Abstract
This paper presents a new technique for the reliable computation of the σ-pseudospectrum defined by Λσ(A)={z∈C : σmin(A−zI)≤σ} where σmin is the smallest singular value. The proposed algorithm builds an orbit of adjacent equilateral triangles to capture the level curve ϒσ(A)={z∈C : σmin(A−zI)=σ} and uses a bisection procedure on specific triangle vertices to compute a numerical approximation to ϒσ. The method is guaranteed to terminate, even in the presence of round-off errors.
Similar content being viewed by others
References
E. Allgower and K. Georg, Continuation and path following, Acta Numerica (1993) 1-64.
C. Bekas and E. Gallopoulos, Cobra: Parallel path following for computing the matrix pseudospectrum, Parallel Computing (to appear).
R. Brent, Algorithms for Minimization without Derivatives(Prentice-Hall, 1973).
M. Brühl, A curve tracing algorithm for computing the pseudospectrum, BIT 36(3) (1996) 441–454.
G. Golub and C.V. Loan, Matrix Computations, 2nd edn. (Johns Hopkins Univ. Press, 1989).
J. Huilfieldt and A. Ruhe, A new algorithm for numerical path following applied to an example from hydrodynamical flow, SIAM J. Sci. Statist. Comput. 11 (1990) 1181–1192.
R. Mejia, Conkub: A conversational path-follower for systems of nonlinear equations, J. Comput. Phys. 63 (1986) 67–84.
D. Mezher and B. Philippe, Parallel computation of the pseudospectrum of large sparse matrices, Parallel Computing (to appear).
S. Lui, Computation of pseudospectra by continuation, SIAM Sci. Comput. 18(2) (1997) 565–573.
B. Philippe and M. Sadkane, Computation of the fundamental singular subspace of a large matrix, Linear Algebra Appl. 257 (1997) 77–104.
W. Rudin, Real and Complex Analysis, 3rd edn. (McGraw-Hill International Editions, 1986).
Y. Saad, Numerical Methods for Large Eigenvalue Problems, Series in Algorithms and Architectures for Advanced Scientific Computing (Manchester Univ. Press, 1992).
H. Schwetlick, G. Timmermann and R. Losche, Path following for large nonlinear equations by implicit block elimination based on recursive projections, Lectures in Appl. Math. 32 (1996) 715–732.
H. Schwetlick and U. Schnabel, Iterative computation of the smallest singular value and the corresponding singular vectors of a matrix, Preprint IOKOMO-06-97, Techn. Univ. Dresden (1997).
L. Trefethen Computation of pseudospectra, Acta Numerica (1999) 247-295, web.comlab.ox.ac.uk/oucl/work/nick.trefethen
L. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39(3) (1997) 383–406.
C. Ueberhuber, Numerical Computation 2, Methods, Software and Analysis(Springer, 1997).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mezher, D., Philippe, B. PAT – a Reliable Path-Following Algorithm. Numerical Algorithms 29, 131–152 (2002). https://doi.org/10.1023/A:1014824425949
Issue Date:
DOI: https://doi.org/10.1023/A:1014824425949