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We prove that the Ring Learning With Errors (RLWE) and the Polynomial Learning With Errors (PLWE) problems over the cyclotomic field Q (ζ n) are not equivalent. Precisely, we show that reducing one problem to the other increases the noise by a factor that is more than polynomial in n. We do so by providing a lower bound, holding for infinitely many positive integers n, for the condition number of the Vandermonde matrix of the nth cyclotomic polynomial. [ABSTRACT FROM AUTHOR]

The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann, and absorbing boundary conditions. This study focuses in particular on the spectral dependence on the polynomial degree p, mesh size h, regularity k, of the IGA discretization and on the time step size Δ t and parameter β of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the number of degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results show that the spectral properties of the IGA collocation matrices are comparable with the available spectral estimates for IGA Galerkin matrices associated with the Poisson problem with Dirichlet boundary conditions, and in some cases, the IGA collocation results are better than the corresponding IGA Galerkin estimates, in particular for increasing p and maximal regularity k = p - 1 . [ABSTRACT FROM AUTHOR]

The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized by a Hessenberg form. We demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial. [ABSTRACT FROM AUTHOR]

Some of the key features of the isogeometric analysis, IGA, are the capacity of exactly representing the problem geometry, the use of the same basis functions to describe the geometry and the solution field, and a straightforward and automatic discretization refining scheme. The higher order continuity of the isogeometric approximation, important to correctly represent the domain geometry, can be a problem to approximate the displacement field in the neighbourhood of a crack. The eXtended Isogeometric Analysis (XIGA) overcomes this obstacle, enlarging the approximate space of IGA. This is achieved by incorporating customized functions, using the enrichment strategy of the Generalized/eXtended Finite Element Method. When these functions are unknown, they can be computed from the solution of local boundary value problems embracing the crack, and a global–local iterative procedure is established. Here this procedure is firstly proposed to combine FEM and isogeometric approximations, denoted XIGA gl . The effectiveness of this approach is investigated in terms of convergence rates and numerical stability. The method is applied to two-dimensional fracture mechanics problems. The numerical experiments show the importance of using the isogeometric approximation to recover more accurate solutions and minimize the deterioration of the conditioning of the related stiffness matrix. [ABSTRACT FROM AUTHOR]

The water distribution system (WDS) hydraulic model is extensively used for design and management of WDS. The nodal water demand is the crucial parameter of the model that requires accurate estimating by the pressure measurements. Proper pressure sampling design is essential for estimating nodal water demand and improving model accuracy. Existing research has emphasized the need to enhance the observability of monitoring systems and mitigate the adverse effects of monitoring noise. However, methods that simultaneously consider both of these factors in sampling design have not been adequately studied. In this study, a novel two-objective sampling design method is developed to improve the system observability and mitigate the adverse effects of monitoring noise. The approach is applied to a realistic network and results demonstrate that the developed approach can effectively improve the observability and robustness of the system especially when considerable measurement noise is considered. [ABSTRACT FROM AUTHOR]

Instrumented indentation is a versatile method of extracting hyper-elastic material parameters, particularly useful for applications where stress-strain data are difficult to be in-situ measured. Because the analytical force-displacement relation is still unavailable for the indentation of hyper-elastic materials, identifying hyper-elastic parameters often requires an iterative optimization strategy that fits finite element simulations with experimental data. However, the optimization strategy is burdened by heavy computation and its prediction accuracy is greatly influenced by the choice of optimization algorithm. To address these challenges in this study, a bidirectional long short-term memory (BLSTM) neural network is presented that directly predicts hyper-elastic material parameters from indentation load-displacement data, focusing on Mooney-Rivlin hyper-elasticity as an example. To improve the predication accuracy, the condition numbers for the inverse identification of the hyper-elastic parameters are investigated. And, a normalization procedure is proposed to treat the input data, which can guarantee the BLSTM network is well-conditioned. During evaluation, the trained BLSTM network significantly outperforms the iterative optimization strategy using a genetic algorithm. Furthermore, the effect of the normalization procedure is demonstrated. [ABSTRACT FROM AUTHOR]

We characterise the sensitivity of several additive tensor decompositions with respect to perturbations of the original tensor. These decompositions include canonical polyadic decompositions, block term decompositions, and sums of tree tensor networks. Our main result shows that the condition number of all these decompositions is invariant under Tucker compression. This result can dramatically speed up the computation of the condition number in practical applications. We give the example of an 265 × 371 × 7 tensor of rank 3 from a food science application whose condition number was computed in 6.9 milliseconds by exploiting our new theorem, representing a speedup of four orders of magnitude over the previous state of the art. [ABSTRACT FROM AUTHOR]

