Nonsingular Linear Systems Research Articles

Fast simulation of planar Clifford circuits

A general quantum circuit can be simulated classically in exponential time. If it has a planar layout, then a tensor-network contraction algorithm due to Markov and Shi has a runtime exponential in the square root of its size, or more generally exponential in the treewidth of the underlying graph. Separately, Gottesman and Knill showed that if all gates are restricted to be Clifford, then there is a polynomial time simulation. We combine these two ideas and show that treewidth and planarity can be exploited to improve Clifford circuit simulation. Our main result is a classical algorithm with runtime scaling asymptotically as nω/2<n1.19 which samples from the output distribution obtained by measuring all n qubits of a planar graph state in given Pauli bases. Here ω is the matrix multiplication exponent. We also provide a classical algorithm with the same asymptotic runtime which samples from the output distribution of any constant-depth Clifford circuit in a planar geometry. Our work improves known classical algorithms with cubic runtime.A key ingredient is a mapping which, given a tree decomposition of some graph G, produces a Clifford circuit with a structure that mirrors the tree decomposition and which emulates measurement of the corresponding graph state. We provide a classical simulation of this circuit with the runtime stated above for planar graphs and otherwise ntω−1 where t is the width of the tree decomposition. Our algorithm incorporates two subroutines which may be of independent interest. The first is a matrix-multiplication-time version of the Gottesman-Knill simulation of multi-qubit measurement on stabilizer states. The second is a new classical algorithm for solving symmetric linear systems over F2 in a planar geometry, extending previous works which only applied to non-singular linear systems in the analogous setting.

On the convergence of randomized and greedy relaxation schemes for solving nonsingular linear systems of equations

We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible. Our main tool is to use weighted ℓ1-norms to measure the residuals. A major result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type. Numerical experiments illustrate the convergence of the method and the bounds obtained. Comparisons with the randomized Kaczmarz method are also presented.

The modified Uzawa methods for solving singular linear systems

For nonsingular linear systems, Yun in 2013 studied three variants of the Uzawa method; see Yun (2013) [18]. These methods contain the Uzawa-AOR method and the Uzawa-SAOR method as special cases. On the basis of the Uzawa-AOR method and the Uzawa-SAOR method, Li in 2017 proposed two modified Uzawa methods by constructing the coefficient matrix of AOR method into a new form of the product of lower triangular matrix and upper triangular matrix; see Li (2017) [26]. In this paper, we present two modified Uzawa methods for solving singular linear systems. The semi-convergence of these methods is analyzed by using the techniques of singular value decomposition and Moore-Penrose inverse. The numerical results are used to verify the theoretical results.

Alternating stationary iterative methods based on double splittings

Matrix double splitting iterations are simple in implementation while solving real non-singular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each ADS scheme induces a preconditioned system which helps in showing the convergence of the ADS schemes. We also show that the classes of matrices for which one ADS scheme is better than the other, are mutually exclusive. Numerical experiments confirm that the proposed ADS schemes have several computational advantages over the existing methods. Though the problems are considered in the rectangular matrix settings, the same problems are even new in non-singular matrix settings.

Fast algorithms for finding the solution of CUPL-Toeplitz linear system from Markov chain

In this paper, the nonsingular CUPL-Toeplitz linear system from Markov chain is solved. We introduce two fast approaches whose complexity could be considered to be O(nlogn) based on the splitting method of the CUPL-Toeplitz matrix which equals to a Toeplitz matrix minus a rank-one matrix. Finally, we confirm the performance of the new algorithms by three numerical experiments.

Convergence theory of iterative methods based on proper splittings and proper multisplittings for rectangular linear systems

Multisplitting methods are useful to solve differential-algebraic equations. In this connection, we discuss the theory of matrix splittings and multisplittings, which can be used for finding the iterative solution of a large class of rectangular (singular) linear system of equations of the form Ax = b. In this direction, many convergence results are proposed for different subclasses of proper splittings in the literature. But, in some practical cases, the convergence speed of the iterative scheme is very slow. To overcome this issue, several comparison results are obtained for different subclasses of proper splittings. This paper also presents a few such results. However, this idea fails to accelerate the speed of the iterative scheme in finding the iterative solution. In this regard, Climent and Perea [J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper multisplittings. With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results. Some of the results obtained in this paper are even new for the iterative theory of nonsingular linear systems.

