### Abstract

Crouzeix's conjecture states that for all polynomials p and matrices A, the inequality (norm of matrix)p(A)(norm of matrix)≤2(norm of matrix)p(norm of matrix)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≡(norm of matrix)p(norm of matrix)W(A)/(norm of matrix)p(A)(norm of matrix), 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.

Original language | English (US) |
---|---|

Journal | Linear Algebra and Its Applications |

DOIs | |

State | Accepted/In press - Oct 18 2016 |

### Fingerprint

### Keywords

- Chebfun
- Field of values
- Nonsmooth optimization
- Numerical range

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*. https://doi.org/10.1016/j.laa.2017.04.035

**Numerical investigation of Crouzeix's conjecture.** / Greenbaum, Anne; Overton, Michael.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Numerical investigation of Crouzeix's conjecture

AU - Greenbaum, Anne

AU - Overton, Michael

PY - 2016/10/18

Y1 - 2016/10/18

N2 - Crouzeix's conjecture states that for all polynomials p and matrices A, the inequality (norm of matrix)p(A)(norm of matrix)≤2(norm of matrix)p(norm of matrix)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≡(norm of matrix)p(norm of matrix)W(A)/(norm of matrix)p(A)(norm of matrix), 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.

AB - Crouzeix's conjecture states that for all polynomials p and matrices A, the inequality (norm of matrix)p(A)(norm of matrix)≤2(norm of matrix)p(norm of matrix)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≡(norm of matrix)p(norm of matrix)W(A)/(norm of matrix)p(A)(norm of matrix), 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.

KW - Chebfun

KW - Field of values

KW - Nonsmooth optimization

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=85018880376&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018880376&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2017.04.035

DO - 10.1016/j.laa.2017.04.035

M3 - Article

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -