Algorithm 674: FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimations. FORTRAN 77 codes SONEST and CONEST are presented for estimating the 1-norm ( or the infinity-norm) of a real or complex matrix, respectively. The codes are of wide applicability in condition estimation since explicit access to the matrix, A, is not required; instead, matrix-vector products Ax and ATx are computed by the calling program via a reverse communication interface. The algorithms are based on a convex optimization method for estimating the 1-norm of a real matrix devised by Hager. We derive new results concerning the behavior of Hager’s method, extend it to complex matrices, and make several algorithmic modifications in order to improve the reliability and efficiency.

References in zbMATH (referenced in 34 articles )

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  1. Petkov, Petko H.: Componentwise perturbation analysis of the Schur decomposition of a matrix (2021)
  2. Sastre, J.; Ibáñez, J.; Defez, E.: Boosting the computation of the matrix exponential (2019)
  3. Sastre, Jorge; Ibáñez, Javier; Alonso-Jordá, Pedro; Peinado, Jesús; Defez, Emilio: Fast Taylor polynomial evaluation for the computation of the matrix cosine (2019)
  4. Diao, Huai-An: Condition numbers for a linear function of the solution of the linear least squares problem with equality constraints (2018)
  5. Alonso, Pedro; Ibáñez, Javier; Sastre, Jorge; Peinado, Jesús; Defez, Emilio: Efficient and accurate algorithms for computing matrix trigonometric functions (2017)
  6. Casacio, Luciana; Lyra, Christiano; Oliveira, Aurelio Ribeiro Leite; Castro, Cecilia Orellana: Improving the preconditioning of linear systems from interior point methods (2017)
  7. Diao, Huai-An; Wei, Yimin; Qiao, Sanzheng: Structured condition numbers of structured Tikhonov regularization problem and their estimations (2016)
  8. Higham, Nicholas J.; Relton, Samuel D.: Estimating the largest elements of a matrix (2016)
  9. Weng, Peter Chang-Yi; Phoa, Frederick Kin Hing: Small-sample statistical condition estimation of large-scale generalized eigenvalue problems (2016)
  10. Diao, Huaian; Wang, Weiguo; Wei, Yimin; Qiao, Sanzheng: On condition numbers for Moore-Penrose inverse and linear least squares problem involving Kronecker products (2013)
  11. Brás, Carmo P.; Hager, William W.; Júdice, Joaquim J.: An investigation of feasible descent algorithms for estimating the condition number of a matrix (2012)
  12. Diao, Huaian; Xiang, Hua; Wei, Yimin: Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations. (2012)
  13. Li, Ren-Cang; Kahan, William: A family of anadromic numerical methods for matrix Riccati differential equations (2012)
  14. Gould, Nicholas I. M.; Robinson, Daniel P.; Thorne, H. Sue: On solving trust-region and other regularised subproblems in optimization (2010)
  15. Petkov, P. Hr.; Konstantinov, M. M.; Christov, N. D.: LAPACK-based condition estimates for the discrete-time LQG design (2009)
  16. Rüberg, Thomas; Schanz, Martin: An alternative collocation boundary element method for static and dynamic problems (2009)
  17. Laub, A. J.; Xia, J.: Applications of statistical condition estimation to the solution of linear systems (2008)
  18. Granat, Robert; Kågström, Bo; Kressner, Daniel: Computing periodic deflating subspaces associated with a specified set of eigenvalues (2007)
  19. Gemignani, Luca: A unitary Hessenberg (QR)-based algorithm via semiseparable matrices (2005)
  20. Shampine, L. F.; Muir, P. H.: Estimating conditioning of BVPs for ODEs (2004)

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