Abstract. Verification methods provide mathematically correct error bounds for the solution of a numerical problem. That includes the proof of solvability of the problem and often uniqueness of the solution within the computed bounds. There are many verification methods for standard problems in numerical analysis, including linear and nonlinear systems of equations, matrix decompositions, eigenproblems, local and global optimization, and ordinary and partial differential equations. Many of those are included in INTLAB, the MATLAB/Octave toolbox for reliable computing. Despite several efforts, the verified solution of general sparse linear systems was an open problem. There are satisfactory algorithms for systems with symmetric positive definite input matrix. To that end error bounds for the solution of [Formula: see text] with general matrix [Formula: see text] could be computed using [Formula: see text], but that reduces the applicability in double precision to matrices with condition number up to [Formula: see text]. We give in this note an algorithm to compute entrywise error bounds for the solution of general real or complex sparse systems with condition number up to the limit [Formula: see text]. Our algorithm splits into three subalgorithms for symmetric positive definite, symmetric indefinite, and general input matrix [Formula: see text]. It is based on a mathematically correct lower bound on the smallest singular value [Formula: see text]. A key point is a factorization [Formula: see text] such that [Formula: see text] and [Formula: see text] have identical sets of singular values with the smallest one close to [Formula: see text]. A mathematically correct lower bound on [Formula: see text] is then computed using [Formula: see text]. Numerical evidence suggests that bounds for the solution of a linear system are computed for condition numbers up to [Formula: see text], and that often the bounds for all entries are close to maximally accurate, i.e., the bounds differ by few bits. We present a new a priori error bound based on Perron–Frobenius theory on the residual of a floating-point Cholesky decomposition which improves on existing ones by some two orders of magnitude. Our three subalgorithms benefit from this new bound. Based on the results in this Part I, an alternative approach will be presented in Part II of this note. Those methods are somewhat simpler, but often slower. However, they seem to be more robust, i.e., produce verified inclusions where the methods of Part I fail. Both approaches for square linear systems will be used in Part II of this note to compute verified error bounds for the solution of least squares problems and for underdetermined linear systems. Moreover, inclusions of the solution of general real or complex systems of nonlinear equations with sparse Jacobi matrix are computed by transforming the problem into a linear system with point matrix and interval right-hand side.
Abstract. Obtaining the inverse of a large symmetric positive definite matrix [Formula: see text] is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be prohibitively expensive, making it infeasible to compute the inverse. In this paper, we present a novel iterative block matrix inversion (IBMI) algorithm, which is designed to approximate the inverse of a large, dense, symmetric positive definite matrix. The matrix is first partitioned into blocks, and an iterative process using block matrix inversion is repeated until the matrix approximation reaches a satisfactory level of accuracy. We demonstrate that the two-block, nonoverlapping approach converges for any positive definite matrix, while numerical results provide strong evidence that the multiblock, overlapping approach also converges for such matrices.
By assigning a nonnegative cost matrix to transitions between states, we propose an extension of the usual mean first passage times for a finite irreducible Markov chain to mean first passage costs. We then prove the existence of a (nonzero) fixed-point cost: i.e., a cost matrix which is, up to an overall multiplicative constant, the mean first passage cost matrix of itself. Moreover, we prove the uniqueness of the fixed-point cost, up to an overall multiplicative constant (and with the minor exception of an n-cycle permutation matrix). The entries of the fixed-point cost matrix thus define natural diffusion distances between the states since these distances are essentially invariant under taking expectations over random walks. We further show that the fixed-point cost matrix can be efficiently computed. In the last part of the paper, we return to general mean first passage costs and decompose them intro three components: one symmetric and two antisymmetric. One of the antisymmetric component vanishes in the case of the heat kernel-based diffusion transition matrix used by R. Coifman and S. Lafon. In this case, the symmetric component has an explicit connection to the Coifman--Lafon diffusion distance and isometric embedding.
Spectral methods are among the most widely used tools for analyzing data derived from real-world systems. Currently, many of the largest available datasets contain on the order of a billion or more vertices---and this number is only expected to grow. As a result, a central challenge is developing spectral methods, particularly eigenvalue solvers, that can handle the massive size and computational demands required to analyze such datasets. A notable characteristic of real-world datasets is the frequent presence of symmetries in the data's underlying graph structures. Our main theoretical contribution is to show that any graph (matrix)---whether weighted or unweighted, directed, or undirected---can be decomposed into a collection of smaller graphs (matrices), each defined with respect to the original graph's symmetries, and that this decomposition can be done while preserving the graph's (matrix's) full set of eigenvalues. As a direct application of this theory, we introduce the LEParD algorithm, which efficiently computes the eigenvalues of graphs exhibiting nontrivial symmetry---common in real-world data. We evaluate the LEParD algorithm on 110 datasets of varying types and sizes and find that, in approximately 82\% of the datasets, it outperforms standard eigenvalue solvers.
Orthogonal nonnegative matrix factorization (ONMF) extends traditional matrix factorization by incorporating nonnegativity and orthogonality constraints. It has gained significant attention for applications in dimension reduction and data clustering. This paper proposes a novel algorithm for solving ONMF within the block coordinate descent (BCD) framework. Both subsequence and whole sequence convergence of the algorithm are rigorously established, representing a major advancement in ONMF research. Extensive experiments on real-world datasets demonstrate the method's effectiveness and efficiency. Compared with several state-of-the-art ONMF techniques, the proposed algorithm achieves superior performance in terms of computational efficiency, relative error, and optimality gap.
