ShaneLogic/chebop-eigenvalue-problems

Chebop Eigenvalue Problems

Basic Eigenvalue Computation

Compute eigenvalues and eigenfunctions for linear operators.

Basic Usage

[V, D] = eigs(L, k)

Parameters

• L: Linear chebop operator
• k: Number of eigenvalues to compute (optional, default: 6)
• V: Quasimatrix of eigenfunctions
• D: Diagonal matrix of eigenvalues

Default Behavior

• Finds 6 eigenvalues if k not specified
• Targets eigenmodes that are 'most readily converged to' (approximately the smoothest ones)

Example

% Define operator: u'' = λu
L = chebop(-1, 1);
L.op = @(x,u) diff(u, 2);
L.bc = 'dirichlet';  % u(-1)=u(1)=0

% Compute 6 eigenvalues
[V, D] = eigs(L, 6);

% Plot eigenfunctions
plot(V)
legend('Mode 1', 'Mode 2', 'Mode 3', 'Mode 4', 'Mode 5', 'Mode 6')

Generalized Eigenvalue Problems

Solve Au = λBu for two operators.

Setup

% Define two linear chebops
A = chebop(-1, 1);
B = chebop(-1, 1);

% Attach ALL boundary conditions to operator A
A.op = @(x,u) diff(u, 2);
A.bc = 'dirichlet';

B.op = @(x,u) u;
% B has no boundary conditions

% Compute generalized eigenvalues
[V, D] = eigs(A, B, k);

Important Rule

Attach all boundary conditions to operator A, not B.

Periodic Problems

L = chebop(-pi, pi);
L.op = @(x,u) diff(u, 2);
L.bc = 'periodic';  % Set periodic boundary conditions

[V, D] = eigs(L, k);

Block Operators and Systems of Equations

Handle coupled variables using block operators.

Define Block Operator

% System: u' = v, v' = -u
L = chebop(0, 2*pi);

% Define operator acting on [u; v]
L.op = @(x, u, v) [diff(u) - v; diff(v) + u];

% Boundary conditions
L.lbc = @(u, v) [u - 1; v];  % u(0)=1, v(0)=0

Solve System

% Solve boundary value problem
U = L \ 0;

% Extract individual variables
u = U(:,1);
v = U(:,2);

% Plot both variables
plot(U)
legend('u', 'v')

Eigenvalue Problems for Systems

% Define block operator for eigenvalue problem
L = chebop(0, 2*pi);
L.op = @(x, u, v) [diff(u) - v; diff(v) + u];
L.bc = 'periodic';

% Compute eigenvalues
[V, D] = eigs(L, k);

% V is quasimatrix with 2 columns per eigenmode
% First column: u component
% Second column: v component

Visualization

% Visualize operator structure
spy(L)

% Shows global vs local dependencies in the operator

Constraints and Limitations

1. Cannot find 'all' eigenvalues: Operators have infinite spectrum
2. Unreasonable requests may fail: Asking for largest eigenvalue of differential operator may not converge
3. Smoothest modes: Default behavior targets smoothest eigenmodes

Output Interpretation

Standard Eigenvalue Problem

• D: Diagonal matrix of eigenvalues λ₁, λ₂, ..., λₖ
• V: Quasimatrix where column j is eigenfunction corresponding to λⱼ

System of Equations

• U: ∞ × n quasimatrix (Chebfun with multiple columns)
• U(:,j): Solution for variable j

Generalized Eigenvalue Problem

• Same structure as standard problem
• Eigenvalues satisfy det(A - λB) = 0

When to Use

• eigs(L, k): Find eigenvalues/eigenfunctions of linear operator
• eigs(A, B, k): Solve generalized eigenvalue problem Au = λBu
• Block operators: Handle systems of coupled ODEs
• Periodic problems: Set L.bc = 'periodic'
• spy(L): Visualize operator structure for systems