Asymptotics of the Solution to the Minimization Problem of the Integral Quadratic Performance Index on Trajectories of a Quasi-Linear System

The problem of minimizing an integral quadratic performance index on the trajectories of a quasi-linear dynamic system with a small parameter multiplying the nonlinearities subject to terminal constraints is considered. Asymptotic approximations of the optimal control in this problem in the form of open loop control and feedback are constructed. The computations are reduced to solving the unperturbed linear-quadratic problem, integrating systems of linear differential equations, and finding the roots of nonsingular linear algebraic systems.

On Euler preconditioned SHSS iterative method for a class of complex symmetric linear systems

In this paper, we propose an Euler preconditioned single-step HSS (EP-SHSS) iterative method for solving a broad class of complex symmetric linear systems. The proposed method can be applied not only to the non-singular complex symmetric linear systems but also to the singular ones. The convergence (semi-convergence) properties of the proposed method are carefully discussed under suitable restrictions. Furthermore, we consider the acceleration of the EP-SHSS method by preconditioned Krylov subspace method and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments verify the effectiveness of the EP-SHSS method either as a solver or as a preconditioner for solving both non-singular and singular complex symmetric linear systems.

Performance analysis of Arithmetic Mean method for solving composite 6-point closed Newton–Cotes quadrature algebraic equation

The main focus of the paper is to study the performance of Arithmetic Mean (AM) iterative method in solving large and nonsingular dense linear system arising from the composite 6-point closed Newton–Cotes quadrature (6-CCNC) approximation of inhomogeneous linear Fredholm integral equations of the second kind. The derivation and implementation of the method are discussed. Some numerical results are also presented.

Convergence Properties of the Single-Step Preconditioned HSS Method for Non-Hermitian Positive Semidefinite Linear Systems

For the nonsingular, non-Hermitian and positive semidefinite linear systems, we derive the convergence results of the single-step preconditioned HSS (SPHSS) method under suitable constraints. Additionally, we consider the acceleration of the SPHSS method by Krylov subspace methods and some spectral properties of the preconditioned matrix are established. Numerical experiments are presented to further examine the effectiveness of the proposed method either as a solver or a preconditioner.

Characterization of the consistent completion of analytic hierarchy process comparison matrices using graph theory

AbstractDecision‐making is frequently affected by uncertainty and/or incomplete information, which turn decision‐making into a complex task. It is often the case that some of the actors involved in decision‐making are not sufficiently familiar with all of the issues to make the appropriate decisions. In this paper, we are concerned about missing information. Specifically, we deal with the problem of consistently completing an analytic hierarchy process comparison matrix and make use of graph theory to characterize such a completion. The characterization includes the degree of freedom of the set of solutions and a linear manifold and, in particular, characterizes the uniqueness of the solution, a result already known in the literature, for which we provide a completely independent proof. Additionally, in the case of nonuniqueness, we reduce the problem to the solution of nonsingular linear systems. In addition to obtaining the priority vector, our investigation also focuses on building the complete pairwise comparison matrix, a crucial step in the necessary process (between synthetic consistency and personal judgement) with the experts. The performance of the obtained results is confirmed.

On the singular Neumann problem in linear elasticity

SummaryThe Neumann problem of linear elasticity is singular with a kernel formed by the rigid motions of the body. There are several tricks that are commonly used to obtain a nonsingular linear system. However, they often cause reduced accuracy or lead to poor convergence of the iterative solvers. In this paper, different well‐posed formulations of the problem are studied through discretization by the finite element method, and preconditioning strategies based on operator preconditioning are discussed. For each formulation, we derive preconditioners that are independent of the discretization parameter. Preconditioners that are robust with respect to the first Lamé constant are constructed for the pure displacement formulations, whereas a preconditioner that is robust in both Lamé constants is constructed for the mixed formulation. It is shown that, for convergence in the first Sobolev norm, it is crucial to respect the orthogonality constraint derived from the continuous problem. On the basis of this observation, a modification to the conjugate gradient method is proposed, which achieves optimal error convergence of the computed solution.

An accurate method for solving a class of fractional Sturm-Liouville eigenvalue problems

Abstract This article is devoted to both theoretical and numerical study of the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. In this paper, we implement a fractional-order Legendre Tau method to approximate the eigenvalues. This method transforms the Sturm-Liouville problem to a sparse nonsingular linear system which is solved using the continuation method. Theoretical results for the considered problem are provided and proved. Numerical results are presented to show the efficiency of the proposed method.

