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Krylov subspace methods foreigenvalue problems
David S. Watkins
watkins@math.wsu.edu
Department of Mathematics
Washington State University
October 16, 2008 – p.
Problem: Linear Elasticity
October 16, 2008 – p.
Problem: Linear ElasticityElastic Deformation(3D, anisotropic, composite materials)
October 16, 2008 – p.
Problem: Linear ElasticityElastic Deformation(3D, anisotropic, composite materials)
Singularities at cracks, interfaces
October 16, 2008 – p.
Problem: Linear ElasticityElastic Deformation(3D, anisotropic, composite materials)
Singularities at cracks, interfaces
Lamé Equations (spherical coordinates)
October 16, 2008 – p.
Problem: Linear ElasticityElastic Deformation(3D, anisotropic, composite materials)
Singularities at cracks, interfaces
Lamé Equations (spherical coordinates)
Separate radial variable.
October 16, 2008 – p.
Problem: Linear ElasticityElastic Deformation(3D, anisotropic, composite materials)
Singularities at cracks, interfaces
Lamé Equations (spherical coordinates)
Separate radial variable.
Get quadratic eigenvalue problem.
(λ2M + λG+K)v = 0
M∗ = M > 0 G∗ = −G K∗ = K < 0
October 16, 2008 – p.
Linear Elasticity, ContinuedDiscretize θ and ϕ variables.
October 16, 2008 – p.
Linear Elasticity, ContinuedDiscretize θ and ϕ variables.(finite element method)
October 16, 2008 – p.
Linear Elasticity, ContinuedDiscretize θ and ϕ variables.(finite element method)
(λ2M + λG+K)v = 0
MT = M > 0 GT = −G KT = K < 0
matrix quadratic eigenvalue problem(large, sparse)
October 16, 2008 – p.
Linear Elasticity, ContinuedDiscretize θ and ϕ variables.(finite element method)
(λ2M + λG+K)v = 0
MT = M > 0 GT = −G KT = K < 0
matrix quadratic eigenvalue problem(large, sparse)
Find few smallest eigenvalues (and correspondingeigenvectors).
October 16, 2008 – p.
Linear Elasticity, ContinuedDiscretize θ and ϕ variables.(finite element method)
(λ2M + λG+K)v = 0
MT = M > 0 GT = −G KT = K < 0
matrix quadratic eigenvalue problem(large, sparse)
Find few smallest eigenvalues (and correspondingeigenvectors).
Respect the structure. (symmetric/skew-symmetric)
October 16, 2008 – p.
Hamiltonian Structure
October 16, 2008 – p.
Hamiltonian Structure
October 16, 2008 – p.
Reduction to First Order
October 16, 2008 – p.
Reduction to First Orderλ2Mv + λGv +Kv = 0
October 16, 2008 – p.
Reduction to First Orderλ2Mv + λGv +Kv = 0
w = λv,
October 16, 2008 – p.
Reduction to First Orderλ2Mv + λGv +Kv = 0
w = λv, Mw = λMv
October 16, 2008 – p.
Reduction to First Orderλ2Mv + λGv +Kv = 0
w = λv, Mw = λMv[
−K 0
0 −M
][
v
w
]
− λ
[
G M
−M 0
][
v
w
]
= 0
October 16, 2008 – p.
Reduction to First Orderλ2Mv + λGv +Kv = 0
w = λv, Mw = λMv[
−K 0
0 −M
][
v
w
]
− λ
[
G M
−M 0
][
v
w
]
= 0
Ax− λBx = 0
October 16, 2008 – p.
Reduction to First Orderλ2Mv + λGv +Kv = 0
w = λv, Mw = λMv[
−K 0
0 −M
][
v
w
]
− λ
[
G M
−M 0
][
v
w
]
= 0
Ax− λBx = 0
symmetric/skew-symmetric
October 16, 2008 – p.
Reduction to Hamiltonian Matrix
October 16, 2008 – p.
Reduction to Hamiltonian MatrixA− λB (symmetric/skew-symmetric)
October 16, 2008 – p.
Reduction to Hamiltonian MatrixA− λB (symmetric/skew-symmetric)
B = RTJR
(
J =
[
0 I
−I 0
])
sometimes easy, always possible
October 16, 2008 – p.
