Investigation of Crouzeixs Conjecture via Nonsmooth Optimization - PowerPoint PPT Presentation

Crouzeix’s Conjecture Let p = p ( z ) be a polynomial and let A be a square matrix. M. Crouzeix conjectured in “Bounds for analytical functions of Crouzeix’s Conjecture matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A , The Field of Values Examples Example, continued � p ( A ) � 2 ≤ 2 � p � W ( A ) . Crouzeix’s Conjecture Crouzeix’s Theorem The left-hand side is the 2-norm (spectral norm, maximum Greatly Improved New Bound from singular value) of the matrix p ( A ) . C´ esar Palencia Special Cases The norm on the right-hand side is the maximum of | p ( z ) | Computing the Field of Values over z ∈ W ( A ) . By the maximum modulus principle, this must Johnson’s Algorithm Finds the Extreme be attained on bd W ( A ) , the boundary of W ( A ) . Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 6 / 50 Optimizing over p and A

Crouzeix’s Conjecture Let p = p ( z ) be a polynomial and let A be a square matrix. M. Crouzeix conjectured in “Bounds for analytical functions of Crouzeix’s Conjecture matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A , The Field of Values Examples Example, continued � p ( A ) � 2 ≤ 2 � p � W ( A ) . Crouzeix’s Conjecture Crouzeix’s Theorem The left-hand side is the 2-norm (spectral norm, maximum Greatly Improved New Bound from singular value) of the matrix p ( A ) . C´ esar Palencia Special Cases The norm on the right-hand side is the maximum of | p ( z ) | Computing the Field of Values over z ∈ W ( A ) . By the maximum modulus principle, this must Johnson’s Algorithm Finds the Extreme be attained on bd W ( A ) , the boundary of W ( A ) . Points Chebfun If p = χ ( A ) , the characteristic polynomial (or minimal Example, continued The Crouzeix Ratio polynomial) of A , then � p ( A ) � 2 = 0 by Cayley-Hamilton, but Computing the Crouzeix Ratio � p � W ( A ) = 0 only if A = λI for λ ∈ C , so that W ( A ) = { λ } . Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 6 / 50 Optimizing over p and A

Crouzeix’s Conjecture Let p = p ( z ) be a polynomial and let A be a square matrix. M. Crouzeix conjectured in “Bounds for analytical functions of Crouzeix’s Conjecture matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A , The Field of Values Examples Example, continued � p ( A ) � 2 ≤ 2 � p � W ( A ) . Crouzeix’s Conjecture Crouzeix’s Theorem The left-hand side is the 2-norm (spectral norm, maximum Greatly Improved New Bound from singular value) of the matrix p ( A ) . C´ esar Palencia Special Cases The norm on the right-hand side is the maximum of | p ( z ) | Computing the Field of Values over z ∈ W ( A ) . By the maximum modulus principle, this must Johnson’s Algorithm Finds the Extreme be attained on bd W ( A ) , the boundary of W ( A ) . Points Chebfun If p = χ ( A ) , the characteristic polynomial (or minimal Example, continued The Crouzeix Ratio polynomial) of A , then � p ( A ) � 2 = 0 by Cayley-Hamilton, but Computing the Crouzeix Ratio � p � W ( A ) = 0 only if A = λI for λ ∈ C , so that W ( A ) = { λ } . Nonsmooth Optimization of If p ( z ) = z and A is a 2 × 2 Jordan block with 0 on the diagonal, the Crouzeix Ratio f then � p ( A ) � 2 = 1 and W ( A ) is a disk centered at 0 with radius Fix p , Optimize over A 0 . 5 , so the left and right-hand sides are equal. Fix A , Optimize over p 6 / 50 Optimizing over p and A

Crouzeix’s Conjecture Let p = p ( z ) be a polynomial and let A be a square matrix. M. Crouzeix conjectured in “Bounds for analytical functions of Crouzeix’s Conjecture matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A , The Field of Values Examples Example, continued � p ( A ) � 2 ≤ 2 � p � W ( A ) . Crouzeix’s Conjecture Crouzeix’s Theorem The left-hand side is the 2-norm (spectral norm, maximum Greatly Improved New Bound from singular value) of the matrix p ( A ) . C´ esar Palencia Special Cases The norm on the right-hand side is the maximum of | p ( z ) | Computing the Field of Values over z ∈ W ( A ) . By the maximum modulus principle, this must Johnson’s Algorithm Finds the Extreme be attained on bd W ( A ) , the boundary of W ( A ) . Points Chebfun If p = χ ( A ) , the characteristic polynomial (or minimal Example, continued The Crouzeix Ratio polynomial) of A , then � p ( A ) � 2 = 0 by Cayley-Hamilton, but Computing the Crouzeix Ratio � p � W ( A ) = 0 only if A = λI for λ ∈ C , so that W ( A ) = { λ } . Nonsmooth Optimization of If p ( z ) = z and A is a 2 × 2 Jordan block with 0 on the diagonal, the Crouzeix Ratio f then � p ( A ) � 2 = 1 and W ( A ) is a disk centered at 0 with radius Fix p , Optimize over A 0 . 5 , so the left and right-hand sides are equal. Fix A , Optimize over p Conjecture extends to analytic functions and to Hilbert space 6 / 50 Optimizing over p and A

Crouzeix’s Theorem � p ( A ) � 2 ≤ 11 . 08 � p � W ( A ) Crouzeix’s Conjecture i.e., the conjecture is true if we replace 2 by 11.08. The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 7 / 50 Optimizing over p and A

Crouzeix’s Theorem � p ( A ) � 2 ≤ 11 . 08 � p � W ( A ) Crouzeix’s Conjecture i.e., the conjecture is true if we replace 2 by 11.08. The Field of Values Examples Example, continued Crouzeix’s “The estimate 11.08 is not optimal. There is no Conjecture doubt that refinements are possible which would Crouzeix’s Theorem Greatly Improved decrease this bound. We are convinced that our New Bound from C´ esar Palencia estimate is very pessimistic, but to improve it Special Cases Computing the Field drastically (recall that our conjecture is that 11.08 of Values Johnson’s Algorithm can be replaced by 2), it is clear that we have to find Finds the Extreme Points a completely different method.” Chebfun Example, continued - Michel Crouzeix, “Numerical range and functional The Crouzeix Ratio calculus in Hilbert space”, J. Funct. Anal. 244 (2007). Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 7 / 50 Optimizing over p and A

