Basic Todaism Carlos Tomei, PUC-Rio Geometry, Dynamics and - - PowerPoint PPT Presentation

Basic Todaism

Carlos Tomei, PUC-Rio

Geometry, Dynamics and Mechanics Seminar Università degli Studi di Padova, 8/2020

An amazing vector field

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum.

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} .

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} . Q∗MQ = RMR−1

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} . Q∗MQ = RMR−1 ⇒ QRM = MQR

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} . Q∗MQ = RMR−1 ⇒ QRM = MQR ⇒ g(M) = QR = [g(M)]Q[g(M)]R

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} . Q∗MQ = RMR−1 ⇒ QRM = MQR ⇒ g(M) = QR = [g(M)]Q[g(M)]R Write g(M) = exp f(M) = [exp f(M)]Q [exp f(M)]R and then S = {[exp f(M)]∗

Q M [exp f(M)]Q , f ∼ g ⇔ f = g + const}. Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} . Q∗MQ = RMR−1 ⇒ QRM = MQR ⇒ g(M) = QR = [g(M)]Q[g(M)]R Write g(M) = exp f(M) = [exp f(M)]Q [exp f(M)]R and then S = {[exp f(M)]∗

Q M [exp f(M)]Q , f ∼ g ⇔ f = g + const}.

The flows M(t) = [exp(t f(M))]∗

Q M [exp(t f(M))]Q solve

M′ = [M, Πskewf(M)], where Πskew + Πupper = I,

Cartan might have discovered the Toda flow like this. 1/10

An amazing vector field

Let M be n × n real, positive, with simple spectrum. Consider the intersection of two orbits, I = {Q∗MQ, Q ∈ SO(n)} ∩ {RMR−1, R ∈ Up+(n)} . Q∗MQ = RMR−1 ⇒ QRM = MQR ⇒ g(M) = QR = [g(M)]Q[g(M)]R Write g(M) = exp f(M) = [exp f(M)]Q [exp f(M)]R and then S = {[exp f(M)]∗

Q M [exp f(M)]Q , f ∼ g ⇔ f = g + const}.

The flows M(t) = [exp(t f(M))]∗

Q M [exp(t f(M))]Q solve

M′ = [M, Πskewf(M)], where Πskew + Πupper = I,

• r, equivalently, M(t) = [exp(t f(M))]R M [exp(t f(M))]−1

R

solve M′ = [M, −Πupperf(M)].

Cartan might have discovered the Toda flow like this. 1/10

Another missed opportunity: the QR method

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0.

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR,

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ.

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T).

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T). Nothing seems to happen: iterate.

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T). Nothing seems to happen: iterate. Amazingly, φn(T) → D.

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T). Nothing seems to happen: iterate. Amazingly, φn(T) → D. Numerical analysts knew that Tk = φk(T) = [T k]∗

QT[T k]Q. The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T). Nothing seems to happen: iterate. Amazingly, φn(T) → D. Numerical analysts knew that Tk = φk(T) = [T k]∗

QT[T k]Q.

Take k ∈ R – an action of R. For T = T0, lim

k→0

1 k (Tk − T0) = [Πskew log T, T] ,

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T). Nothing seems to happen: iterate. Amazingly, φn(T) → D. Numerical analysts knew that Tk = φk(T) = [T k]∗

QT[T k]Q.

Take k ∈ R – an action of R. For T = T0, lim

k→0

1 k (Tk − T0) = [Πskew log T, T] , a Toda flow is born.

The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

Another missed opportunity: the QR method

Following Rutishauser, Francis had a great idea to compute σ(T), T > 0. Factor T = [T]Q[T]R = QR, and define the QR step T = QR → φ(T) = RQ. As φ(T) = RQ = Q∗QRQ = Q∗TQ, σ(φ(T)) = σ(T). Nothing seems to happen: iterate. Amazingly, φn(T) → D. Numerical analysts knew that Tk = φk(T) = [T k]∗

QT[T k]Q.

