The Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, 2008) is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the eigenvalues and eigenvectors. Using the Ritz values, we propose a self-contained variant of the Ritz-IDR(s) method (Simoncini and Szyld, 2010) for solving a system of linear equations. In addition, the Ritz vectors are used to speed-up IDR(s) for the solution of sequence of systems of linear equations.

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently using the preconditioned IDR(s) method. We present an efficient and robust way to apply a single preconditioner using MSSS matrix computations. For 3D problems, we present a memory-efficient implementation that exploits the solution of a sequence of 2D problems. Realistic examples in two and three spatial dimensions demonstrate the performance of the new algorithm.

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations. This thesis is devoted to the development of new algorithms to solve this kind of problems. In particular, when the matrices involved are sparse and non-symmetric. The new algorithms are based on the Induced Dimension Reduction method [IDR(s)]. IDR(s) is a Krylov subspace method originally proposed in 2008 to solve systems of linear equations. IDR(s) has received considerable attention due to its stable and fast convergence. It is, therefore, natural to ask if it is possible to extend IDR(s) to solve other matrix problems, and if so, to compare those extensions with other well-established methods. This work aims to answer these questions. The main matrix problems considered in this dissertation are: the standard eigenvalue problem, the quadratic eigenvalue problem, the solution of systems of linear equations, the solution of sequences of systems of linear equations, and linear matrix equations. We focus on examples that arise from the discretization of partial differential equations.

In this work we are interested in the numerical solution of the Quadratic Eigenvalue Problem (QEP) (λ²M + λD + K)x = 0, where M, D, and K are given matrices of order N. Particularly, we study the applicability of the IDR(s) for eigenvalues to solve QEP. We present an IDR(s) algorithm that exploits the special block structure of the lin-ealized QEP to compute its eigenpairs. To this end we incorporate ideas from Second Order Arnoldi method proposed in [3].

The Induced Dimension Reduction method (IDR(s)) [1] is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using the spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the eigenvalues and eigenvectors. Using the Ritz values, we propose a self-contained variant of the Ritz-IDR((s) method [2] for solving a system of linear equations. In addition, the Ritz vectors are used to speed-up IDR(s) in the solution of a sequence of linear systems.

This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization allows us to develop IDR(s) algorithms to approximate the solution of linear matrix equations. The IDR(s) method presented here has two main advantages; firstly, it does not require the computation of inverses of any matrix, and secondly, it allows incorporation of preconditioners. Additionally, we present a simple preconditioner to solve the Sylvester equation based on a fixed point iteration. Several numerical examples illustrate the performance of IDR(s) for solving linear matrix equations. We also present the software implementation.

This work presents a new algorithm to compute eigenpairs of large unsymmetric matrices. Using the Induced Dimension Reduction method (IDR(ss)), which was originally proposed for solving systems of linear equations, we obtain a Hessenberg decomposition, from which we approximate the eigenvalues and eigenvectors of a matrix. This decomposition has two main advantages. First, IDR(ss) is a short-recurrence method, which is attractive for large scale computations. Second, the IDR(ss) polynomial used to create this Hessenberg decomposition is also used as a filter to discard the unwanted eigenvalues. Additionally, we incorporate the implicitly restarting technique proposed by D.C. Sorensen, in order to approximate specific portions of the spectrum and improve the convergence.

Discretization of (linearized) convection-diffusion-reaction problems yields a large and sparse non symmetric linear system of equations, Ax=b.
In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, SIAM J Sci Comput 31(2):1035–1062, 2008), with other short-recurrences Krylov methods, specifically the Bi-Conjugate Gradient Method (Bi-CG) (Fletcher, Conjugate gradient methods for indefinite systems. In: Proceedings of the Dundee conference on numerical analysis, pp 73–89, 1976), restarted Generalized Minimal Residual (GMRES(m)) (Saad and Schultz, SIAM J Sci Stat Comput 7:856–869, 1986), and Bi-Conjugate Gradient Stabilized method (Bi-CGSTAB) (van der Vorst, SIAM J Sci Stat Comput 13(2):631–644, 1992).

Discretization of (linearized) convection-diusion-reaction problems yields a large and sparse non symmetric linear system of equations, Ax = b: (1) In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) [10], with other short-recurrences Krylov methods, specically the Bi-Conjugate Gradient Method (Bi-CG) [1], restarted Generalized Minimal Residual (GMRES(m)) [4], and Bi-Conjugate Gradient Stabilized method (Bi-CGSTAB) [11].