Reduction to Hamiltonian MatrixA− λB (symmetric/skew-symmetric)
B = RTJR
(
J =
[
0 I
−I 0
])
sometimes easy, always possible
A− λRTJR
October 16, 2008 – p.
Reduction to Hamiltonian MatrixA− λB (symmetric/skew-symmetric)
B = RTJR
(
J =
[
0 I
−I 0
])
sometimes easy, always possible
A− λRTJR
R−TAR−1 − λJ
October 16, 2008 – p.
Reduction to Hamiltonian MatrixA− λB (symmetric/skew-symmetric)
B = RTJR
(
J =
[
0 I
−I 0
])
sometimes easy, always possible
A− λRTJR
R−TAR−1 − λJ
JTR−TAR−1 − λI
October 16, 2008 – p.
Reduction to Hamiltonian MatrixA− λB (symmetric/skew-symmetric)
B = RTJR
(
J =
[
0 I
−I 0
])
sometimes easy, always possible
A− λRTJR
R−TAR−1 − λJ
JTR−TAR−1 − λI
H = JTR−TAR−1 is Hamiltonian.
October 16, 2008 – p.
in our case . . .
October 16, 2008 – p.
in our case . . .B =
[
G M
−M 0
]
October 16, 2008 – p.
in our case . . .B =
[
G M
−M 0
]
B = RTJR =
[
I −1
2G
0 M
][
0 I
−I 0
][
I 01
2G M
]
October 16, 2008 – p.
in our case . . .B =
[
G M
−M 0
]
B = RTJR =
[
I −1
2G
0 M
][
0 I
−I 0
][
I 01
2G M
]
H =
[
I 0
−1
2G I
][
0 M−1
−K 0
][
I 0
−1
2G I
]
October 16, 2008 – p.
in our case . . .B =
[
G M
−M 0
]
B = RTJR =
[
I −1
2G
0 M
][
0 I
−I 0
][
I 01
2G M
]
H =
[
I 0
−1
2G I
][
0 M−1
−K 0
][
I 0
−1
2G I
]
Do not compute H explicitly.
October 16, 2008 – p.
in our case . . .B =
[
G M
−M 0
]
B = RTJR =
[
I −1
2G
0 M
][
0 I
−I 0
][
I 01
2G M
]
H =
[
I 0
−1
2G I
][
0 M−1
−K 0
][
I 0
−1
2G I
]
Do not compute H explicitly. (nor M−1)
October 16, 2008 – p.
Working with H
October 16, 2008 – p.
Working with HKrylov subspace methods
October 16, 2008 – p.
Working with HKrylov subspace methods
just need to apply the operator:
October 16, 2008 – p.
Working with HKrylov subspace methods
just need to apply the operator: x 7→ Hx
October 16, 2008 – p.
Working with HKrylov subspace methods
just need to apply the operator: x 7→ Hx
H =
[
I 0
−1
2G I
][
0 M−1
−K 0
][
I 0
−1
2G I
]
October 16, 2008 – p.
Working with HKrylov subspace methods
just need to apply the operator: x 7→ Hx
H =
[
I 0
−1
2G I
][
0 M−1
−K 0
][
I 0
−1
2G I
]
October 16, 2008 – p.
Working with HKrylov subspace methods
just need to apply the operator: x 7→ Hx
H =
[
I 0
−1
2G I
][
0 M−1
−K 0
][
I 0
−1
2G I
]
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
October 16, 2008 – p.
Working with H−1
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
x 7→ H−1x
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
x 7→ H−1x
Do not compute (−K)−1
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
x 7→ H−1x
Do not compute (−K)−1
Cholesky decomposition:
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
x 7→ H−1x
Do not compute (−K)−1
Cholesky decomposition: (−K) = RTR
To compute w = −K−1v,
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
x 7→ H−1x
Do not compute (−K)−1
Cholesky decomposition: (−K) = RTR
To compute w = −K−1v, Solve (−K)w = v.
October 16, 2008 – p.
Working with H−1
H−1 =
[
I 01
2G I
][
0 (−K)−1
M 0
][
I 01
2G I
]
x 7→ H−1x
Do not compute (−K)−1
Cholesky decomposition: (−K) = RTR
To compute w = −K−1v, Solve (−K)w = v.
RTRw = v Backsolve!
October 16, 2008 – p.
