Numerical investigation of Crouzeix’s conjecture. (English) Zbl 1415.15021
Summary: Crouzeix’s conjecture states that for all polynomials \(p\) and matrices \(A\), the inequality \(\| p(A) \| \leq 2 \| p \|_{W(A)}\) holds, where the quantity on the left is the 2-norm of the matrix \(p(A)\) and the norm on the right is the maximum modulus of the polynomial \(p\) on \(W(A)\), the field of values of \(A\). We report on some extensive numerical experiments investigating the conjecture via nonsmooth minimization of the Crouzeix ratio \(f \equiv \| p \|_{W(A)} / \| p(A) \|\), using Chebfun to evaluate this quantity accurately and efficiently and the BFGS method to search for its minimal value, which is 0.5 if Crouzeix’s conjecture is true. Almost all of our optimization searches deliver final polynomial-matrix pairs that are very close to nonsmooth stationary points of \(f\) with stationary value 0.5 (for which \(W(A)\) is a disk) or smooth stationary points of \(f\) with stationary value 1 (for which \(W(A)\) has a corner). Our observations have led us to some additional conjectures as well as
some new theorems. We hope that these give insight into Crouzeix’s conjecture, which is strongly supported by our results.
MSC:
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 47N10 | Applications of operator theory in optimization, convex analysis, mathematical programming, economics |
References:
| [1] | Berger, C., A strange dilation theorem, Notices Amer. Math. Soc., 12, 590 (1965), abstract 625-152 |
| [2] | Borwein, J. M.; Lewis, A. S., Convex Analysis and Nonlinear Optimization: Theory and Examples (2000), Springer: Springer New York · Zbl 0953.90001 |
| [3] | Burke, J. V.; Lewis, A. S.; Overton, M. L., A robust gradient sampling algorithm for nonsmooth, nonconvex optimization, SIAM J. Optim., 15, 751-779 (2005) · Zbl 1078.65048 |
| [4] | Choi, D.; Greenbaum, A., Roots of matrices in the study of GMRES convergence and Crouzeix’s conjecture, SIAM J. Matrix Anal. Appl., 36, 289-301 (2015) · Zbl 1314.15015 |
| [5] | Cowen, C.; Harel, E., An effective algorithm for computing the numerical range (1995) |
| [6] | Choi, D., A proof of Crouzeix’s conjecture for a class of matrices, Linear Algebra Appl., 438, 8, 3247-3257 (2013) · Zbl 1281.15021 |
| [7] | Clarke, F. H., Optimization and Nonsmooth Analysis (1990), John Wiley: John Wiley New York: SIAM: John Wiley: John Wiley New York: SIAM Philadelphia, reprinted by · Zbl 0696.49002 |
| [8] | Crouzeix, M., Bounds for analytical functions of matrices, Integral Equations Operator Theory, 48, 461-477 (2004) · Zbl 1069.47004 |
| [9] | Crouzeix, M., Numerical range and functional calculus in Hilbert space, J. Funct. Anal., 244, 2, 668-690 (2007) · Zbl 1116.47004 |
| [10] | Crouzeix, M., Spectral sets and \(3 \times 3\) nilpotent matrices, (Topics in Functional and Harmonic Analysis. Topics in Functional and Harmonic Analysis, Theta Ser. Adv. Math., vol. 14 (2013), Theta: Theta Bucharest), 27-42 · Zbl 1538.15024 |
| [11] | M. Crouzeix, 2015, private communication.; M. Crouzeix, 2015, private communication. |
| [12] | Crouzeix, M.; Palencia, C., The numerical range is a \((1 + \sqrt{2})\)-spectral set, SIAM J. Matrix Anal. Appl. (2017), to appear · Zbl 1368.47006 |
| [13] | Driscoll, T. A.; Hale, N.; Trefethen, L. N., Chebfun Guide (2014), Pafnuty Publications: Pafnuty Publications Oxford |
| [14] | Greenbaum, A.; Choi, D., Crouzeix’s conjecture and perturbed Jordan blocks, Linear Algebra Appl., 436, 7, 2342-2352 (2012) · Zbl 1390.15074 |
| [15] | Greenbaum, A.; Caldwell, T.; Li, K., Near normal dilations of nonnormal matrices and linear operators, SIAM J. Matrix Anal. Appl., 37, 4, 1365-1381 (2016) · Zbl 1388.15019 |
| [16] | Greenbaum, A.; Lewis, A. S.; Overton, M. L., Variational analysis of the Crouzeix ratio, Math. Program. (2016), in press |
| [17] | Guglielmi, N.; Overton, M. L.; Stewart, G. W., An efficient algorithm for computing the generalized null space decomposition, SIAM J. Matrix Anal. Appl., 36, 1, 38-54 (2015) · Zbl 1327.65072 |
| [18] | Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press: Cambridge University Press Cambridge, U.K. · Zbl 0729.15001 |
| [19] | Johnson, C. R., Numerical determination of the field of values of a general complex matrix, SIAM J. Numer. Anal., 15, 3, 595-602 (1978) · Zbl 0389.65018 |
| [20] | Kippenhahn, R., Über den Wertevorrat einer Matrix, Math. Nachr.. Math. Nachr., Linear Multilinear Algebra, 56, 185-225 (2008), English translation by P.F. Zachlin and M.E. Hochstenbach · Zbl 1137.47003 |
| [21] | Kiwiel, K. C., Convergence of the gradient sampling algorithm for nonsmooth nonconvex optimization, SIAM J. Optim., 18, 379-388 (2007) · Zbl 1149.65043 |
| [22] | Kublanovskaja, V. N., A method for solving the complete problem of eigenvalues of a degenerate matrix, Zh. Vychisl. Mat. Mat. Fiz., 6, 611-620 (1966) · Zbl 0171.36003 |
| [23] | Lewis, A. S.; Overton, M. L., Nonsmooth optimization via quasi-Newton methods, Math. Program., 141, 1-2, Ser. A, 135-163 (2013) · Zbl 1280.90118 |
| [24] | Mergelyan, S. N., On the representation of functions by series of polynomials on closed sets, Amer. Math. Soc. Transl., 1953, 85, 8 (1953) |
| [25] | Mergelyan, S. N., Uniform approximations to functions of a complex variable, Amer. Math. Soc. Transl., 1954, 101, 99 (1954) · Zbl 0059.05902 |
| [26] | Okubo, K.; Ando, T., Constants related to operators of class \(C_\rho \), Manuscripta Math., 16, 4, 385-394 (1975) · Zbl 0325.47008 |
| [27] | Pearcy, C., An elementary proof of the power inequality for the numerical radius, Michigan Math. J., 13, 289-291 (1966) · Zbl 0143.16205 |
| [28] | Tso, S.-H.; Wu, P.-Y., Matricial ranges of quadratic operators, Rocky Mountain J. Math., 29, 3, 1139-1152 (1999) · Zbl 0957.47005 |
| [29] | von Neumann, J., Eine Spektraltheorie fuer allgemeine Operatoren eines unitaeren Raumes, Math. Nachr.. (Collected Works, vol. IV (1962), Pergamon: Pergamon Oxford), 4, 341-364 (1951) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