Take k ∈ R – an action of R. For T = T0, lim

k→0

1 k (Tk − T0) = [Πskew log T, T] , a Toda flow is born. For t → ∞, [T k]Q → Q∞, where T0 = Q∗

∞ΛQ∞ (the power method). The fifties and early sixties. The Toda flow is from the late sixties, seventies. 2/10

The physical track – a completely integrable system

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) ,

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) .

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

)

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

) and cleverly arrange them in matrices, J =     a1 b1 b1 a2 b2 b2 a3 b3 b3 a4     ,

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

) and cleverly arrange them in matrices, J =     a1 b1 b1 a2 b2 b2 a3 b3 b3 a4     , ΠskewJ =     −b1 b1 −b2 b2 −b3 b3    

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

) and cleverly arrange them in matrices, J =     a1 b1 b1 a2 b2 b2 a3 b3 b3 a4     , ΠskewJ =     −b1 b1 −b2 b2 −b3 b3     J′ = [J, ΠskewJ], a Lax pair.

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

) and cleverly arrange them in matrices, J =     a1 b1 b1 a2 b2 b2 a3 b3 b3 a4     , ΠskewJ =     −b1 b1 −b2 b2 −b3 b3     J′ = [J, ΠskewJ], a Lax pair. The two conjugations preserve spectrum, symmetry, profile.

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

) and cleverly arrange them in matrices, J =     a1 b1 b1 a2 b2 b2 a3 b3 b3 a4     , ΠskewJ =     −b1 b1 −b2 b2 −b3 b3     J′ = [J, ΠskewJ], a Lax pair. The two conjugations preserve spectrum, symmetry, profile. (Adler 79) Jacobis of zero trace form a coadjoint orbit.

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

The physical track – a completely integrable system

(Toda 67) For n particles in R, take H(x, y) = 1 2

n

• k=1

y 2

k + n−1

• k=1

exp(xk − xk+1) , x′

k = ∂H

∂yk = yk, y ′

k = − ∂H

∂xk = exp(xk−1 − xk) − exp(xk − xk+1) . (Flaschka 74) Change variables, ak = −yk/2 , bk = 1

2 exp( xk −xk+1 2

) and cleverly arrange them in matrices, J =     a1 b1 b1 a2 b2 b2 a3 b3 b3 a4     , ΠskewJ =     −b1 b1 −b2 b2 −b3 b3     J′ = [J, ΠskewJ], a Lax pair. The two conjugations preserve spectrum, symmetry, profile. (Adler 79) Jacobis of zero trace form a coadjoint orbit. The vector fields induced by Hi(x, y) = λi(a, b) commute.

(Lax 68) The celebrated KdV equation is a Lax pair. For S(t) = −D2

x + Mu(x, t), KdV is S′ = [Πf(S), S].

3/10

Scattering and inverse variables (Moser 79)

GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1)

GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1) For fixed spectra, the resulting sets are Liouville tori JΛ ≃ Rn−1,

GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1) For fixed spectra, the resulting sets are Liouville tori JΛ ≃ Rn−1, parameterized by inverse variables (norming constants) JΛ ∼ = {c = (c1 > 0, . . . , cn > 0)}. The c′

ks are the first coordinates of the eigenvectors. GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1) For fixed spectra, the resulting sets are Liouville tori JΛ ≃ Rn−1, parameterized by inverse variables (norming constants) JΛ ∼ = {c = (c1 > 0, . . . , cn > 0)}. The c′

ks are the first coordinates of the eigenvectors.

Under Toda, λ′

i = 0 and c(t) = exp(tΛ)c(0)/ exp(tΛ)c(0). GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1) For fixed spectra, the resulting sets are Liouville tori JΛ ≃ Rn−1, parameterized by inverse variables (norming constants) JΛ ∼ = {c = (c1 > 0, . . . , cn > 0)}. The c′

ks are the first coordinates of the eigenvectors.