Crouzeix’s Theorem � p ( A ) � 2 ≤ 11 . 08 � p � W ( A ) Crouzeix’s Conjecture i.e., the conjecture is true if we replace 2 by 11.08. The Field of Values Examples Example, continued Crouzeix’s “The estimate 11.08 is not optimal. There is no Conjecture doubt that refinements are possible which would Crouzeix’s Theorem Greatly Improved decrease this bound. We are convinced that our New Bound from C´ esar Palencia estimate is very pessimistic, but to improve it Special Cases Computing the Field drastically (recall that our conjecture is that 11.08 of Values Johnson’s Algorithm can be replaced by 2), it is clear that we have to find Finds the Extreme Points a completely different method.” Chebfun Example, continued - Michel Crouzeix, “Numerical range and functional The Crouzeix Ratio calculus in Hilbert space”, J. Funct. Anal. 244 (2007). Computing the Crouzeix Ratio Nonsmooth Optimization of Remarkably broad impact: the norm of an analytic function of a the Crouzeix Ratio f matrix A is bounded by a modest constant times its norm on the Fix p , Optimize over A field of values W ( A ) . Fix A , Optimize over p 7 / 50 Optimizing over p and A

Greatly Improved New Bound from C´ esar Palencia √ Crouzeix’s � � � p ( A ) � 2 ≤ 1 + 2 � p � W ( A ) Conjecture The Field of Values √ Examples i.e., the conjecture is true if we replace 2 by 1 + 2 Example, continued Crouzeix’s Conjecture Presented at a conference in Greece, summer 2016 Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 8 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved W ( A ) = D : ■ New Bound from C´ esar Palencia • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Special Cases • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 Computing the Field of Values (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved W ( A ) = D : ■ New Bound from C´ esar Palencia • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Special Cases • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 Computing the Field of Values (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Johnson’s Algorithm n = 2 (Crouzeix, 2004), and, more generally, the minimum Finds the Extreme ■ Points polynomial of A has degree 2 (follows from Tso and Wu, 1999) Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved W ( A ) = D : ■ New Bound from C´ esar Palencia • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Special Cases • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 Computing the Field of Values (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Johnson’s Algorithm n = 2 (Crouzeix, 2004), and, more generally, the minimum Finds the Extreme ■ Points polynomial of A has degree 2 (follows from Tso and Wu, 1999) Chebfun n = 3 and A 3 = 0 (Crouzeix, 2013) Example, continued ■ The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved W ( A ) = D : ■ New Bound from C´ esar Palencia • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Special Cases • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 Computing the Field of Values (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Johnson’s Algorithm n = 2 (Crouzeix, 2004), and, more generally, the minimum Finds the Extreme ■ Points polynomial of A has degree 2 (follows from Tso and Wu, 1999) Chebfun n = 3 and A 3 = 0 (Crouzeix, 2013) Example, continued ■ The Crouzeix Ratio A is an upper Jordan block with a perturbation in the bottom left ■ Computing the Crouzeix Ratio corner (Choi and Greenbaum, 2012) or any diagonal scaling of such Nonsmooth A (Choi, 2013) Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved W ( A ) = D : ■ New Bound from C´ esar Palencia • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Special Cases • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 Computing the Field of Values (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Johnson’s Algorithm n = 2 (Crouzeix, 2004), and, more generally, the minimum Finds the Extreme ■ Points polynomial of A has degree 2 (follows from Tso and Wu, 1999) Chebfun n = 3 and A 3 = 0 (Crouzeix, 2013) Example, continued ■ The Crouzeix Ratio A is an upper Jordan block with a perturbation in the bottom left ■ Computing the Crouzeix Ratio corner (Choi and Greenbaum, 2012) or any diagonal scaling of such Nonsmooth A (Choi, 2013) Optimization of A = TDT − 1 with D diagonal and � T �� T − 1 � ≤ 2 (easy) the Crouzeix Ratio f ■ Fix p , Optimize over A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Special Cases The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ C n × n . Crouzeix’s Conjecture Let r ( A ) = max ζ ∈ W ( A ) | ζ | (numerical radius) and D = open unit disk The Field of Values Examples p ( ζ ) = ζ m : ■ Example, continued � A m � ≤ 2 r ( A m ) ≤ 2 r ( A ) m = 2 max ζ ∈ W ( A ) | ζ m | Crouzeix’s Conjecture (power inequality, Berger 1965, Pearcy 1966) Crouzeix’s Theorem Greatly Improved W ( A ) = D : ■ New Bound from C´ esar Palencia • if � B � ≤ 1 , then � p ( B ) � ≤ sup ζ ∈D | p ( ζ ) | (von Neumann, 1951) Special Cases • if r ( A ) ≤ 1 , then A = TBT − 1 with � B � ≤ 1 and � T �� T − 1 � ≤ 2 Computing the Field of Values (Okubo and Ando, 1975), so � p ( A ) � ≤ 2 � p ( B ) � Johnson’s Algorithm n = 2 (Crouzeix, 2004), and, more generally, the minimum Finds the Extreme ■ Points polynomial of A has degree 2 (follows from Tso and Wu, 1999) Chebfun n = 3 and A 3 = 0 (Crouzeix, 2013) Example, continued ■ The Crouzeix Ratio A is an upper Jordan block with a perturbation in the bottom left ■ Computing the Crouzeix Ratio corner (Choi and Greenbaum, 2012) or any diagonal scaling of such Nonsmooth A (Choi, 2013) Optimization of A = TDT − 1 with D diagonal and � T �� T − 1 � ≤ 2 (easy) the Crouzeix Ratio f ■ AA ∗ = A ∗ A (then the constant 2 can be improved to 1 ). Fix p , Optimize over ■ A Fix A , Optimize over p 9 / 50 Optimizing over p and A

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 10 / 50 Optimizing over p and A

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed Example, continued Crouzeix’s that the extreme points of W ( A ) can be characterized as Conjecture Crouzeix’s Theorem Greatly Improved ext W ( A ) = { z θ = v ∗ θ Av θ : θ ∈ [0 , 2 π ) } New Bound from C´ esar Palencia Special Cases where v θ is a normalized eigenvector corresponding to the largest Computing the Field of Values eigenvalue of the Hermitian matrix Johnson’s Algorithm Finds the Extreme Points H θ = 1 Chebfun � e iθ A + e − iθ A ∗ � . Example, continued 2 The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 10 / 50 Optimizing over p and A

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed Example, continued Crouzeix’s that the extreme points of W ( A ) can be characterized as Conjecture Crouzeix’s Theorem Greatly Improved ext W ( A ) = { z θ = v ∗ θ Av θ : θ ∈ [0 , 2 π ) } New Bound from C´ esar Palencia Special Cases where v θ is a normalized eigenvector corresponding to the largest Computing the Field of Values eigenvalue of the Hermitian matrix Johnson’s Algorithm Finds the Extreme Points H θ = 1 Chebfun � e iθ A + e − iθ A ∗ � . Example, continued 2 The Crouzeix Ratio Computing the Crouzeix Ratio The proof uses a supporting hyperplane argument. Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 10 / 50 Optimizing over p and A

