of continuous-time algebraic Riccati equations. This collection may serve for testing purposes in the construction of new numerical methods, but may also be used as a reference set for the comparison of methods. The collected examples focus mainly on applications in linear-quadratic optimal control theory. This version updates an earlier benchmark collection and includes one new example. 0 Introduction We present a collection of examples for continuous-time algebraic Riccati equations (CARE) of the form 0 = Q+A T X +XA \Gamma XGX (1) where A; G; Q; X 2 R n\Thetan . The matrices Q = Q T and G = G T may be given in factored form Q = C T ~ QC, G = BR \Gamma1 B T with C 2 R p\Thetan , B 2 R n\Thetam , ~ Q = ~ Q T 2 R p\Thetap , and R = R T 2 R m\Thetam . The corresponding Hamiltonian matrix is defined by H = A \GammaG \GammaQ \GammaA T = A \GammaBR \Gamma1 B T \GammaC T ~...

We present a Newton--like method for solving algebraic Riccati equations that uses exact line search to improve the sometimes erratic convergence behavior of Newton's method. It avoids the problem of a disastrously large first step and accelerates convergence when Newton steps are too small or too long. The additional work to perform the line search is small relative to the work needed to calculate the Newton step. 1 Introduction We study the generalized continuous--time algebraic Riccati equation (CARE) 0 = R(X) = C T QC +A T XE + E T XA \Gamma (B T XE + S T C) T R \Gamma1 (B T XE + S T C) (1) Here A; E; X 2 IR n\Thetan , B 2 IR n\Thetam , R = R T 2 IR m\Thetam , Q = Q T 2 IR p\Thetap , C 2 IR p\Thetan , and S 2 IR p\Thetam . We will assume that E is nonsingular, Q \Gamma SR \Gamma1 S T 0, and R ? 0 where M ? 0 (M 0) denotes positive (semi-) definite matrices M . In principle, by inverting E, (1) may be reduced to the case E = I . This is conve...

This paper describes LAPACK-based Fortran 77 subroutines for the reduction of a Hamiltonian matrix to square-reduced form and the approximation of all its eigenvalues using the implicit version of Van Loan's method. The transformation of the Hamiltonian matrix to a square-reduced Hamiltonian uses only orthogonal symplectic similarity transformations. The eigenvalues can then be determined by applying the Hessenberg QR iteration to a matrix of half the order of the Hamiltonian matrix and taking the square roots of the computed values. Using scaling strategies similar to those suggested for algebraic Riccati equations can in some cases improve the accuracy of the computed eigenvalues. We demonstrate the performance of the subroutines for several examples and show how they can be used to solve some control-theoretic problems.

This paper describes three numerical methods to collapse a formal product of p pairs of matrices P = Q p\Gamma1 k=0 E \Gamma1 k A k down to the product of a single pair E \Gamma1 A. In the setting of linear relations, the product formally extends to the case in which some of the E k 's are singular and it is impossible to explicitly form P as a single matrix. The methods differ in flop count, work space, and inherent parallelism. They have in common that they are immune to overflows and use no matrix inversions. A rounding error analysis shows that the special case of collapsing two pairs is numerically backward stable.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (log p)&sup2:. The paper consists...