The nonlinear programming makes use of quasi-Newton methods, a collection of optimization approaches when traditional Newton's method are challenging due to the calculation of the Jacobian matrix and its inverse. Since the Jacobian matrix is computationally difficult to compute and sometimes not available specifically when dealing with non-smooth monotone systems, quasi-Newton methods with superlinear convergence are preferred for solving nonlinear system of equations. This paper provides a new version of the derivative-free David–Fletcher–Powell (DFP) approach for dealing with nonlinear monotone system of equations with convex constraints. The optimal value of the scaling parameter is found by minimizing the condition number of the DFP matrix. Under certain assumptions, the proposed method has global convergence, required minimal storage and is derivative-free. When compared to standard methods, the proposed method requires less iteration, function evaluations, and CPU time. The image restoration test problems demonstrate the method's reliability and efficiency. [ABSTRACT FROM AUTHOR]

Theoretical results and numerical experiments show that the linear systems originating from the last iterations of interior point methods (IPM) are very ill-conditioned. For this reason, preconditioners are necessary to approach this problem. In addition to that, in large-scale problems, the use of iterative methods and implicit preconditioners is essential because we only compute matrix–vector multiplications. Preconditioners with a lower computational cost than the splitting preconditioner only have good performance in the initial iterations of the IPM, so this preconditioner has become very important in the last iterations. The study of improvements thereof is justified. This paper studies the variation of the diagonal matrix D entries that appear in the linear systems to be solved to try to reuse or recycle some linearly independent columns of the splitting preconditioner base previously computed in a given IPM iteration to build another basis in the next one. It is justified by the fact that a subset of linearly independent columns remains linearly independent, and from that available subset, one may complete the number of columns necessary to form the new base. The numerical results show that the column recycling proposal improves the speed and robustness of the original approach for a test set, especially for large-scale problems. [ABSTRACT FROM AUTHOR]

We analyze two algorithms for computing the symplectic factorization A = LLT of a given symmetric positive definite symplectic matrix A. The first algorithm W1 is an implementation of the HHT factorization from Dopico and Johnson (SIAM J. Matrix Anal. Appl. 31(2):650–673, 2009), see Theorem 5.2. The second one is a new algorithm W2 that uses both Cholesky and Reverse Cholesky decompositions of symmetric positive definite matrices. We present a comparison of these algorithms and illustrate their properties by numerical experiments in MATLAB. A particular emphasis is given on symplecticity properties of the computed matrices in floating-point arithmetic. [ABSTRACT FROM AUTHOR]

The demands of accuracy in measurements and engineering models today render the condition number of problems larger. While a corresponding increase in the precision of floating point numbers ensured a stable computing, the uncertainty in convergence when using residue as a stopping criterion has increased. We present an analysis of the uncertainty in convergence when using relative residue as a stopping criterion for iterative solution of linear systems, and the resulting over/under computation for a given tolerance in error. This shows that error estimation is significant for an efficient or accurate solution even when the condition number of the matrix is not large. An error estimator for iterations of the CG algorithm was proposed more than two decades ago. Recently, an error estimator was described for the GMRES algorithm which allows for non-symmetric linear systems as well, where is the iteration number. We suggest a minor modification in this GMRES error estimation for increased stability. In this work, we also propose an error estimator for -norm and norm of the error vector in the Bi-CG algorithm. The robust performance of these estimates as a stopping criterion results in increased savings and accuracy in computation, as the condition number and size of problems increase. [ABSTRACT FROM AUTHOR]

Gaussian processes (GPs) are used for numerous different applications, including uncertainty quantification and optimization. Ill-conditioning of the covariance matrix for GPs is common with the use of various kernels, including the Gaussian, rational quadratic, and Matérn kernels. A common approach to overcome this problem is to add a nugget along the diagonal of the covariance matrix. For GPs that are not constructed with gradients, it is straightforward to derive a nugget value that guarantees the condition number of the covariance matrix to be below a user-set threshold. However, for gradient-enhanced GPs, there are no existing practical bounds to select a nugget that guarantee that the condition number of the gradient-enhanced covariance matrix is below a user-set threshold. In this paper a novel approach is taken to bound the condition number of the covariance matrix for GPs that use the Gaussian kernel. This is achieved by using non-isotropic rescaling for the data and a modest nugget value. This non-intrusive method works for GPs applied to problems of any dimension and it allows all data points to be kept. The method is applied to a Bayesian optimizer using a gradient-enhanced GP to achieve deep convergence. Without this method, the high condition number constrains the hyperparameters for the GP and this is shown to impede the convergence of the optimizer. It is also demonstrated that applying this method to the rational quadratic and Matérn kernels alleviates the ill-conditioning of their gradient-enhanced covariance matrices. Implementation of the method is straightforward and clearly described in the paper. [ABSTRACT FROM AUTHOR]