Computing the Field of Values The extreme points of a convex set are those that cannot be Crouzeix’s Conjecture expressed as a convex combination of two other points in the set. The Field of Values Examples Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed Example, continued Crouzeix’s that the extreme points of W ( A ) can be characterized as Conjecture Crouzeix’s Theorem Greatly Improved ext W ( A ) = { z θ = v ∗ θ Av θ : θ ∈ [0 , 2 π ) } New Bound from C´ esar Palencia Special Cases where v θ is a normalized eigenvector corresponding to the largest Computing the Field of Values eigenvalue of the Hermitian matrix Johnson’s Algorithm Finds the Extreme Points H θ = 1 Chebfun � e iθ A + e − iθ A ∗ � . Example, continued 2 The Crouzeix Ratio Computing the Crouzeix Ratio The proof uses a supporting hyperplane argument. Nonsmooth Optimization of Thus, we can compute as many extreme points as we like. the Crouzeix Ratio f Continuing with the previous example... Fix p , Optimize over A Fix A , Optimize over p 10 / 50 Optimizing over p and A

Johnson’s Algorithm Finds the Extreme Points Crouzeix’s Conjecture 3 θ ∈ [3.99,5.3] The Field of Values Examples Example, continued 2 Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved 1 New Bound from θ ∈ [5.3,2 π ] C´ esar Palencia Special Cases θ ∈ [2.29,3.99] 0 Computing the Field of Values Johnson’s Algorithm θ ∈ [0,0.96] Finds the Extreme −1 Points Chebfun Example, continued The Crouzeix Ratio −2 Computing the Crouzeix Ratio θ ∈ [0.96,2.29] Nonsmooth −3 Optimization of the Crouzeix Ratio f −1 0 1 2 3 4 5 6 Fix p , Optimize over A The extreme points of W(A) lie in the union of 5 connected sets Fix A , Optimize over p 11 / 50 Optimizing over p and A

Johnson’s Algorithm Finds the Extreme Points Crouzeix’s Conjecture 3 θ ∈ [3.99,5.3] The Field of Values Examples Example, continued 2 Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved 1 New Bound from θ ∈ [5.3,2 π ] C´ esar Palencia Special Cases θ ∈ [2.29,3.99] 0 Computing the Field of Values Johnson’s Algorithm θ ∈ [0,0.96] Finds the Extreme −1 Points Chebfun Example, continued The Crouzeix Ratio −2 Computing the Crouzeix Ratio θ ∈ [0.96,2.29] Nonsmooth −3 Optimization of the Crouzeix Ratio f −1 0 1 2 3 4 5 6 Fix p , Optimize over A The extreme points of W(A) lie in the union of 5 connected sets Fix A , Optimize But how can we do this accurately, automatically and efficiently? over p 11 / 50 Optimizing over p and A

Chebfun Crouzeix’s Chebfun (Trefethen et al, 2004–present) represents real- or Conjecture complex-valued functions on real intervals to machine precision The Field of Values Examples accuracy using Chebyshev interpolation. Example, continued Crouzeix’s Conjecture The necessary degree of the polynomial is determined Crouzeix’s Theorem Greatly Improved automatically. For example, representing sin( πx ) on [ − 1 , 1] to New Bound from C´ esar Palencia machine precision requires degree 19. Special Cases Computing the Field of Values Most Matlab functions are overloaded to work with chebfun’s. Johnson’s Algorithm Finds the Extreme Points Chebfun Applying Chebfun’s fov to compute the boundary of W ( A ) for Example, continued the previous example... The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 12 / 50 Optimizing over p and A

Example, continued Field of Values of A = diag(J,B,D): J is Jordan block, B full, D diagonal 2.5 Crouzeix’s Conjecture 2 The Field of Values Examples Example, continued 1.5 Crouzeix’s Conjecture 1 Crouzeix’s Theorem Greatly Improved W(A) New Bound from 0.5 break points of W(A) C´ esar Palencia || χ (A)|| W(A) attained Special Cases 0 Computing the Field W(J) of Values W(B) Johnson’s Algorithm −0.5 Finds the Extreme W(D) Points eig(A) Chebfun −1 Example, continued The Crouzeix Ratio −1.5 Computing the Crouzeix Ratio −2 Nonsmooth Optimization of the Crouzeix Ratio f −2.5 0 1 2 3 4 5 Fix p , Optimize over A Internal points shown are chebfun interpolation points Fix A , Optimize over p 13 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix’s Theorem order n . Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix’s Theorem order n . The Crouzeix ratio f is Greatly Improved New Bound from C´ esar Palencia A mapping from C m +1 × C n × n to R (associating ■ Special Cases polynomials p ∈ P m with their vectors of coefficients Computing the Field of Values c ∈ C m +1 using the monomial basis) Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix’s Theorem order n . The Crouzeix ratio f is Greatly Improved New Bound from C´ esar Palencia A mapping from C m +1 × C n × n to R (associating ■ Special Cases polynomials p ∈ P m with their vectors of coefficients Computing the Field of Values c ∈ C m +1 using the monomial basis) Johnson’s Algorithm Finds the Extreme Points Not convex ■ Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix’s Theorem order n . The Crouzeix ratio f is Greatly Improved New Bound from C´ esar Palencia A mapping from C m +1 × C n × n to R (associating ■ Special Cases polynomials p ∈ P m with their vectors of coefficients Computing the Field of Values c ∈ C m +1 using the monomial basis) Johnson’s Algorithm Finds the Extreme Points Not convex ■ Chebfun Not defined if p ( A ) = 0 Example, continued ■ The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix’s Theorem order n . The Crouzeix ratio f is Greatly Improved New Bound from C´ esar Palencia A mapping from C m +1 × C n × n to R (associating ■ Special Cases polynomials p ∈ P m with their vectors of coefficients Computing the Field of Values c ∈ C m +1 using the monomial basis) Johnson’s Algorithm Finds the Extreme Points Not convex ■ Chebfun Not defined if p ( A ) = 0 Example, continued ■ The Crouzeix Ratio Lipschitz continuous at all other points, but not necessarily ■ Computing the Crouzeix Ratio differentiable Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

