Abstract
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.
1. 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 , , 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 [1–6]. 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 . 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.
2. Indefinite Arnoldi and Lanczos Algorithms
Let be the -dimensional space consisting of all column vectors with complex coordinates , . The standard inner product in is denoted by . Thus where and and the bar denotes the complex conjugation. Remember that a function from to is called an indefinite inner product in if the following axioms are satisfied: (i)Linearity in the first argument: for all and all complex numbers , .(ii)Antisymmetry:
Example 1. Let , and define where is a permutation for which and . It is easy to see that is an indefinite inner product. Indeed, properties are clearly confirmed. But for nondegeneracy property, if for all , then in special case, for , where by choosing an appropriate permutation .
Thus,and this yields that , for . So, .
By [3], we know that, for every invertible Hermitian matrix , the formula determines an indefinite inner product on and conversely for every indefinite inner product on 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 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 wherein is the number of and is the number of . This is because if then and conversely if then it is clear that , for all .
Note. For , consider the relation .
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 subspacewhere is matrix and ’s are vectors, is shown below: (1)Choose a vector such that (2)Define (3)For Do:(4)For Do:(5)Compute and (6)Compute (7)(8)if then stop(9)(10)EndDo(11)EndDo
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 form a -orthonormal basis of the Krylov subspace .
Proof. Consider the following expression: Certainly that vectors are -orthonormal.
Proposition 4. Define (i), the Hessenberg matrix whose nonzero entries are defined by indefinite Arnoldi’s algorithm,(ii), the matrix with column vectors ,(iii). Then the following relations are valid: In particular, if then
Proof. By considering lines , , and of indefinite Arnoldi’s algorithm, relation (13) is verified straightforwardly. Indeed, in general, (13) is the matrix representation of (12): Now, to see (14), left-multiply the relation (16) in . We earn On the other hand, given that the vectors build a -orthonormal basis, then, we have the following. (1)According to the definition of , or it is orthogonal to , that is, , for . Thus, (2)We have In other words, Therefore, relation (17) can be summarized as follows: and, by left-multiplying in , Particularly, if , relation (19) yields that 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 [7–10]. 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.
The indefinite version of the Hermitian Lanczos algorithm can be formulated as follows:(1)Choose an initial vector such that (2)Set (3)For Do:(4)(5)(6)(7)if then stop(8)(9)(10)EndDo
3. Indefinite Iterative Arnoldi’s Method for Eigenvalue Problem
Definition 6. Suppose that , and are the same as those in Proposition (8), let be an eigenpair of ,
is called Ritz value of and provides an approximation for eigenvalue of .
is called Ritz vector of and provides an approximation for eigenvector of .
Proposition 7. Let be an eigenvector of associated with the eigenvalue and the Ritz approximate eigenvector, so that . Then,
Proof. Let and . We have
One of way to circumvent the difficulty is to restart the algorithm. After a run with Arnoldi and Lanczos’ vectors, we compute the approximate eigenvector and use it as an initial vector for the next run with Arnoldi and Lanczos’ methods. This process, which is the simplest of this kind, is iterated to convergence. The iterative indefinite Arnoldi algorithm can be formulated as follows:(1)Start: choose an initial vector and a dimension .(2)Iterate: perform steps of indefinite Arnoldi’s algorithm.(3)Restart: compute the approximate eigenvector associated with the rightmost eigenvalue . If satisfied stop; else set and go to .
Of course, it should be noted that this algorithm is not beneficial for us, because in any case it is better to use the definite algorithm.
In the following we have the same iterative indefinite Lanczos algorithm which can be formulated as follows:(1)Start: choose an initial vector and a dimension .(2)Iterate: perform steps of indefinite Lanczos’ algorithm.(3)Restart: compute the approximate eigenvector associated with the rightmost eigenvalue . If satisfied stop; else set and go to (2).
This algorithm is used for -symmetric matrices, since the number of flops in the performance of algorithm is certainly less than that of the Arnoldi algorithm. This is illustrated by the examples in the next section.
4. Numerical Example
In this section, we present numerical examples and compare the methods modified by the Ritz approximation in the Arnoldi and indefinite Lanczos process.
Algorithms are written with personal computer using MATLAB software and the following examples are done using these codes.
Example 1. Consider the matrices and , where are diagonal matrices with random elements in and also , are tridiagonal matrices with random elements in such that and is vector with random entries in .
Let ; in this case , . The results are in Table 1.
Example 2. Consider the matriceswhere is identity matrix and in the form such thatwherein , , are diagonal matrices with random elements (entries) in .
In this case, (i)if , the results are in Table 2; (ii)if ; the results are in Table 3.
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.
5. 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.
Conflicts of Interest
The authors declare that they have no conflicts of interest.