Under Toda, λ′

i = 0 and c(t) = exp(tΛ)c(0)/ exp(tΛ)c(0).

(Symes 82, Deift, Nanda, T. 83) For standard QR, Jn = J(n), where J′ = [J, Πskew log J], J(0) = J0 .

GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1) For fixed spectra, the resulting sets are Liouville tori JΛ ≃ Rn−1, parameterized by inverse variables (norming constants) JΛ ∼ = {c = (c1 > 0, . . . , cn > 0)}. The c′

ks are the first coordinates of the eigenvectors.

Under Toda, λ′

i = 0 and c(t) = exp(tΛ)c(0)/ exp(tΛ)c(0).

(Symes 82, Deift, Nanda, T. 83) For standard QR, Jn = J(n), where J′ = [J, Πskew log J], J(0) = J0 . More generally, the solution of J′ = [J, Πskew f(J)], J(0) = J0 is J(t) = [exp(f(J0))]∗

Q J(0) [exp(f(J0))]Q . GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Scattering and inverse variables (Moser 79)

diag(λ1 < . . . < λn)

−∞

← − J(t)

∞

− → diag(λn > . . . > λ1) For fixed spectra, the resulting sets are Liouville tori JΛ ≃ Rn−1, parameterized by inverse variables (norming constants) JΛ ∼ = {c = (c1 > 0, . . . , cn > 0)}. The c′

ks are the first coordinates of the eigenvectors.

Under Toda, λ′

i = 0 and c(t) = exp(tΛ)c(0)/ exp(tΛ)c(0).

(Symes 82, Deift, Nanda, T. 83) For standard QR, Jn = J(n), where J′ = [J, Πskew log J], J(0) = J0 . More generally, the solution of J′ = [J, Πskew f(J)], J(0) = J0 is J(t) = [exp(f(J0))]∗

Q J(0) [exp(f(J0))]Q .

A Hamiltonian flow with limit points...

GGKM, AKNS, Lax, Fadeev... Reyman, Semenov-Tian-Shansky, Adler, Kostant... 4/10

Shifts

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !).

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators:

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n. Rayleigh’s shift strategy generates periodic orbits.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n. Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n. Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n. Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. (Leite, Saldanha, T., 10) If Λ has no three eigenvalues in arithmetic progression, Wilkinson iteration leads to cubic convergence of Jn,n−1. Otherwise, there may be a Cantor-like set in which iteration is quadratic.

According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n. Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. (Leite, Saldanha, T., 10) If Λ has no three eigenvalues in arithmetic progression, Wilkinson iteration leads to cubic convergence of Jn,n−1. Otherwise, there may be a Cantor-like set in which iteration is quadratic. And remember: |λ − Jn,n| = O(J2

n,n−1) ! According to Parlett, there are shifts for all seasons. 5/10

Shifts

Algorithms are usually performed on Jacobi matrices (drop the signs of the (k, k + 1)-entries of J and the spectrum does not change !). Shifts are remarkable accelerators: J(0) − sI = QR → J(1) = Q∗ J(0) Q = R J(0) R−1. Rayleigh would take s to be Jn,n. For Wilkinson, s is the eigenvalue of the bottom 2 × 2 block closer to Jn,n. Rayleigh’s shift strategy generates periodic orbits. Wilkinson’s does not. (Leite, Saldanha, T., 12) No continuous shift strategy yields an always convergent iteration. (Leite, Saldanha, T., 10) If Λ has no three eigenvalues in arithmetic progression, Wilkinson iteration leads to cubic convergence of Jn,n−1. Otherwise, there may be a Cantor-like set in which iteration is quadratic. And remember: |λ − Jn,n| = O(J2

n,n−1) ! Deflate ! According to Parlett, there are shifts for all seasons. 5/10

The isospectral manifold TΛ

The eye wants its part (Italian proverb). 6/10

The isospectral manifold TΛ

The eye wants its part (Italian proverb). 6/10

The isospectral manifold TΛ

The eye wants its part (Italian proverb). 6/10

The isospectral manifold TΛ

(5,7,4) (5,4,7) (4,5,7) (4,7,5) (7,5,4) (4,5,7) (5,4,7) (7,5,4) (5,7,4) (5,4,7) (4,5,7) (4,7,5) (4,5,7) (5,4,7) (5,7,4)

g a c c a g h h

(7,4,5) (5,7,4)