Accelerating the Solution of Linear Systems by Iterative Refinement in Three Precisions

We propose a general algorithm for solving a $n\times n$ nonsingular linear system $Ax = b$ based on iterative refinement with three precisions. The working precision is combined with possibly different precisions for solving for the correction term and for computing the residuals. Via rounding error analysis of the algorithm we derive sufficient conditions for convergence and bounds for the attainable normwise forward error and normwise and componentwise backward errors. Our results generalize and unify many existing rounding error analyses for iterative refinement. With single precision as the working precision, we show that by using LU factorization in IEEE half precision as the solver and calculating the residuals in double precision it is possible to solve $Ax = b$ to full single precision accuracy for condition numbers $\kappa_2(A) \le 10^4$, with the $O(n^3)$ part of the computations carried out entirely in half precision. We show further that by solving the correction equations by GMRES preconditioned by the LU factors the restriction on the condition number can be weakened to $\kappa_2(A) \le 10^8$, although in general there is no guarantee that GMRES will converge quickly. Taking for comparison a standard $Ax = b$ solver that uses LU factorization in single precision, these results suggest that on architectures for which half precision is efficiently implemented it will be possible to solve certain linear systems $Ax = b$ up to twice as fast \emph{and} to greater accuracy. Analogous results are given with double precision as the working precision.

Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces

We use some results from polarity theory to recast several geometric properties of Conjugate Gradient-based methods, for the solution of nonsingular symmetric linear systems. This approach allows us to pursue three main theoretical objectives. First, we can provide a novel geometric perspective on the generation of conjugate directions, in the context of positive definite systems. Second, we can extend the above geometric perspective to treat the generation of conjugate directions for handling indefinite linear systems. Third, by exploiting the geometric insight suggested by polarity theory, we can easily study the possible degeneracy (pivot breakdown) of Conjugate Gradient-based methods on indefinite linear systems. In particular, we prove that the degeneracy of the standard Conjugate Gradient on nonsingular indefinite linear systems can occur only once in the execution of the Conjugate Gradient.

Nearly optimal computations with structured matrices

We estimate the Boolean complexity of multiplication of structured matrices by a vector and the solution of nonsingular linear systems of equations with these matrices. We study four basic and most popular classes, that is, Toeplitz, Hankel, Cauchy and Vandermonde matrices, for which the cited computational problems are equivalent to the task of polynomial multiplication and division and polynomial and rational multipoint evaluation and interpolation. The Boolean cost estimates for the latter problems have been obtained by Kirrinnis in [10], except for rational interpolation. We supply them now as well as the Boolean complexity estimates for the important problems of multiplication of transposed Vandermonde matrix and its inverse by a vector. All known Boolean cost estimates from [10] for such problems rely on using Kronecker product. This implies the d-fold precision increase for the d-th degree output, but we avoid such an increase by relying on distinct techniques based on employing FFT. Furthermore we simplify the analysis and make it more transparent by combining the representations of our tasks and algorithms both via structured matrices and via polynomials and rational functions. This also enables further extensions of our estimates to cover Trummer's important problem and computations with the popular classes of structured matrices that generalize the four cited basic matrix classes, as well as the transposed Vandermonde matrices. It is known that the solution of Toeplitz, Hankel, Cauchy, Vandermonde, and transposed Vandermonde linear systems of equations is generally prone to numerical stability problems, and numerical problems arise even for multiplication of Cauchy, Vandermonde, and transposed Vandermonde matrices by a vector. Thus our FFT-based results on the Boolean complexity of these important computations could be quite interesting because our estimates are reasonable even for more general classes of structured matrices, showing rather moderate growth of the complexity as the input size increases.

Multipreconditioning for nonsymmetric problems: The case of orthomin and biCG

Preconditioned Krylov subspace methods [7] are powerful tools for solving linear systems but sometimes they converge very slowly, and often after a long stagnation. A natural way to fix this is by enlarging the space in which the solution is computed at each iteration. Following this idea, we propose in this note two multipreconditioned algorithms: multipreconditioned orthomin and multipreconditioned biCG, which aim at solving general nonsingular linear systems in a small number of iterations. After describing the algorithms, we illustrate their behaviour on systems arising from the FETI domain decomposition method, where in order to enlarge the search space, each local component in the usual preconditioner is kept as a separate preconditioner.