The Crouzeix Ratio Define the Crouzeix ratio Crouzeix’s f ( p, A ) = � p � W ( A ) Conjecture . The Field of Values � p ( A ) � 2 Examples Example, continued The conjecture states that f ( p, A ) is bounded below by 0 . 5 Crouzeix’s independently of the polynomial degree m and the matrix Conjecture Crouzeix’s Theorem order n . The Crouzeix ratio f is Greatly Improved New Bound from C´ esar Palencia A mapping from C m +1 × C n × n to R (associating ■ Special Cases polynomials p ∈ P m with their vectors of coefficients Computing the Field of Values c ∈ C m +1 using the monomial basis) Johnson’s Algorithm Finds the Extreme Points Not convex ■ Chebfun Not defined if p ( A ) = 0 Example, continued ■ The Crouzeix Ratio Lipschitz continuous at all other points, but not necessarily ■ Computing the Crouzeix Ratio differentiable Nonsmooth Semialgebraic (its graph is a finite union of sets, each of Optimization of ■ the Crouzeix Ratio f which is defined by a finite system of polynomial inequalities) Fix p , Optimize over A Fix A , Optimize over p 14 / 50 Optimizing over p and A

Computing the Crouzeix Ratio Crouzeix’s Numerator: use Chebfun’s fov (modified to return any line Conjecture segments in the boundary) combined with its overloaded polyval The Field of Values Examples and norm( · ,inf) . Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 15 / 50 Optimizing over p and A

Computing the Crouzeix Ratio Crouzeix’s Numerator: use Chebfun’s fov (modified to return any line Conjecture segments in the boundary) combined with its overloaded polyval The Field of Values Examples and norm( · ,inf) . Example, continued Crouzeix’s Denominator: use Matlab ’s standard polyvalm and norm( · ,2) . Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 15 / 50 Optimizing over p and A

Computing the Crouzeix Ratio Crouzeix’s Numerator: use Chebfun’s fov (modified to return any line Conjecture segments in the boundary) combined with its overloaded polyval The Field of Values Examples and norm( · ,inf) . Example, continued Crouzeix’s Denominator: use Matlab ’s standard polyvalm and norm( · ,2) . Conjecture Crouzeix’s Theorem Greatly Improved The main cost is the construction of the chebfun defining the New Bound from C´ esar Palencia field of values. Special Cases Computing the Field of Values Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix A , Optimize over p 15 / 50 Optimizing over p and A

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Nonsmooth Optimization of Fix p , Optimize over A the Crouzeix Ratio f Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 16 / 50

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in f Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 17 / 50

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in f Crouzeix’s Conjecture When the max value of | p ( z ) | on bd W ( A ) is attained at Nonsmooth ■ Optimization of more than one point z (the most important, as this the Crouzeix Ratio f Nonsmoothness of frequently occurs at apparent minimizers) the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 17 / 50

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in f Crouzeix’s Conjecture When the max value of | p ( z ) | on bd W ( A ) is attained at Nonsmooth ■ Optimization of more than one point z (the most important, as this the Crouzeix Ratio f Nonsmoothness of frequently occurs at apparent minimizers) the Crouzeix Ratio BFGS Even if such z is unique, when the normalized vector v for ■ Experiments which v ∗ Av = z is not unique up to a scalar, implying that Fix p , Optimize over A the maximum eigenvalue of the corresponding H θ matrix has Fix A , Optimize multiplicity two or more (does not seem to occur at over p Optimizing over p minimizers) and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 17 / 50

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in f Crouzeix’s Conjecture When the max value of | p ( z ) | on bd W ( A ) is attained at Nonsmooth ■ Optimization of more than one point z (the most important, as this the Crouzeix Ratio f Nonsmoothness of frequently occurs at apparent minimizers) the Crouzeix Ratio BFGS Even if such z is unique, when the normalized vector v for ■ Experiments which v ∗ Av = z is not unique up to a scalar, implying that Fix p , Optimize over A the maximum eigenvalue of the corresponding H θ matrix has Fix A , Optimize multiplicity two or more (does not seem to occur at over p Optimizing over p minimizers) and A When the maximum singular value of p ( A ) has multiplicity ■ Nonsmooth Analysis of the Crouzeix two or more (does not seem to occur at minimizers) Ratio Concluding Remarks 17 / 50

Nonsmoothness of the Crouzeix Ratio There are three possible sources of nonsmoothness in f Crouzeix’s Conjecture When the max value of | p ( z ) | on bd W ( A ) is attained at Nonsmooth ■ Optimization of more than one point z (the most important, as this the Crouzeix Ratio f Nonsmoothness of frequently occurs at apparent minimizers) the Crouzeix Ratio BFGS Even if such z is unique, when the normalized vector v for ■ Experiments which v ∗ Av = z is not unique up to a scalar, implying that Fix p , Optimize over A the maximum eigenvalue of the corresponding H θ matrix has Fix A , Optimize multiplicity two or more (does not seem to occur at over p Optimizing over p minimizers) and A When the maximum singular value of p ( A ) has multiplicity ■ Nonsmooth Analysis of the Crouzeix two or more (does not seem to occur at minimizers) Ratio Concluding Remarks In all of these cases the gradient of f is not defined. But in practice, none of these cases ever occur, except the first one in the limit . 17 / 50

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 18 / 50

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio f It works by building an approximation to the Hessian of the Nonsmoothness of the Crouzeix Ratio function using gradient differences, and has a well known BFGS superlinear convergence property under a regularity condition. Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 18 / 50

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio f It works by building an approximation to the Hessian of the Nonsmoothness of the Crouzeix Ratio function using gradient differences, and has a well known BFGS superlinear convergence property under a regularity condition. Experiments Fix p , Optimize over Although its global convergence theory is limited to the convex A Fix A , Optimize case (Powell, 1976), it generally finds local minimizers efficiently over p in the nonconvex case too, although there are pathological Optimizing over p and A counterexamples. Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 18 / 50

BFGS BFGS (Broyden, Fletcher, Goldfarb and Shanno, all Crouzeix’s independently in 1970), is the standard quasi-Newton algorithm Conjecture Nonsmooth for minimizing smooth (continuously differentiable) functions. Optimization of the Crouzeix Ratio f It works by building an approximation to the Hessian of the Nonsmoothness of the Crouzeix Ratio function using gradient differences, and has a well known BFGS superlinear convergence property under a regularity condition. Experiments Fix p , Optimize over Although its global convergence theory is limited to the convex A Fix A , Optimize case (Powell, 1976), it generally finds local minimizers efficiently over p in the nonconvex case too, although there are pathological Optimizing over p and A counterexamples. Nonsmooth Analysis of the Crouzeix Remarkably, this property seems to extend to nonsmooth Ratio functions too, with a linear rate of local convergence, although Concluding Remarks the convergence theory is extremely limited (Lewis and Overton, 2013). It builds a very ill conditioned “Hessian” approximation, with “infinitely large” curvature in some directions and finite curvature in other directions. 18 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A ◆ degree ≤ m = n − 1 Fix A , Optimize over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A ◆ degree ≤ m = n − 1 Fix A , Optimize ◆ unbounded degree (different method) over p Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A ◆ degree ≤ m = n − 1 Fix A , Optimize ◆ unbounded degree (different method) over p Optimize over both p with degree ≤ m = n − 1 and A with order n ■ Optimizing over p and A Nonsmooth Analysis of the Crouzeix Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A ◆ degree ≤ m = n − 1 Fix A , Optimize ◆ unbounded degree (different method) over p Optimize over both p with degree ≤ m = n − 1 and A with order n ■ Optimizing over p and A We restrict p to have real coefficients and A to be real, in Nonsmooth Analysis of the Crouzeix Hessenberg form (all but one superdiagonal is zero). Ratio Concluding Remarks 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A ◆ degree ≤ m = n − 1 Fix A , Optimize ◆ unbounded degree (different method) over p Optimize over both p with degree ≤ m = n − 1 and A with order n ■ Optimizing over p and A We restrict p to have real coefficients and A to be real, in Nonsmooth Analysis of the Crouzeix Hessenberg form (all but one superdiagonal is zero). Ratio We have obtained similar results for p with complex coefficients Concluding Remarks and complex A (then can take A to be triangular) 19 / 50