−+ ++ −− +−

b b f f d e d e

The eye wants its part (Italian proverb). 6/10

The isospectral manifold TΛ

(5,7,4) (5,4,7) (4,5,7) (4,7,5) (7,5,4) (4,5,7) (5,4,7) (7,5,4) (5,7,4) (5,4,7) (4,5,7) (4,7,5) (4,5,7) (5,4,7) (5,7,4)

g a c c a g h h

(7,4,5) (5,7,4)

−+ ++ −− +−

b b f f d e d e

(7,4,5) (4,7,5) (4,5,7) (5,4,7) (5,7,4) (7,5,4) The eye wants its part (Italian proverb). 6/10

The isospectral manifold TΛ

(5,7,4) (5,4,7) (4,5,7) (4,7,5) (7,5,4) (4,5,7) (5,4,7) (7,5,4) (5,7,4) (5,4,7) (4,5,7) (4,7,5) (4,5,7) (5,4,7) (5,7,4)

g a c c a g h h

(7,4,5) (5,7,4)

−+ ++ −− +−

b b f f d e d e

(7,4,5) (4,7,5) (4,5,7) (5,4,7) (5,7,4) (7,5,4)

Four hexagons, three (black) deflation components.

The eye wants its part (Italian proverb). 6/10

The isospectral manifold TΛ

(5,7,4) (5,4,7) (4,5,7) (4,7,5) (7,5,4) (4,5,7) (5,4,7) (7,5,4) (5,7,4) (5,4,7) (4,5,7) (4,7,5) (4,5,7) (5,4,7) (5,7,4)

g a c c a g h h

(7,4,5) (5,7,4)

−+ ++ −− +−

b b f f d e d e

(7,4,5) (4,7,5) (4,5,7) (5,4,7) (5,7,4) (7,5,4)

Four hexagons, three (black) deflation components. This is why continuous shift strategies are problematic.

The eye wants its part (Italian proverb). 6/10

TΛ — an isospectral manifold

A conformal diffeomorphic projection of TΛ ⊂ R5. 7/10

TΛ — an isospectral manifold

A conformal diffeomorphic projection of TΛ ⊂ R5. 7/10

TΛ — an isospectral manifold

Real, tridiagonal symmetric matrices with eigenvalues 4, 5 e 7.

A conformal diffeomorphic projection of TΛ ⊂ R5. 7/10

TΛ — an isospectral manifold

Real, tridiagonal symmetric matrices with eigenvalues 4, 5 e 7. For points in red, T2,1 = 0; black stands for T3,2 = 0.

A conformal diffeomorphic projection of TΛ ⊂ R5. 7/10

Some symplectic and topological properties of TΛ

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR. An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is Rn−1

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is Rn−1 (from results on Coxeter groups, Davis 83).

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is Rn−1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of TΛ.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is Rn−1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of TΛ. (Casian, Kodama, 01) Twisted isospectral manifolds.

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is Rn−1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of TΛ. (Casian, Kodama, 01) Twisted isospectral manifolds. What about charts?

An interesting object indeed. 8/10

Some symplectic and topological properties of TΛ

(T. 84) The closure of Jacobi matrices with fixed spectrum JΛ is a permutohedron P = conv {(λπ(1), . . . , λπ(n)), π ∈ Sn}. (Bloch, Flaschka, Ratiu 90) It is the image of a moment map. (Gibson, Saldanha, T) It is an ‘extreme projectivization’ of (0, ∞)n. In TΛ, h(T) = n

i=1 i Sii is a height function for Toda and QR.