K =
October 16, 2008 – p. 10
K =
0 20 40 60 80 100 120 140
0
20
40
60
80
100
120
140
nz = 670
October 16, 2008 – p. 10
R =
October 16, 2008 – p. 11
R =
0 20 40 60 80 100 120 140
0
20
40
60
80
100
120
140
nz = 896
October 16, 2008 – p. 11
K−1=
October 16, 2008 – p. 12
K−1=
0 20 40 60 80 100 120 140
0
20
40
60
80
100
120
140
nz = 21316
October 16, 2008 – p. 12
Second Application
October 16, 2008 – p. 13
Second ApplicationNonlinear Optics
October 16, 2008 – p. 13
Second ApplicationNonlinear Optics
Schrödinger eigenvalue problem
October 16, 2008 – p. 13
Second ApplicationNonlinear Optics
Schrödinger eigenvalue problem
−~2
2m∇2ψ + V ψ = λψ
October 16, 2008 – p. 13
Second ApplicationNonlinear Optics
Schrödinger eigenvalue problem
−~2
2m∇2ψ + V ψ = λψ
Solve numerically (finite elements)
October 16, 2008 – p. 13
Second ApplicationNonlinear Optics
Schrödinger eigenvalue problem
−~2
2m∇2ψ + V ψ = λψ
Solve numerically (finite elements)
Kv = λMv K = KT > 0, M = MT > 0
October 16, 2008 – p. 13
Second ApplicationNonlinear Optics
Schrödinger eigenvalue problem
−~2
2m∇2ψ + V ψ = λψ
Solve numerically (finite elements)
Kv = λMv K = KT > 0, M = MT > 0
Matrices are large and sparse.
October 16, 2008 – p. 13
Kv = λMv
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR RTRv = λMv
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR RTRv = λMv
R−TMR−1(Rv) = λ−1(Rv)
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR RTRv = λMv
R−TMR−1(Rv) = λ−1(Rv)
A = R−TMR−1, AT = A > 0
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR RTRv = λMv
R−TMR−1(Rv) = λ−1(Rv)
A = R−TMR−1, AT = A > 0
x 7→ Ax
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR RTRv = λMv
R−TMR−1(Rv) = λ−1(Rv)
A = R−TMR−1, AT = A > 0
x 7→ Ax backsolve
October 16, 2008 – p. 14
Kv = λMv
Want few smallest eigenvaluesand associated eigenvectors.
Invert the problem.
K = RTR RTRv = λMv
R−TMR−1(Rv) = λ−1(Rv)
A = R−TMR−1, AT = A > 0
x 7→ Ax backsolve
Do not form A explicitly.
October 16, 2008 – p. 14
What our applications have incommon
October 16, 2008 – p. 15
What our applications have incommon
large, sparse matrices
October 16, 2008 – p. 15
What our applications have incommon
large, sparse matrices
use of matrix factorization (Cholesky decomposition)
October 16, 2008 – p. 15
What our applications have incommon
large, sparse matrices
use of matrix factorization (Cholesky decomposition)
some kind of structure
October 16, 2008 – p. 15
What our applications have incommon
large, sparse matrices
use of matrix factorization (Cholesky decomposition)
some kind of structure
. . . not enough time to discuss this
October 16, 2008 – p. 15
Classificationof Eigenvalue Problems
October 16, 2008 – p. 16
Classificationof Eigenvalue Problems
small
October 16, 2008 – p. 16
Classificationof Eigenvalue Problems
small
medium
October 16, 2008 – p. 16
Classificationof Eigenvalue Problems
small
medium
large
October 16, 2008 – p. 16
Small Matrices
October 16, 2008 – p. 17
Small Matricesstore conventionally
October 16, 2008 – p. 17
Small Matricesstore conventionally
similarity transformations
October 16, 2008 – p. 17
Small Matricesstore conventionally
similarity transformations
QR algorithm
October 16, 2008 – p. 17
Small Matricesstore conventionally
similarity transformations
QR algorithm
get all eigenvalues/vectors
October 16, 2008 – p. 17
Small Matricesstore conventionally
similarity transformations
QR algorithm
get all eigenvalues/vectors
n ≈ 103
October 16, 2008 – p. 17
Medium Matrices
October 16, 2008 – p. 18
Medium Matricesstore as sparse matrix
October 16, 2008 – p. 18
Medium Matricesstore as sparse matrix
no similarity transformations
October 16, 2008 – p. 18
Medium Matricesstore as sparse matrix
no similarity transformations
matrix factorization okay
October 16, 2008 – p. 18
