Book Review
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3307767
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Book Information:
Authors:
Jörg Liesen and
Zdeněk Stra-koš
Title:
Krylov subspace methods: principles and analysis
Additional book information:
Numerical Methods and Scientific Computation,
Oxford University Press,
Oxford,
2013,
xiv+391 pp.,
ISBN 978-0-19-965541-0
SIGNUM Newsletter, 16 (1984), p. 7.
Athanasios C. Antoulas, Approximation of large-scale dynamical systems, Advances in Design and Control, vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. With a foreword by Jan C. Willems. MR 2155615, DOI 10.1137/1.9780898718713
W. E. Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951), 17–29. MR 42792, DOI 10.1090/S0033-569X-1951-42792-9
Owe Axelsson, Iterative solution methods, Cambridge University Press, Cambridge, 1994. MR 1276069, DOI 10.1017/CBO9780511624100
Bernhard Beckermann and Lothar Reichel, The Arnoldi process and GMRES for nearly symmetric matrices, SIAM J. Matrix Anal. Appl. 30 (2008), no. 1, 102–120. MR 2399571, DOI 10.1137/060668274
Russell Carden and Derek J. Hansen, Ritz values of normal matrices and Ceva’s theorem, Linear Algebra Appl. 438 (2013), no. 11, 4114–4129. MR 3034520, DOI 10.1016/j.laa.2012.12.030
Michel Crouzeix, Numerical range and functional calculus in Hilbert space, J. Funct. Anal. 244 (2007), no. 2, 668–690. MR 2297040, DOI 10.1016/j.jfa.2006.10.013
L. Elsner and Kh. D. Ikramov, On a condensed form for normal matrices under finite sequences of elementary unitary similarities, Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), 1997, pp. 79–98. MR 1436675, DOI 10.1016/S0024-3795(96)00526-5
Mark Embree, The tortoise and the hare restart GMRES, SIAM Rev. 45 (2003), no. 2, 259–266. MR 2010378, DOI 10.1137/S003614450139961
Vance Faber and Thomas Manteuffel, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. Numer. Anal. 21 (1984), no. 2, 352–362. MR 736337, DOI 10.1137/0721026
Bernd Fischer, Polynomial based iteration methods for symmetric linear systems, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester; B. G. Teubner, Stuttgart, 1996. MR 1449136, DOI 10.1007/978-3-663-11108-5
R. Fletcher, Conjugate gradient methods for indefinite systems, Numerical analysis (Proc 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, pp. 73–89. MR 0461857
Anne Greenbaum, Iterative methods for solving linear systems, Frontiers in Applied Mathematics, vol. 17, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR 1474725, DOI 10.1137/1.9781611970937
Anne Greenbaum, On the role of the left starting vector in the two-sided Lanczos algorithm and nonsymmetric linear system solvers, in Numerical Analysis 1997, D. F. Griffiths, D. J. Higham, and G. A. Watson, eds., Addison Wesley Longman, Harlow, Essex, UK, 1998, pp. 124–132.
Anne Greenbaum, Vlastimil Pták, and Zdeněk Strakoš, Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl. 17 (1996), no. 3, 465–469. MR 1397238, DOI 10.1137/S0895479894275030
Magnus R. Hestenes and Eduard Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards 49 (1952), 409–436 (1953). MR 0060307
Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards 45 (1950), 255–282. MR 0042791
S. M. Malamud, Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4043–4064. MR 2159699, DOI 10.1090/S0002-9947-04-03649-9
G. Meurant, Computer solution of large linear systems, Studies in Mathematics and its Applications, vol. 28, North-Holland Publishing Co., Amsterdam, 1999. MR 1700774
C. C. Paige and M. A. Saunders, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 617–629. MR 383715, DOI 10.1137/0712047
Yousef Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. MR 1990645, DOI 10.1137/1.9780898718003
Youcef Saad and Martin H. Schultz, Conjugate gradient-like algorithms for solving nonsymmetric linear systems, Math. Comp. 44 (1985), no. 170, 417–424. MR 777273, DOI 10.1090/S0025-5718-1985-0777273-9
Youcef Saad and Martin H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869.
Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088, DOI 10.1090/coll054.1
V. Simoncini, Computational methods for linear matrix equations. Manuscript, January 2014.
Kim-Chuan Toh, GMRES vs. ideal GMRES, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 30–36. MR 1428196, DOI 10.1137/S089547989427909X
Lloyd N. Trefethen and David Bau III, Numerical linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.
Henk A. van der Vorst, Iterative Krylov methods for large linear systems, Cambridge Monographs on Applied and Computational Mathematics, vol. 13, Cambridge University Press, Cambridge, 2003. MR 1990752, DOI 10.1017/CBO9780511615115
Yu. V. Vorobyev, Method of moments in applied mathematics, Gordon and Breach Science Publishers, New York-London-Paris, 1965. Translated from the Russian by Bernard Seckler. MR 0184400
J. Wallis, Opera Mathematica, vol. 1, Oxford University Press, 1695.
References
- SIGNUM Newsletter, 16 (1984), p. 7.
