A Method of Indefinite Krylov Subspace for Eigenvalue Problem

We describe an indefinite state of Arnoldi’s method for solving the eigenvalues problems. In the following, we scrutinize the indefinite state of Lanczos’ method for solving the eigenvalue problems and we show that this method for the -Hermitian matrices works much better than Arnoldi’s method.


Introduction
Around the early 1950s the idea of Krylov subspace iteration was established by Cornelius Lanczos and Walter Arnoldi.Lanczos' method was based on two mutually orthogonal vector sequences and his motivation came from eigenvalue problems.In that context, the most prominent feature of the method is that it reduces the original matrix to tridiagonal form.Lanczos later applied his method to solve linear systems, in particular symmetric ones.Krylov subspace iterations or Krylov subspace methods are iterative methods which are used as linear system solvers and also iterative solvers of eigenvalue problems.
On the other hand, the indefinite inner product defined by  = diag( 1 , . . .,   ),   ∈ {−1,+1}, arises frequently in applications.It is used, for example, in the theory of relativity and in the research of the polarized light.More on the applications of such products can be found in [1][2][3][4][5][6].These applications in other fields of science inspired us to resume Lanczos, FOM, and Arnoldi's methods in this mentioned indefinite inner space.Indefinite Arnoldi's method is a process that constructs a -orthonormal basis for the nondegenerated Krylov subspace; the basis that we proved have a particular common property, about the structure of the product of their vectors.In the following, iterative Arnoldi's method for eigenvalue problem has been renovated in the mentioned indefinite inner product space and also a process is made which is useful for solving eigenvalue problem with -Hermitian coefficient matrices.Indefinite iterative Lanczos' method for eigenvalue problem is the same as Lanczos' method that has been restored in the indefinite inner product space.
This paper is organized as follows: In Section 2, we recall the standard inner product and indefinite inner product in C  .In the following part of this section we show some examples on indefinite inner product that will guide us to the objectives of this article.We express the indefinite Arnoldi algorithm for the computation of a -orthonormal basis of the Krylov subspace and describe property of the basis in the form of several propositions to review and in the end of this section we exhibit the indefinite version of the Hermitian Lanczos algorithm.
In Section 3, we present a definition of the Ritz eigenpair of the a matrix and new algorithms from the indefinite Arnoldi and Lanczos' algorithm that is named modified methods by the Ritz approximation in the indefinite Arnoldi and Lanczos' process.This is our main goal and initiative in this article.
Since the initial vectors are very important at the start of each iteration to the above algorithms, in this article, we consider the Ritz eigenvector instead of the common eigenvector in the start of the each repetition algorithm.In Section 4 we present numerical examples and compare the modified method by the Ritz approximation in the Arnoldi and indefinite Lanczos algorithm.Algorithms are written with personal computer using MATLAB software and we illustrate examples by using these codes.

Indefinite Arnoldi and Lanczos Algorithms
Let C  be the -dimensional space consisting of all column vectors  with complex coordinates   ,  = 1, 2, . . ., .The standard inner product in C  is denoted by (⋅, ⋅).Thus where  = ( 1 , . . .,   )  and  = ( 1 , . . .,   )  and the bar denotes the complex conjugation.Remember that a function [⋅, ⋅] from C  × C  to C is called an indefinite inner product in C  if the following axioms are satisfied: (i) Linearity in the first argument: for all  1 ,  2 ,  ∈ C  and all complex numbers , .
By [3], we know that, for every  ×  invertible Hermitian matrix , the formula determines an indefinite inner product on C  and conversely for every indefinite inner product [⋅, ⋅] on C  there exists an  ×  invertible and Hermitian matrix  such that (7) holds and the established correspondence [⋅, ⋅] ↔  is a bijection between the set of all indefinite inner products on C  and the set of all  ×  invertible Hermitian matrices.
Example 2. In Example (7), the corresponding nonsingular Hermitian matrix to that indefinite inner product is written in the form  = diag(±1) wherein  is the number of +1 and  −  is the number of −1.This is because if then and conversely if then it is clear that [, ] = (, ), for all ,  ∈ C  .
That the result becomes zero is a very special case and depends on how we choose the matrix .
Recently, new algorithms have been presented in this regard and have been compared in terms of the number of replays and the time required to run algorithms.
These methods could solve the linear systems of equations with -Hermitian coefficient matrices and also considered Arnoldi, FOM, and Lanczos' methods for solving linear system of equations and resumed these methods in a space that is equipped with a special indefinite inner product and proposed new algorithms to run these methods.The indefinite Arnoldi algorithm for the computation of a orthonormal basis of the Krylov subspace where  is  ×  matrix and V  's are  × 1 vectors, is shown below: (1) Choose a vector  such that [, ] ̸ = 0 For  = 1, . . .,  Do: (4) For  = 1, . . .,  Do: Here   is an upper Hessenberg matrix.
The following proposition expressed indefinite Arnoldi's algorithm.(You can see [7,8].)Proposition 3. Assume that the indefinite Arnoldi algorithm does not stop before the th step.Then the vectors V 1 , . . ., V  form a -orthonormal basis of the Krylov subspace K  (, V).
(1) According to the definition of (2) We have In other words, J  *    = .
Therefore, relation (17) can be summarized as follows: and, by left-multiplying in J , Particularly, if  = , relation (19) yields that J  *   =  −1  and thereby Remark 5.It should be noted that a permutation matrix  is available such That We know that the indefinite Hermitian Lanczos algorithm is a simplified version of the indefinite Arnoldi algorithm that is applied to -Hermitian matrices.The th step of the algorithm transforms matrix  into a Hessenberg and -Hermitian matrix, in other words, a tridiagonal matrix   ; when  is equal to the dimension of ,   is similar to .To see more, refer to [7][8][9][10].As can be seen in the following, this method is expressed as a special case of the indefinite Arnoldi method in the complex space for the special case when matrix  is -Hermitian.
It should be noted that obviously the number of external iterations depends on the number of internal iterations selected.The larger the  value is taken, the more the external iterations will decrease.

Conclusion
We know that the matrices of -Hermitian are important and functional matrices.We can find a way to find the eigenvalues of these types of matrices that are more efficient than Arnoldi's method.To do this, we first constructed an indefinite Arnoldi method in order to be able to also make that indefinite Lanczos method.We tried to study and compare the Arnoldi and indefinite Lanczos methods.It was observed that Arnoldi's method is not efficient in indefinite status, but the indefinite Lanczos method for -symmetric (Hermitian) matrices is much better than Arnoldi's method.