Medium Matricesstore as sparse matrix
no similarity transformations
matrix factorization okay
shift and invert
October 16, 2008 – p. 18
0 20 40 60 80 100 120 140
0
20
40
60
80
100
120
140
nz = 670
October 16, 2008 – p. 19
0 20 40 60 80 100 120 140
0
20
40
60
80
100
120
140
nz = 1815
October 16, 2008 – p. 20
Medium Matrices, Continued
October 16, 2008 – p. 21
Medium Matrices, Continuedstore matrix factor as sparse matrix
October 16, 2008 – p. 21
Medium Matrices, Continuedstore matrix factor as sparse matrix
n ≈ 105
October 16, 2008 – p. 21
Medium Matrices, Continuedstore matrix factor as sparse matrix
n ≈ 105
get selected eigenvalues/vectors
October 16, 2008 – p. 21
Medium Matrices, Continuedstore matrix factor as sparse matrix
n ≈ 105
get selected eigenvalues/vectors
Krylov subspace methods
October 16, 2008 – p. 21
Medium Matrices, Continuedstore matrix factor as sparse matrix
n ≈ 105
get selected eigenvalues/vectors
Krylov subspace methods
Jacobi-Davidson methods
October 16, 2008 – p. 21
Large Matrices
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
no similarity transformations
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
no similarity transformations
no shift-and-invert
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
no similarity transformations
no shift-and-invert
n ≈ 107
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
no similarity transformations
no shift-and-invert
n ≈ 107
get selected eigenvalues/vectors
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
no similarity transformations
no shift-and-invert
n ≈ 107
get selected eigenvalues/vectors
Krylov subspace methods
October 16, 2008 – p. 22
Large Matricesstore as sparse matrix
no similarity transformations
no shift-and-invert
n ≈ 107
get selected eigenvalues/vectors
Krylov subspace methods
Jacobi-Davidson methods
October 16, 2008 – p. 22
Krylov Subspace Methods
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
Example: A = R−TMR−1
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
Example: A = R−TMR−1
Pick a vector v
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
Example: A = R−TMR−1
Pick a vector v
v, Av, A2v, A3v, . . .
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
Example: A = R−TMR−1
Pick a vector v
v, Av, A2v, A3v, . . .
Krylov subspace:
Kj(A, v) = span{
v,Av,A2v, . . . , Aj−1v}
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
Example: A = R−TMR−1
Pick a vector v
v, Av, A2v, A3v, . . .
Krylov subspace:
Kj(A, v) = span{
v,Av,A2v, . . . , Aj−1v}
Look in here for approximate eigenvectors.
October 16, 2008 – p. 23
Krylov Subspace Methodsx 7→ Ax
Example: A = R−TMR−1
Pick a vector v
v, Av, A2v, A3v, . . .
Krylov subspace:
Kj(A, v) = span{
v,Av,A2v, . . . , Aj−1v}
Look in here for approximate eigenvectors.
. . . but need better basis
October 16, 2008 – p. 23
Arnoldi Process
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
jth step: v̂j+1 = Avj −∑j
i=1vihij
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
jth step: v̂j+1 = Avj −∑j
i=1vihij
hij = 〈Avj , vi〉 (Gram-Schmidt)
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
jth step: v̂j+1 = Avj −∑j
i=1vihij
hij = 〈Avj , vi〉 (Gram-Schmidt)
Normalization: hj+1,j = ‖ v̂j+1 ‖, vj+1 = v̂j+1/hj+1,j
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
jth step: v̂j+1 = Avj −∑j
i=1vihij
hij = 〈Avj , vi〉 (Gram-Schmidt)
Normalization: hj+1,j = ‖ v̂j+1 ‖, vj+1 = v̂j+1/hj+1,j
Collect coefficients hij
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
jth step: v̂j+1 = Avj −∑j
i=1vihij
hij = 〈Avj , vi〉 (Gram-Schmidt)
Normalization: hj+1,j = ‖ v̂j+1 ‖, vj+1 = v̂j+1/hj+1,j
Collect coefficients hij
H4 =
h11 h12 h13 h14
h21 h22 h23 h24
h32 h33 h34
h43 h44
October 16, 2008 – p. 24
Arnoldi Processbuild orthonormal basis v1, v2, v3, . . .