A New Analysis of Iterative Refinement and Its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems

Iterative refinement is a long-standing technique for improving the accuracy of a computed solution to a nonsingular linear system $Ax = b$ obtained via LU factorization. It makes use of residuals computed in extra precision, typically at twice the working precision, and existing results guarantee convergence if the matrix $A$ has condition number safely less than the reciprocal of the unit roundoff, $u$. We identify a mechanism that allows iterative refinement to produce solutions with normwise relative error of order $u$ to systems with condition numbers of order $u^{-1}$ or larger, provided that the update equation is solved with a relative error sufficiently less than $1$. A new rounding error analysis is given and its implications are analyzed. Building on the analysis, we develop a GMRES-based iterative refinement method (GMRES-IR) that makes use of the computed LU factors as preconditioners. GMRES-IR exploits the fact that even if $A$ is extremely ill conditioned the LU factors contain enough information that preconditioning can greatly reduce the condition number of $A$. Our rounding error analysis and numerical experiments show that GMRES-IR can succeed where standard refinement fails, and that it can provide accurate solutions to systems with condition numbers of order $u^{-1}$ and greater. Indeed in our experiments with such matrices---both random and from the University of Florida Sparse Matrix Collection---GMRES-IR yields a normwise relative error of order $u$ in at most $3$ steps in every case.

Minimal polynomial and reduced rank extrapolation methods are related

Minimal Polynomial Extrapolation (MPE) and Reduced Rank Extrapolation (RRE) are two polynomial methods used for accelerating the convergence of sequences of vectors {xm}. They are applied successfully in conjunction with fixed-point iterative schemes in the solution of large and sparse systems of linear and nonlinear equations in different disciplines of science and engineering. Both methods produce approximations sk to the limit or antilimit of {xm} that are of the form sk=źi=0kźixi$\boldsymbol {s}_{k}={\sum }^{k}_{i=0}\gamma _{i}\boldsymbol {x}_{i}$ with źi=0kźi=1${\sum }^{k}_{i=0}\gamma _{i}=1$, for some scalars źi. The way the two methods are derived suggests that they might, somehow, be related to each other; this has not been explored so far, however. In this work, we tackle this issue and show that the vectors skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$ and skRRE$\boldsymbol {s}_{k}^{\textit {{\tiny {RRE}}}}$ produced by the two methods are related in more than one way, and independently of the way the xm are generated. One of our results states that RRE stagnates, in the sense that skRRE=skź1RRE$\boldsymbol {s}_{k}^{\textit {{\tiny {RRE}}}}=\boldsymbol {s}_{k-1}^{\textit {{\tiny {RRE}}}}$, if and only if skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$ does not exist. Another result states that, when skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$ exists, there holds μkskRRE=μkź1skź1RRE+źkskMPEwithμk=μkź1+źk,$$\mu_{k}\boldsymbol{s}_{k}^{\textit{{\tiny{RRE}}}}=\mu_{k-1}\boldsymbol{s}_{k-1}^{\textit{{\tiny{RRE}}}}+ \nu_{k}\boldsymbol{s}_{k}^{\textit{{\tiny{MPE}}}}\quad \text{with}\quad \mu_{k}=\mu_{k-1}+\nu_{k}, $$for some positive scalars μk, μkź1, and źk that depend only on skRRE$\boldsymbol {s}_{k}^{\textit {{\tiny {RRE}}}}$, skź1RRE$\boldsymbol {s}_{k-1}^{\textit {{\tiny {RRE}}}}$, and skMPE$\boldsymbol {s}_{k}^{\textit {{\tiny {MPE}}}}$, respectively. Our results are valid when MPE and RRE are defined in any weighted inner product and the norm induced by it. They also contain as special cases the known results pertaining to the connection between the method of Arnoldi and the method of generalized minimal residuals, two important Krylov subspace methods for solving nonsingular linear systems.

A parallel sparse linear system solver based on Hermitian/skew-Hermitian splitting

In this paper we describe a parallel algorithm for solving large sparse nonsingular linear systems Ax=f, of order n, using the Hermitian Skew-Hermitian splitting approach for handling the augmented linear system, of order 2n, that arises from the linear least problem of minimizing the 2-norm of (f−Ax). We use the restarted GMRES as the outer iteration with the Hermitian Skew-Hermitian Splitting (HSS) preconditioner. In solving systems involving this preconditioner, the most time consuming part deals with handling shifted skew-symmetric systems. We solve such systems using the successive overrelaxation (SOR). Theoretical analysis shows that our solver always converges to the unique solution of Ax=f. We present several numerical experiments that demonstrate the robustness of our solver compared to other schemes, and show its parallel scalability on a single multicore node.