Extending the Gauss–Huard method for the solution of Lyapunov matrix equations and matrix inversion

Abstract

The solution of linear systems is a recurrent operation in scientific and engineering applications, traditionally addressed via the LU factorization. The Gauss-Huard (GH) algorithm has been introduced as an efficient alternative in modern platforms equipped with accelerators, although this approach presented some functional constraints. In particular, it was not possible to reuse part of the computations in the solution of delayed linear systems or in the inversion of the matrix. Here, we adapt GH to overcome these two deficiencies of GH, yielding new algorithms that exhibit the same computational cost as their corresponding counterparts based on the LU factorization of the matrix. We evaluate the novel GH extensions on the solution of Lyapunov matrix equations via the LRCF-ADI method, validating our approach via experiments with three benchmarks from model order reduction. Figure 2. Blocked Gauss-Huard (GH) for the solution of Ax D b. On entry, O A D OEA; b, and upon completion, the last column of O A is overwritten with the solution x.Figure 3. Unblocked algorithm for the reutilization of Gauss-Huard (GH) in the solution of Ax D b. On entry, N A is the matrix resulting from the application of the GH algorithm to A, and upon completion, b is overwritten with the solution x.A related problem appears when A has been employed to solve a linear system via GH, and therefore, its contents have been overwritten with the factorization, and the inverse of this matrix is required next. Under special circumstances, this scenario can be of interest, for example, to avoid explicit multiplication with the matrix inverse in the solution of Lyapunov equations via the matrix sign function [10].

From Penrose Equations to Zhang Neural Network, Getz–Marsden Dynamic System, and DDD (Direct Derivative Dynamics) Using Substitution Technique

“…is novel model differs from any previous TVMI solution model. Note that the right-hand side of (10) involves multiple matrix-multiplication, so the model is termed as continuous-time ZMMMI model (10). To prepare for the following discussion, a lemma is given below [28].…”

“…Proof. Let α(t) � X(t) − A − 1 (t) represent the difference between the solution generated by ZMMMI model (10) and the theoretical inverse of A(t). Substituting X(t) � α(t) + A − 1 (t) into (8), we obtain…”

“…Generally speaking, the matrix inversion problem can be formulated as AX � I, where A ∈ R n×n is a known constant matrix, X ∈ R n×n is the unknown matrix to be computed, and I ∈ R n×n is an identity matrix. Over the years, efforts were directed towards computational issues of time-variant matrix inversion (TVMI) and a wealth of algorithms were proposed and applied to solving this problem [9][10][11]. For example, Yeung and Kumbi [9] developed an inversion method based on the multidimensional discrete Fourier transform of matrix sequences and applied it to an electron amplifier.…”

“…For example, Yeung and Kumbi [9] developed an inversion method based on the multidimensional discrete Fourier transform of matrix sequences and applied it to an electron amplifier. In [10], Benner et al adopted Gauss-Huard algorithm to solve TVMI problem. In [11], Xiao et al designed a complexvalued nonlinear recurrent neural network to solve timevarying complex matrix inversion.…”

Algorithmic advances for parallel architectures

“…Paper focuses on LU factorization, which is a recurrent operation in scientific and engineering applications used to solve linear systems. In particular, the authors adapt the Gauss‐Huard algorithm that has been introduced as an efficient solution for modern platforms equipped with accelerators to improve reusing of computations in this algorithm.…”