The Wielandt-Hoffman theorem. (T. 84) The cohomology ring of TΛ has simple, explicit generators. (Fried 86) It is free. (T. 84) Its covering is Rn−1 (from results on Coxeter groups, Davis 83). (Gaifullin, 14) Any homology class of a manifold has a multiple which is the image of a finite covering of TΛ. (Casian, Kodama, 01) Twisted isospectral manifolds. What about charts? Larger isospectral manifolds?

An interesting object indeed. 8/10

Charts – bidiagonal variables

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ. Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg):

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ !

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1. Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1.

Simple evolutions: B′ = [B, f(Λ)]

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1.

Simple evolutions: B′ = [B, f(Λ)] , i.e.,

• new βπ

i

• = exp(f(λπ

i+1)/f(λπ i ))

• ld βπ

i

• .

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1.

Simple evolutions: B′ = [B, f(Λ)] , i.e.,

• new βπ

i

• = exp(f(λπ

i+1)/f(λπ i ))

• ld βπ

i

• .

Dropping signs (going Jacobi) is as bad as inserting absolute values.

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1.

Simple evolutions: B′ = [B, f(Λ)] , i.e.,

• new βπ

i

• = exp(f(λπ

i+1)/f(λπ i ))

• ld βπ

i

• .

Dropping signs (going Jacobi) is as bad as inserting absolute values. Limits now belong to the charts: asymptotics is local theory.

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1.

Simple evolutions: B′ = [B, f(Λ)] , i.e.,

• new βπ

i

• = exp(f(λπ

i+1)/f(λπ i ))

• ld βπ

i

• .

Dropping signs (going Jacobi) is as bad as inserting absolute values. Limits now belong to the charts: asymptotics is local theory. Bidiagonal variables are stabler than norming constants.

Way beyond integrability. 9/10

Charts – bidiagonal variables

(Leite, Saldanha, T. 08) For π ∈ Sn and T ∈ TΛ , write T = Q∗

π Λπ Qπ.

A matrix T ∈ U π

Λ ⊂ TΛ admits an LU factorization of Qπ,

Qπ = LπUπ: Lπ uni-lower and Uπ pos-upper, so that T = U−1

π

• L−1

π

Λπ Lπ

• Uπ.

Set Bπ = L−1

π ΛπLπ (lower) = UπTU−1 π

(upper Hessenberg): Bπ is lower bidiagonal with diagonal Λπ ! The entries βπ

j = (Bπ)j+1, j define a chart ψπ : U π Λ → Rn−1.

Simple evolutions: B′ = [B, f(Λ)] , i.e.,

• new βπ

i

• = exp(f(λπ

i+1)/f(λπ i ))

• ld βπ

i

• .

Dropping signs (going Jacobi) is as bad as inserting absolute values. Limits now belong to the charts: asymptotics is local theory. Bidiagonal variables are stabler than norming constants. This begs for a new inverse algorithm: Gragg and Harrod, ’84.

Way beyond integrability. 9/10

Larger matrices

Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices

(Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits.

Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices

(Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20)

Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices

(Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) A staircase induces a matrix profile p.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices

(Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) A staircase induces a matrix profile p.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

(Torres-T. 20) Charts extend to isospectral manifolds of given profile.

Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

Larger matrices

(Deift, Li, Nanda, T., 86, 89) Toda flows are completely integrable on symmetric and non-symmetric generic orbits. The Toda flow is Morse-Smale on isospectral matrices of full matrices (for sl(n), Shub-Vasquez 87; for other special cases, Chernyakov,Sharigin, Sorin 14, 17, 19; for non-compact semisimple lie algebras, Torres-T. 20) A staircase induces a matrix profile p.

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

(Torres-T. 20) Charts extend to isospectral manifolds of given profile.

Thank you !

