TABLE OF CONTENTS Arnoldi's method [9] is an orthogonal projection method onto 
for general non-Hermitian matrices. The procedure was first introduced in 1951 as a means of reducing a dense matrix into Hessenberg form with a unitary transformation. In his paper, Arnoldi hinted that the eigenvalues of the Hessenberg matrix obtained from a number of steps smaller than n could provide accurate approximations to some eigenvalues of the original matrix. It was later discovered that this strategy leads to an efficient technique for approximating eigenvalues of large sparse matrices, which was then extended to the solution of large sparse linear systems of equations. The method will first be described theoretically, i.e., assuming exact arithmetic, then implementation details will be addressed.
Arnoldi's procedure is an algorithm for building an orthogonal basis of the Krylov subspace 
. In exact arithmetic, one variant of the algorithm is as follows.
Choose a vector v 1 such that ?? 1 ? 2 = 1
For j = 1, 2, , m, Do
Compute h ij = ( A ? j, ? i) for i = 1, 2, , j
Compute w j := A ? j ? ? j i = 1 h ij ? i
h j + 1, j = ? w j ? 2
If h j + 1, j = 0 then Stop
? j+ 1
