Arnoldi Nonsymmetric Krylov Hessenberg
Orthogonalize a nonsymmetric Krylov sequence and see when its Hessenberg projection closes exactly.
Free theory articleOrthogonalize a nonsymmetric Krylov sequence and see when its Hessenberg projection closes exactly.
Concept & interpretation
The Arnoldi iteration is an algorithm for computing an orthogonal basis of the Krylov subspace generated by a matrix A and a vector v. Unlike the symmetric Lanczos process, which produces a tridiagonal matrix, the Arnoldi process handles general, possibly nonnormal matrices and produces an upper Hessenberg matrix H.
experiment
You can interactively explore the Arnoldi algorithm using theArnoldi Nonsymmetric Krylov Hessenberg module.
- Set the Matrix control to Nonnormal and the Start vector to ones.
- Set the Steps control to 2 to observe the dimension of the subspace and the upper Hessenberg matrix elements.
- Change the Matrix control to Shift and the Start vector to e1. Observe the Happy Breakdown field change to Yes at step 1.