Duistermaat, Kolk, Varadarajan, Shub, Vasquez... 10/10

QR steps commute

Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R,

Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q. Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

= [g(S) [f(S)]Q)]∗

Q S [g(S) [f(S)]Q]Q Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

= [g(S) [f(S)]Q)]∗

Q S [g(S) [f(S)]Q]Q = [g(S) f(S)]∗ Q S [g(S) f(S)]]Q Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

= [g(S) [f(S)]Q)]∗

Q S [g(S) [f(S)]Q]Q = [g(S) f(S)]∗ Q S [g(S) f(S)]]Q

= [gf(S)]∗

Q S [ gf(S)]]Q Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

= [g(S) [f(S)]Q)]∗

Q S [g(S) [f(S)]Q]Q = [g(S) f(S)]∗ Q S [g(S) f(S)]]Q

= [gf(S)]∗

Q S [ gf(S)]]Q = [fg(S)]∗ Q S [ fg(S)]]Q Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

= [g(S) [f(S)]Q)]∗

Q S [g(S) [f(S)]Q]Q = [g(S) f(S)]∗ Q S [g(S) f(S)]]Q

= [gf(S)]∗

Q S [ gf(S)]]Q = [fg(S)]∗ Q S [ fg(S)]]Q

The steps commute.

Giant steps. 11/10

QR steps commute

A general QR step: for f(S) = QR = [f(S)]Q [f(S)]R, ˜ S = [f(S)]∗

Q S [f(S)]Q.

Take two steps, S → [f(S)]∗

Q S [f(S)]Q

→ [g

• [f(S)]∗

Q S [f(S)]Q

• ]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [g
• [f(S)]∗

Q S [f(S)]Q

• ]Q

= [[f(S)]∗

Q g

• S
• [f(S)]Q)]∗

Q

• [f(S)]∗

Q S [f(S)]Q

• [[f(S)]∗

Q g

• S
• [f(S)]Q]Q

= [g

• S
• [f(S)]Q)]∗

Q [f(S)]Q[f(S)]∗ Q S [f(S)]Q[f(S)]∗ Q [g

• S
• [f(S)]Q]Q

= [g(S) [f(S)]Q)]∗

Q S [g(S) [f(S)]Q]Q = [g(S) f(S)]∗ Q S [g(S) f(S)]]Q

= [gf(S)]∗

Q S [ gf(S)]]Q = [fg(S)]∗ Q S [ fg(S)]]Q

The steps commute. An action of Rn.

Giant steps. 11/10

A few papers

Bloch, A. M., Flaschka, H. and Ratiu, T., A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra, Duke

• Math. J., 61, 41-65, 1990.

Deift, P ., Nanda, T., Tomei, C., Differential equations for the symmetric eigenvalue problem, SIAM J. Num. Anal. 20, 1-22, 1983. Flaschka, H., The Toda lattice, Phys. Rev. B 9, 1924-1925, 1974.

• W. Gragg e W. Harrod, The numerically stable reconstruction of Jacobi

matrices from spectral data, Numer. Math. 44, 317-335, 1984. Leite, R. S., Saldanha, N.C. and Tomei, C., An atlas for tridiagonal isospectral manifolds, Lin. Alg. Appl. 429, 387-402, 2008; The Asymptotics of Wilkinson’s Shift: Loss of Cubic Convergence, FoCM, 10, 15-36, 2010; Dynamics of the Symmetric Eigenvalue Problem with Shift Strategies, IMRN 2013, 4382-4412. Moser, J., Finitely many mass points on the line under the influence of an exponential potential, Lecture Notes in Phys. 38, 467-497, 1975. Parlett, B. N., The Symmetric Eigenvalue Problem, SIAM. Symes, W., The QR algorithm and scattering for the finite nonperiodic Toda lattice, Physica 4D, 275-280, 1982; Hamiltonian group actions and integrable systems, Physica 1D, 339-374, 1980.

Very few. */10