jth step: v̂j+1 = Avj −∑j
i=1vihij
hij = 〈Avj , vi〉 (Gram-Schmidt)
Normalization: hj+1,j = ‖ v̂j+1 ‖, vj+1 = v̂j+1/hj+1,j
Collect coefficients hij
H4 =
h11 h12 h13 h14
h21 h22 h23 h24
h32 h33 h34
h43 h44
eigenvalues are Ritz values
October 16, 2008 – p. 24
Example: 479 × 479 matrix
−150 −100 −50 0 50 100 150−2000
−1500
−1000
−500
0
500
1000
1500
2000
October 16, 2008 – p. 25
Example: 479 × 479 matrix
−150 −100 −50 0 50 100 150−2000
−1500
−1000
−500
0
500
1000
1500
2000
October 16, 2008 – p. 26
Example: 479 × 479 matrix
−150 −100 −50 0 50 100 150−2000
−1500
−1000
−500
0
500
1000
1500
2000
October 16, 2008 – p. 27
Example: 479 × 479 matrix
−150 −100 −50 0 50 100 150−2000
−1500
−1000
−500
0
500
1000
1500
2000
October 16, 2008 – p. 28
For better accuracy . . .
October 16, 2008 – p. 29
For better accuracy . . .
. . . take more steps.
October 16, 2008 – p. 29
For better accuracy . . .
. . . take more steps.
but,
October 16, 2008 – p. 29
For better accuracy . . .
. . . take more steps.
but, vectors take up space.
October 16, 2008 – p. 29
For better accuracy . . .
. . . take more steps.
but, vectors take up space.
Alternate plan:
October 16, 2008 – p. 29
For better accuracy . . .
. . . take more steps.
but, vectors take up space.
Alternate plan:
Start over with a better vector.
October 16, 2008 – p. 29
Implicit Restarts
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
Get 30 Ritz values.
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
Get 30 Ritz values.
Keep the best ones (e.g. 10) . . .
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
Get 30 Ritz values.
Keep the best ones (e.g. 10) . . .
. . . and associated invariant subspace.
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
Get 30 Ritz values.
Keep the best ones (e.g. 10) . . .
. . . and associated invariant subspace.
Restart at step 11.
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
Get 30 Ritz values.
Keep the best ones (e.g. 10) . . .
. . . and associated invariant subspace.
Restart at step 11.
(neat details omitted)
October 16, 2008 – p. 30
Implicit RestartsTake, say, 30 steps.
Get 30 Ritz values.
Keep the best ones (e.g. 10) . . .
. . . and associated invariant subspace.
Restart at step 11.
(neat details omitted)
Build back up to 30 and then restart again.
October 16, 2008 – p. 30
Example
October 16, 2008 – p. 31
Example460 × 460 Hamiltonian matrix (toy problem)
October 16, 2008 – p. 31
Example460 × 460 Hamiltonian matrix (toy problem)
asking for 24 eigenpairs
October 16, 2008 – p. 31
Example460 × 460 Hamiltonian matrix (toy problem)
asking for 24 eigenpairs
building to dimension 84
October 16, 2008 – p. 31
Example460 × 460 Hamiltonian matrix (toy problem)
asking for 24 eigenpairs
building to dimension 84
restarting with 28
October 16, 2008 – p. 31
Example460 × 460 Hamiltonian matrix (toy problem)
asking for 24 eigenpairs
building to dimension 84
restarting with 28
Hamiltonian Lanczos process
October 16, 2008 – p. 31
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 32
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 33
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 34
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 35
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 36
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 37
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 38
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 39
−3 −2 −1 0 1 2 3
−3
−2
−1
0
1
2
3
October 16, 2008 – p. 40
Concluding Remarks
October 16, 2008 – p. 41
Concluding RemarksI used these methods to solve my “medium” sizedeigenvalue problems.
October 16, 2008 – p. 41
Concluding RemarksI used these methods to solve my “medium” sizedeigenvalue problems.
Simple ideas lead to powerful methods.
October 16, 2008 – p. 41
Concluding RemarksI used these methods to solve my “medium” sizedeigenvalue problems.
Simple ideas lead to powerful methods.
Thank you for your attention.
October 16, 2008 – p. 41
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