Experiments We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Crouzeix’s Conjecture Several scenarios: Nonsmooth Optimization of Fix p with degree m , optimize over A with fixed order n ■ the Crouzeix Ratio f Nonsmoothness of with m = n − 1 ◆ the Crouzeix Ratio with m = n ◆ BFGS Experiments Fix A with order n , optimize over p with ■ Fix p , Optimize over A ◆ degree ≤ m = n − 1 Fix A , Optimize ◆ unbounded degree (different method) over p Optimize over both p with degree ≤ m = n − 1 and A with order n ■ Optimizing over p and A We restrict p to have real coefficients and A to be real, in Nonsmooth Analysis of the Crouzeix Hessenberg form (all but one superdiagonal is zero). Ratio We have obtained similar results for p with complex coefficients Concluding Remarks and complex A (then can take A to be triangular) Subsequent slides show the sorted final values of the Crouzeix ratio after running BFGS for a maximum of 1000 iterations from each of 100 randomly generated starting points. 19 / 50

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , Optimize over A order n Fix p , Optimize over A Final Fields of Values for Lowest Computed f Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 20 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ z n − 1 , Optimize over A order n n=3 n=4 n=5 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of the Crouzeix Ratio f 0.7 0.7 0.7 Fix p , Optimize over 0.6 0.6 0.6 A 0.5 0.5 0.5 Fix p ( z ) ≡ z n − 1 , Optimize over A 0.4 0.4 0.4 0 50 100 0 50 100 0 50 100 order n Final Fields of Values for Lowest n=6 n=7 n=8 Computed f 1.1 1.1 1.1 Is the Ratio 0 . 5 1 1 1 Attained? A Local Minimizer 0.9 0.9 0.9 with f = 1 0.8 0.8 0.8 Why is the Crouzeix Ratio One? 0.7 0.7 0.7 Fix p ( z ) ≡ z n , 0.6 0.6 0.6 Optimize over A with order n 0.5 0.5 0.5 Fix p ( z ) ≡ ( z − 0.4 0.4 0.4 1)( z − 2)( z 2 + 1) , 0 50 100 0 50 100 0 50 100 Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Apparently 0.5, 1 and a few other values are all locally minimal Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 21 / 50 (* n = 4 *) Is the Ratio 0 . 5

Final Fields of Values for Lowest Computed f n=3 n=4 n=5 Crouzeix’s Conjecture 3 1 1.5 2 1 Nonsmooth 0.5 1 0.5 Optimization of 0 0 0 the Crouzeix Ratio f -0.5 -1 -0.5 Fix p , Optimize over -1 -2 A -1 -1.5 -3 Fix p ( z ) ≡ z n − 1 , -1 -0.5 0 0.5 1 -2 0 2 -1 0 1 Optimize over A order n Final Fields of Values for Lowest Computed f n=6 n=7 n=8 Is the Ratio 0 . 5 4 Attained? 8 6 3 6 A Local Minimizer 4 2 4 with f = 1 2 1 2 Why is the Crouzeix 0 0 0 Ratio One? -2 -1 -2 Fix p ( z ) ≡ z n , -4 -2 -4 -6 Optimize over A -3 -8 -6 with order n -4 -4 -2 0 2 4 -5 0 5 -5 0 5 Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 22 / 50 (* n = 4 *) Is the Ratio 0 . 5

Final Fields of Values for Lowest Computed f n=3 n=4 n=5 Crouzeix’s Conjecture 3 1 1.5 2 1 Nonsmooth 0.5 1 0.5 Optimization of 0 0 0 the Crouzeix Ratio f -0.5 -1 -0.5 Fix p , Optimize over -1 -2 A -1 -1.5 -3 Fix p ( z ) ≡ z n − 1 , -1 -0.5 0 0.5 1 -2 0 2 -1 0 1 Optimize over A order n Final Fields of Values for Lowest Computed f n=6 n=7 n=8 Is the Ratio 0 . 5 4 Attained? 8 6 3 6 A Local Minimizer 4 2 4 with f = 1 2 1 2 Why is the Crouzeix 0 0 0 Ratio One? -2 -1 -2 Fix p ( z ) ≡ z n , -4 -2 -4 -6 Optimize over A -3 -8 -6 with order n -4 -4 -2 0 2 4 -5 0 5 -5 0 5 Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Note: eigs( A ) → 0 = root( p ) . Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 22 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , Optimize over A order n Final Fields of Values for Lowest Computed f Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 23 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Independently, Crouzeix and Choi showed that the ratio 0 . 5 is attained if p ( z ) = z m and A is the m + 1 by m + 1 matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio f   · ·   � � 0 2 Fix p , Optimize over   if m = 1 , or · · if m > 1 A   0 0 Fix p ( z ) ≡ z n − 1 ,   · 1 √   Optimize over A   order n · 2   Final Fields of 0 Values for Lowest Computed f for which W ( A ) is the unit disk. We call this the C -matrix . Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 23 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Independently, Crouzeix and Choi showed that the ratio 0 . 5 is attained if p ( z ) = z m and A is the m + 1 by m + 1 matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio f   · ·   � � 0 2 Fix p , Optimize over   if m = 1 , or · · if m > 1 A   0 0 Fix p ( z ) ≡ z n − 1 ,   · 1 √   Optimize over A   order n · 2   Final Fields of 0 Values for Lowest Computed f for which W ( A ) is the unit disk. We call this the C -matrix . Is the Ratio 0 . 5 Attained? A Local Minimizer We conjecture that, when p ( z ) = z m , this is essentially the only with f = 1 Why is the Crouzeix case where 0 . 5 can be attained ( A can be changed via unitary Ratio One? Fix p ( z ) ≡ z n , similarity transformations, multiplying A by a scalar, and Optimize over A appending another diagonal block whose field of values is with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , contained in that of the first block). Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 23 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Independently, Crouzeix and Choi showed that the ratio 0 . 5 is attained if p ( z ) = z m and A is the m + 1 by m + 1 matrix Crouzeix’s √ Conjecture   0 2 Nonsmooth Optimization of · 1   the Crouzeix Ratio f   · ·   � � 0 2 Fix p , Optimize over   if m = 1 , or · · if m > 1 A   0 0 Fix p ( z ) ≡ z n − 1 ,   · 1 √   Optimize over A   order n · 2   Final Fields of 0 Values for Lowest Computed f for which W ( A ) is the unit disk. We call this the C -matrix . Is the Ratio 0 . 5 Attained? A Local Minimizer We conjecture that, when p ( z ) = z m , this is essentially the only with f = 1 Why is the Crouzeix case where 0 . 5 can be attained ( A can be changed via unitary Ratio One? Fix p ( z ) ≡ z n , similarity transformations, multiplying A by a scalar, and Optimize over A appending another diagonal block whose field of values is with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , contained in that of the first block). Opt. over A ( n = 5 ) We base this on the computation of the generalized null space Is the Ratio 0 . 5 decomposition (staircase decomposition) of the computed A . Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , (Kublanovskaya 1966, Ruhe 1970, Golub-Wilkinson 1976, Van Dooren 1979, Opt. over A 23 / 50 K˚ agstr¨ om et al, Edelman-Ma 2000, Guglielmi-Overton-Stewart 2015.) (* n = 4 *) Is the Ratio 0 . 5