- Athanasios C. Antoulas, Approximation of large-scale dynamical systems, Advances in Design and Control, vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. With a foreword by Jan C. Willems. MR 2155615 (2006c:93001), DOI 10.1137/1.9780898718713
- W. E. Arnoldi, The principle of minimized iteration in the solution of the matrix eigenvalue problem, Quart. Appl. Math. 9 (1951), 17–29. MR 0042792 (13,163e)
- Owe Axelsson, Iterative solution methods, Cambridge University Press, Cambridge, 1994. MR 1276069 (95f:65005), DOI 10.1017/CBO9780511624100
- Bernhard Beckermann and Lothar Reichel, The Arnoldi process and GMRES for nearly symmetric matrices, SIAM J. Matrix Anal. Appl. 30 (2008), no. 1, 102–120. MR 2399571 (2009k:65053), DOI 10.1137/060668274
- Russell Carden and Derek J. Hansen, Ritz values of normal matrices and Ceva’s theorem, Linear Algebra Appl. 438 (2013), no. 11, 4114–4129. MR 3034520, DOI 10.1016/j.laa.2012.12.030
- Michel Crouzeix, Numerical range and functional calculus in Hilbert space, J. Funct. Anal. 244 (2007), no. 2, 668–690. MR 2297040 (2008f:47020), DOI 10.1016/j.jfa.2006.10.013
- L. Elsner and Kh. D. Ikramov, On a condensed form for normal matrices under finite sequences of elementary unitary similarities, Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), 1997, pp. 79–98. MR 1436675 (98b:15011), DOI 10.1016/S0024-3795(96)00526-5
- Mark Embree, The tortoise and the hare restart GMRES, SIAM Rev. 45 (2003), no. 2, 259–266 (electronic). MR 2010378 (2004j:65043), DOI 10.1137/S003614450139961
- Vance Faber and Thomas Manteuffel, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. Numer. Anal. 21 (1984), no. 2, 352–362. MR 736337 (85i:65038), DOI 10.1137/0721026
- Bernd Fischer, Polynomial based iteration methods for symmetric linear systems, Wiley-Teubner Series Advances in Numerical Mathematics, John Wiley & Sons, Ltd., Chichester; B. G. Teubner, Stuttgart, 1996. MR 1449136 (98e:65017), DOI 10.1007/978-3-663-11108-5
- R. Fletcher, Conjugate gradient methods for indefinite systems, Numerical analysis (Proc 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Springer, Berlin, 1976, pp. 73–89. Lecture Notes in Math., Vol. 506. MR 0461857 (57 \#1841)
- Anne Greenbaum, Iterative methods for solving linear systems, Frontiers in Applied Mathematics, vol. 17, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. MR 1474725 (98j:65023), DOI 10.1137/1.9781611970937
- Anne Greenbaum, On the role of the left starting vector in the two-sided Lanczos algorithm and nonsymmetric linear system solvers, in Numerical Analysis 1997, D. F. Griffiths, D. J. Higham, and G. A. Watson, eds., Addison Wesley Longman, Harlow, Essex, UK, 1998, pp. 124–132.
- Anne Greenbaum, Vlastimil Pták, and Zdeněk Strakoš, Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl. 17 (1996), no. 3, 465–469. MR 1397238 (97c:65057), DOI 10.1137/S0895479894275030
- Magnus R. Hestenes and Eduard Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards 49 (1952), 409–436 (1953). MR 0060307 (15,651a)
- Cornelius Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards 45 (1950), 255–282. MR 0042791 (13,163d)
- S. M. Malamud, Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4043–4064. MR 2159699 (2006e:15018), DOI 10.1090/S0002-9947-04-03649-9
- G. Meurant, Computer solution of large linear systems, Studies in Mathematics and its Applications, vol. 28, North-Holland Publishing Co., Amsterdam, 1999. MR 1700774 (2000f:65003)
- C. C. Paige and M. A. Saunders, Solutions of sparse indefinite systems of linear equations, SIAM J. Numer. Anal. 12 (1975), no. 4, 617–629. MR 0383715 (52 \#4595)
- Yousef Saad, Iterative methods for sparse linear systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. MR 1990645 (2004h:65002), DOI 10.1137/1.9780898718003
- Youcef Saad and Martin H. Schultz, Conjugate gradient-like algorithms for solving nonsymmetric linear systems, Math. Comp. 44 (1985), no. 170, 417–424. MR 777273 (86d:65047), DOI 10.2307/2007961
- Youcef Saad and Martin H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856–869.
- Barry Simon, Orthogonal polynomials on the unit circle. Part 1, American Mathematical Society Colloquium Publications, vol. 54, American Mathematical Society, Providence, RI, 2005. Classical theory. MR 2105088 (2006a:42002a)
- V. Simoncini, Computational methods for linear matrix equations. Manuscript, January 2014.
- Kim-Chuan Toh, GMRES vs. ideal GMRES, SIAM J. Matrix Anal. Appl. 18 (1997), no. 1, 30–36. MR 1428196 (97i:65054), DOI 10.1137/S089547989427909X
- Lloyd N. Trefethen and David Bau III, Numerical linear algebra, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1997.
- Henk A. van der Vorst, Iterative Krylov methods for large linear systems, Cambridge Monographs on Applied and Computational Mathematics, vol. 13, Cambridge University Press, Cambridge, 2003. MR 1990752 (2005k:65075), DOI 10.1017/CBO9780511615115
- Yu. V. Vorobyev, Method of moments in applied mathematics, Gordon and Breach Science Publishers, New York-London-Paris, 1965. Translated from the Russian by Bernard Seckler. MR 0184400 (32 \#1872)
- J. Wallis, Opera Mathematica, vol. 1, Oxford University Press, 1695.
Review Information:
Reviewer:
Mark Embree
Affiliation:
Department of Mathematics, Virginia Tech
Email:
embree@vt.edu
Journal:
Bull. Amer. Math. Soc.
52 (2015), 151-158
DOI:
https://doi.org/10.1090/S0273-0979-2014-01473-9
Published electronically:
September 10, 2014
Review copyright:
© Copyright 2014
American Mathematical Society
