Arnoldi iteration is an important numerical method used in linear algebra for approximating the eigenvalues and eigenvectors of a large, sparse matrix. It is particularly useful for solving problems in fields such as scientific computing, quantum mechanics, and engineering, where one may encounter large systems that cannot be solved directly due to computational limitations. ### Overview The Arnoldi iteration algorithm builds an orthonormal basis for the Krylov subspace generated by the matrix in question.
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Starting with , the Arnoldi iteration constructs an orthonormal basis of successive Krylov subspaces. At step , orthogonalize against , and normalize the remainder as if it is nonzero. The coefficients form an upper Hessenberg matrix and satisfy . A zero remainder means the current Krylov subspace is invariant under .