A Local Minimizer with f = 1 A smooth local min, p(z)=z 4 (fixed), dim(A) = 5 (var), ratio = 9.999999999999996e−01 Crouzeix’s W(A) Conjecture 2.5 eigenvalues(A) Nonsmooth roots(p) 2 Optimization of the Crouzeix Ratio f 1.5 Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , 1 Optimize over A order n 0.5 Final Fields of Values for Lowest 0 Computed f Is the Ratio 0 . 5 −0.5 Attained? A Local Minimizer with f = 1 −1 Why is the Crouzeix Ratio One? −1.5 Fix p ( z ) ≡ z n , Optimize over A with order n −2 Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , −2.5 Opt. over A ( n = 5 ) −1 0 1 2 3 4 5 Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 24 / 50 (* n = 4 *) Is the Ratio 0 . 5

A Local Minimizer with f = 1 A smooth local min, p(z)=z 4 (fixed), dim(A) = 5 (var), ratio = 9.999999999999996e−01 Crouzeix’s W(A) Conjecture 2.5 eigenvalues(A) Nonsmooth roots(p) 2 Optimization of the Crouzeix Ratio f 1.5 Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , 1 Optimize over A order n 0.5 Final Fields of Values for Lowest 0 Computed f Is the Ratio 0 . 5 −0.5 Attained? A Local Minimizer with f = 1 −1 Why is the Crouzeix Ratio One? −1.5 Fix p ( z ) ≡ z n , Optimize over A with order n −2 Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , −2.5 Opt. over A ( n = 5 ) −1 0 1 2 3 4 5 Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , “Ice cream cone” shape Opt. over A 24 / 50 (* n = 4 *) Is the Ratio 0 . 5

Why is the Crouzeix Ratio One? Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , Optimize over A order n Final Fields of Values for Lowest Computed f Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 25 / 50 (* n = 4 *) Is the Ratio 0 . 5

Why is the Crouzeix Ratio One? Because for this computed local minimizer, Crouzeix’s A = U diag( λ, B ) U ∗ + E Conjecture Nonsmooth Optimization of with U unitary, � E � very small and λ ∈ R , so the Crouzeix Ratio f Fix p , Optimize over A W ( A ) ≈ conv( λ, W ( B )) Fix p ( z ) ≡ z n − 1 , Optimize over A order n with λ active and the block B inactive , that is: Final Fields of Values for Lowest Computed f � p � W ( A ) is attained only at λ Is the Ratio 0 . 5 ■ Attained? | p ( λ ) | > � p ( B ) � 2 A Local Minimizer ■ with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 25 / 50 (* n = 4 *) Is the Ratio 0 . 5

Why is the Crouzeix Ratio One? Because for this computed local minimizer, Crouzeix’s A = U diag( λ, B ) U ∗ + E Conjecture Nonsmooth Optimization of with U unitary, � E � very small and λ ∈ R , so the Crouzeix Ratio f Fix p , Optimize over A W ( A ) ≈ conv( λ, W ( B )) Fix p ( z ) ≡ z n − 1 , Optimize over A order n with λ active and the block B inactive , that is: Final Fields of Values for Lowest Computed f � p � W ( A ) is attained only at λ Is the Ratio 0 . 5 ■ Attained? | p ( λ ) | > � p ( B ) � 2 A Local Minimizer ■ with f = 1 Why is the Crouzeix So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 25 / 50 (* n = 4 *) Is the Ratio 0 . 5

Why is the Crouzeix Ratio One? Because for this computed local minimizer, Crouzeix’s A = U diag( λ, B ) U ∗ + E Conjecture Nonsmooth Optimization of with U unitary, � E � very small and λ ∈ R , so the Crouzeix Ratio f Fix p , Optimize over A W ( A ) ≈ conv( λ, W ( B )) Fix p ( z ) ≡ z n − 1 , Optimize over A order n with λ active and the block B inactive , that is: Final Fields of Values for Lowest Computed f � p � W ( A ) is attained only at λ Is the Ratio 0 . 5 ■ Attained? | p ( λ ) | > � p ( B ) � 2 A Local Minimizer ■ with f = 1 Why is the Crouzeix So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . Ratio One? Fix p ( z ) ≡ z n , Furthermore, the gradient of f ( p, · ) is zero at such A , although Optimize over A with order n showing this is more work. Thus, such A is a smooth stationary Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , point of f ( z m , · ) . Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 25 / 50 (* n = 4 *) Is the Ratio 0 . 5

