Abstract
The indefinite inner product defined by J=diagj1,…,jn, jk∈−1,+1, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite iterative methods: indefinite Arnoldi’s method, indefinite Lanczos method (ILM), and indefinite full orthogonalization method (IFOM). The indefinite Arnoldi’s method is introduced as a process that constructs a J-orthonormal basis for the nondegenerated Krylov subspace. The ILM method is introduced as a special case of the indefinite Arnoldi’s method for J-Hermitian matrices. IFOM is mentioned as a process for solving linear systems of equations with J-Hermitian coefficient matrices. Finally, by providing numerical examples, the FOM, IFOM, and ILM processes have been compared with each other in terms of the required time for solving linear systems and also from the point of the number of iterations.
Highlights
Nowadays, iterative methods are used extensively for solving general large sparse linear systems in many areas of scientific computing because they are easier to implement efficiently on high-performance computers than direct methods
We introduce three iterative methods in the space with hyperbolic inner product. ese methods are indefinite Arnoldi, indefinite Lanczos (ILM), and indefinite full orthogonalization (IFOM), and we define new algorithms to run these hyperbolic versions
E advantage of indefinite Lanczos method (ILM) is that it solves some classes of linear systems with different coefficient matrices, for different choices of matrix J. e following examples explain more in which we use n instead of m
Summary
Iterative methods are used extensively for solving general large sparse linear systems in many areas of scientific computing because they are easier to implement efficiently on high-performance computers than direct methods. Projection methods for solving systems of linear equations have been known for some time. 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 methods which built up Krylov subspaces look for good approximations to eigenvectors. Ese methods are indefinite Arnoldi, indefinite Lanczos (ILM), and indefinite full orthogonalization (IFOM), and we define new algorithms to run these hyperbolic versions. We will compare these indefinite algorithms with their common definite modes, from the point of the number of iterations and the required time to run the algorithms. Counting the arithmetic act of multiplication in FOM, IFOM, and ILM algorithms and conclusion are the last two sections, respectively
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