chebop-eigenvalue-problems

# Chebop Eigenvalue Problems ## Basic Eigenvalue Computation Compute eigenvalues and eigenfunctions for linear operators. ### Basic Usage ```matlab [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 ```matlab % 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 ```matlab % 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 ```matlab 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 ```matlab % 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 ```matlab % 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 ```matlab % 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 ```matlab % 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