Why is the Crouzeix Ratio One? Because for this computed local minimizer, Crouzeix’s A = U diag( λ, B ) U ∗ + E Conjecture Nonsmooth Optimization of with U unitary, � E � very small and λ ∈ R , so the Crouzeix Ratio f Fix p , Optimize over A W ( A ) ≈ conv( λ, W ( B )) Fix p ( z ) ≡ z n − 1 , Optimize over A order n with λ active and the block B inactive , that is: Final Fields of Values for Lowest Computed f � p � W ( A ) is attained only at λ Is the Ratio 0 . 5 ■ Attained? | p ( λ ) | > � p ( B ) � 2 A Local Minimizer ■ with f = 1 Why is the Crouzeix So, � p � W ( A ) = | p ( λ ) | = � p ( A ) � 2 and hence f ( p, A ) = 1 . Ratio One? Fix p ( z ) ≡ z n , Furthermore, the gradient of f ( p, · ) is zero at such A , although Optimize over A with order n showing this is more work. Thus, such A is a smooth stationary Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , point of f ( z m , · ) . Opt. over A ( n = 5 ) This doesn’t imply that it is a local minimizer, but the numerical Is the Ratio 0 . 5 Attained? results make this evident. Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 25 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ z n , Optimize over A with order n n=3 n=4 n=5 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of the Crouzeix Ratio f 0.7 0.7 0.7 Fix p , Optimize over 0.6 0.6 0.6 A 0.5 0.5 0.5 Fix p ( z ) ≡ z n − 1 , 0.4 0.4 0.4 Optimize over A 0 50 100 0 50 100 0 50 100 order n Final Fields of Values for Lowest n=6 n=7 n=8 Computed f 1.1 1.1 1.1 Is the Ratio 0 . 5 1 1 1 Attained? A Local Minimizer 0.9 0.9 0.9 with f = 1 0.8 0.8 0.8 Why is the Crouzeix 0.7 0.7 0.7 Ratio One? Fix p ( z ) ≡ z n , 0.6 0.6 0.6 Optimize over A 0.5 0.5 0.5 with order n Fix p ( z ) ≡ ( z − 0.4 0.4 0.4 1)( z − 2)( z 2 + 1) , 0 50 100 0 50 100 0 50 100 Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 26 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ z n , Optimize over A with order n n=3 n=4 n=5 1.1 1.1 1.1 Crouzeix’s 1 1 1 Conjecture 0.9 0.9 0.9 Nonsmooth 0.8 0.8 0.8 Optimization of the Crouzeix Ratio f 0.7 0.7 0.7 Fix p , Optimize over 0.6 0.6 0.6 A 0.5 0.5 0.5 Fix p ( z ) ≡ z n − 1 , 0.4 0.4 0.4 Optimize over A 0 50 100 0 50 100 0 50 100 order n Final Fields of Values for Lowest n=6 n=7 n=8 Computed f 1.1 1.1 1.1 Is the Ratio 0 . 5 1 1 1 Attained? A Local Minimizer 0.9 0.9 0.9 with f = 1 0.8 0.8 0.8 Why is the Crouzeix 0.7 0.7 0.7 Ratio One? Fix p ( z ) ≡ z n , 0.6 0.6 0.6 Optimize over A 0.5 0.5 0.5 with order n Fix p ( z ) ≡ ( z − 0.4 0.4 0.4 1)( z − 2)( z 2 + 1) , 0 50 100 0 50 100 0 50 100 Opt. over A ( n = 5 ) Is the Ratio 0 . 5 No value found near 0.5. We conjecture this is not possible. Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 26 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Best solution found, p(z)=z(z−1)(z 2 +1) (fixed), dim(A) = 5 (var), ratio = 5.000003853159926e−01 Crouzeix’s W(A) 8 Conjecture eigenvalues(A) Nonsmooth roots(p) Optimization of 6 the Crouzeix Ratio f Fix p , Optimize over 4 A Fix p ( z ) ≡ z n − 1 , Optimize over A 2 order n Final Fields of Values for Lowest 0 Computed f Is the Ratio 0 . 5 Attained? −2 A Local Minimizer with f = 1 Why is the Crouzeix −4 Ratio One? Fix p ( z ) ≡ z n , Optimize over A −6 with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , −8 Opt. over A ( n = 5 ) −10 −5 0 5 10 Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 27 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Best solution found, p(z)=z(z−1)(z 2 +1) (fixed), dim(A) = 5 (var), ratio = 5.000003853159926e−01 Crouzeix’s W(A) 8 Conjecture eigenvalues(A) Nonsmooth roots(p) Optimization of 6 the Crouzeix Ratio f Fix p , Optimize over 4 A Fix p ( z ) ≡ z n − 1 , Optimize over A 2 order n Final Fields of Values for Lowest 0 Computed f Is the Ratio 0 . 5 Attained? −2 A Local Minimizer with f = 1 Why is the Crouzeix −4 Ratio One? Fix p ( z ) ≡ z n , Optimize over A −6 with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , −8 Opt. over A ( n = 5 ) −10 −5 0 5 10 Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , W ( A ) is approximately a large disk around all roots of p Opt. over A 27 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s At first we thought so: we computed f ( p, A ) agreeing to 0.5 to Conjecture six digits. Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , Optimize over A order n Final Fields of Values for Lowest Computed f Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 28 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s At first we thought so: we computed f ( p, A ) agreeing to 0.5 to Conjecture six digits. Nonsmooth Optimization of the Crouzeix Ratio f However, the closer we get to 0.5, the more W ( A ) blows up. So, Fix p , Optimize over 0.5 is not attained. A Fix p ( z ) ≡ z n − 1 , Optimize over A order n Final Fields of Values for Lowest Computed f Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 28 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s At first we thought so: we computed f ( p, A ) agreeing to 0.5 to Conjecture six digits. Nonsmooth Optimization of the Crouzeix Ratio f However, the closer we get to 0.5, the more W ( A ) blows up. So, Fix p , Optimize over 0.5 is not attained. A Fix p ( z ) ≡ z n − 1 , Optimize over A Theorem 1. For any fixed polynomial p of degree m ≥ 1 , there order n exists a divergent sequence { A ( k ) } of order n = m + 1 for which Final Fields of Values for Lowest f ( p, A ( k ) ) → 0 . 5 as k → ∞ . Furthermore, we can choose A ( k ) Computed f Is the Ratio 0 . 5 so { W ( A ( k ) ) } are disks. Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 28 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s At first we thought so: we computed f ( p, A ) agreeing to 0.5 to Conjecture six digits. Nonsmooth Optimization of the Crouzeix Ratio f However, the closer we get to 0.5, the more W ( A ) blows up. So, Fix p , Optimize over 0.5 is not attained. A Fix p ( z ) ≡ z n − 1 , Optimize over A Theorem 1. For any fixed polynomial p of degree m ≥ 1 , there order n exists a divergent sequence { A ( k ) } of order n = m + 1 for which Final Fields of Values for Lowest f ( p, A ( k ) ) → 0 . 5 as k → ∞ . Furthermore, we can choose A ( k ) Computed f Is the Ratio 0 . 5 so { W ( A ( k ) ) } are disks. Attained? A Local Minimizer with f = 1 However, 0.5 is not attained. Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 28 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s At first we thought so: we computed f ( p, A ) agreeing to 0.5 to Conjecture six digits. Nonsmooth Optimization of the Crouzeix Ratio f However, the closer we get to 0.5, the more W ( A ) blows up. So, Fix p , Optimize over 0.5 is not attained. A Fix p ( z ) ≡ z n − 1 , Optimize over A Theorem 1. For any fixed polynomial p of degree m ≥ 1 , there order n exists a divergent sequence { A ( k ) } of order n = m + 1 for which Final Fields of Values for Lowest f ( p, A ( k ) ) → 0 . 5 as k → ∞ . Furthermore, we can choose A ( k ) Computed f Is the Ratio 0 . 5 so { W ( A ( k ) ) } are disks. Attained? A Local Minimizer with f = 1 However, 0.5 is not attained. Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Observation. When we fix p to be any polynomial of degree m Optimize over A with order n except a monomial, and we optimize over ( m + 1) × ( m + 1) Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , matrices A , we frequently generate a sequence as described in Opt. over A Theorem 1, except that W ( A ) are not exactly disks. ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 28 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A (* n = 4 *) Best solution found, p(z)=z(z−1)(z 2 +1) (fixed), dim(A) = 4 (var), ratio = 5.000198002813829e−01 1 Crouzeix’s W(A) Conjecture eigenvalues(A) 0.8 Nonsmooth roots(p) Optimization of the Crouzeix Ratio f 0.6 Fix p , Optimize over A 0.4 Fix p ( z ) ≡ z n − 1 , Optimize over A 0.2 order n Final Fields of Values for Lowest 0 Computed f Is the Ratio 0 . 5 Attained? −0.2 A Local Minimizer with f = 1 Why is the Crouzeix −0.4 Ratio One? Fix p ( z ) ≡ z n , −0.6 Optimize over A with order n Fix p ( z ) ≡ ( z − −0.8 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) −1 0 0.5 1 1.5 2 Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 29 / 50 (* n = 4 *) Is the Ratio 0 . 5

Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A (* n = 4 *) Best solution found, p(z)=z(z−1)(z 2 +1) (fixed), dim(A) = 4 (var), ratio = 5.000198002813829e−01 1 Crouzeix’s W(A) Conjecture eigenvalues(A) 0.8 Nonsmooth roots(p) Optimization of the Crouzeix Ratio f 0.6 Fix p , Optimize over A 0.4 Fix p ( z ) ≡ z n − 1 , Optimize over A 0.2 order n Final Fields of Values for Lowest 0 Computed f Is the Ratio 0 . 5 Attained? −0.2 A Local Minimizer with f = 1 Why is the Crouzeix −0.4 Ratio One? Fix p ( z ) ≡ z n , −0.6 Optimize over A with order n Fix p ( z ) ≡ ( z − −0.8 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) −1 0 0.5 1 1.5 2 Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , W ( A ) is approximately a tiny disk around some root of p Opt. over A 29 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s No. The closer we get to 0.5, the more W ( A ) shrinks to a point. Conjecture In the limit, we get f ( p, A ) = 0 / 0 so it is not defined. Nonsmooth Optimization of the Crouzeix Ratio f Fix p , Optimize over A Fix p ( z ) ≡ z n − 1 , Optimize over A order n Final Fields of Values for Lowest Computed f Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 30 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s No. The closer we get to 0.5, the more W ( A ) shrinks to a point. Conjecture In the limit, we get f ( p, A ) = 0 / 0 so it is not defined. Nonsmooth Optimization of the Crouzeix Ratio f Theorem 2. Fix p to have degree m with at least two distinct Fix p , Optimize over roots. Then, for all integers n with 2 ≤ n ≤ m , there exists a A Fix p ( z ) ≡ z n − 1 , convergent sequence of n × n matrices { A ( k ) } for which the Optimize over A Crouzeix ratio f ( p, A ( k ) ) → 0 . 5 . Furthermore, we can choose order n Final Fields of A ( k ) so { W ( A ( k ) ) } is a sequence of disks shrinking to a root of Values for Lowest Computed f p . Is the Ratio 0 . 5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 30 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s No. The closer we get to 0.5, the more W ( A ) shrinks to a point. Conjecture In the limit, we get f ( p, A ) = 0 / 0 so it is not defined. Nonsmooth Optimization of the Crouzeix Ratio f Theorem 2. Fix p to have degree m with at least two distinct Fix p , Optimize over roots. Then, for all integers n with 2 ≤ n ≤ m , there exists a A Fix p ( z ) ≡ z n − 1 , convergent sequence of n × n matrices { A ( k ) } for which the Optimize over A Crouzeix ratio f ( p, A ( k ) ) → 0 . 5 . Furthermore, we can choose order n Final Fields of A ( k ) so { W ( A ( k ) ) } is a sequence of disks shrinking to a root of Values for Lowest Computed f p . Is the Ratio 0 . 5 Attained? A Local Minimizer However, 0.5 is not attained. with f = 1 Why is the Crouzeix Ratio One? Fix p ( z ) ≡ z n , Optimize over A with order n Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 30 / 50 (* n = 4 *) Is the Ratio 0 . 5

Is the Ratio 0 . 5 Attained? Crouzeix’s No. The closer we get to 0.5, the more W ( A ) shrinks to a point. Conjecture In the limit, we get f ( p, A ) = 0 / 0 so it is not defined. Nonsmooth Optimization of the Crouzeix Ratio f Theorem 2. Fix p to have degree m with at least two distinct Fix p , Optimize over roots. Then, for all integers n with 2 ≤ n ≤ m , there exists a A Fix p ( z ) ≡ z n − 1 , convergent sequence of n × n matrices { A ( k ) } for which the Optimize over A Crouzeix ratio f ( p, A ( k ) ) → 0 . 5 . Furthermore, we can choose order n Final Fields of A ( k ) so { W ( A ( k ) ) } is a sequence of disks shrinking to a root of Values for Lowest Computed f p . Is the Ratio 0 . 5 Attained? A Local Minimizer However, 0.5 is not attained. with f = 1 Why is the Crouzeix Ratio One? Observation. When we fix p to be any polynomial of degree Fix p ( z ) ≡ z n , Optimize over A m > 1 with at least two roots, and we optimize over m × m with order n Fix p ( z ) ≡ ( z − matrices A , we sometimes generate a sequence as described in 1)( z − 2)( z 2 + 1) , Opt. over A Theorem 2, except that W ( A ) are not exactly disks. ( n = 5 ) Is the Ratio 0 . 5 Attained? Fix p ( z ) ≡ ( z − 1)( z − 2)( z 2 + 1) , Opt. over A 30 / 50 (* n = 4 *) Is the Ratio 0 . 5