In this paper, strategies are provided for a powerful numerical manifold method (NMM) with h and p refinement in analyses of elasticity and elasto-plasticity. The new NMM is based on the concept of the independent cover, which gets rid of NMM's important defect of rank deficiency when using higher-order local approximation functions. Several techniques are presented. In terms of mesh generation, a relationship between the quadtree structure and the mathematical mesh is established to allow a robust h-refinement. As to the condition number, a scaling based on the physical patch is much better than the classical scaling based on the mathematical patch; an overlapping width of 1%–10% can ensure a good condition number for 2nd, 3rd, and 4th order local approximation functions; the small element issue can be overcome after the local approximation on small patch is replaced by that on a regular patch. On numerical accuracy, local approximation using complete polynomials is necessary for the optimal convergence rate. Two issues that may damage the convergence rate should be prevented. The first is to approximate the curved boundary of a higher-order element by overly few straight lines, and the second is excessive overlapping width. Finally, several refinement strategies are verified by numerical examples. [ABSTRACT FROM AUTHOR]

The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with respect to structured perturbations. Those are perturbations preserving the rank of the tensor that is decomposed. On the other hand, the angular condition number measures the perturbations of the rank-1 summands up to scaling. We show for random rank-2 tensors that the expected value of the condition number is infinite for a wide range of choices of the density. Under a mild additional assumption, we show that the same is true for most higher ranks r ≥ 3 as well. In fact, as the dimensions of the tensor tend to infinity, asymptotically all ranks are covered by our analysis. On the contrary, we show that rank-2 tensors have finite expected angular condition number. Based on numerical experiments, we conjecture that this could also be true for higher ranks. Our results underline the high computational complexity of computing tensor rank decompositions. We discuss consequences of our results for algorithm design and for testing algorithms computing tensor rank decompositions. [ABSTRACT FROM AUTHOR]

Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations. [ABSTRACT FROM AUTHOR]

The problem discussed is how to obtain a normal matrix from a binormal one and, conversely, a binormal matrix from a normal one via the right multiplication on a suitable unitary matrix. Let be a normal matrix badly conditioned with respect to inversion, that is, having a large condition number . We show that, among the binormal matrices that can be obtained from , there is a matrix whose eigenvalues have individual condition numbers of order . [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast solver for the linear system-induced from applying the finite difference method. For this purpose, we provide a new decomposition of the matrix consisting of a second-order central finite difference matrix and a small condition number matrix. We analyze the fourth-order finite difference matrix using this decomposition and the good properties of the two decomposed matrices. In addition, we compare the upper bounds of the condition numbers of the three matrices, which are closely related to the number of iterations. In terms of computational cost, we show the superiority of the proposed solver by solving the one-dimensional Poisson's equations. We also demonstrated the efficiency of using the proposed solver by numerically observing the factors, such as condition number and spectral radius. [ABSTRACT FROM AUTHOR]

We study approximate K-optimal designs for various regression models by minimizing the condition number of the information matrix. This minimizes the error sensitivity in the computation of the least squares estimator of regression parameters and also avoids the multicollinearity in regression. Using matrix and optimization theory, we derive several theoretical results of K-optimal designs, including convexity of K-optimality criterion, lower bounds of the condition number, and symmetry properties of K-optimal designs. A general numerical method is developed to find K-optimal designs for any regression model on a discrete design space. In addition, specific results are obtained for polynomial, trigonometric and second-order response models. [ABSTRACT FROM AUTHOR]

Numerical stability of the Filon–Clenshaw–Curtis rules is considered, when applied to oscillatory integrals with the linear oscillator. The following results are proved: (1) the coefficients of the ( N + 1 )-point rule, for any N > 2 , never lie in a right sector of the complex plane; (2) the coefficients of the 2-point rule lie in a right sector only when k ∈ [ d π - 3 π / 4 , d π - π / 4) , for any integer d > 0 large enough; and (3) the coefficients of the 3-point rule lie in a right sector only when k ∈ (d π - π / 2 , d π - π / 4) , for any integer d > 0 large enough. These results imply that the condition numbers associated with the 2-point and the 3-point rules are bounded by π / 2 when k satisfies the aforementioned conditions. Then, we extend the stability intervals for k and show that in the following cases, the FCC rules can be applied in a stable manner: (1) the 2-point rule with k far enough from d π for any integer d > 0 ; (2) the 3-point rule with k ∈ [ d π - π / 2 , d π) far enough from d π ; and (3) the 4-point rule with k ∈ [ d π - π / 2 , d π) far enough from both d π - π / 2 and d π . [ABSTRACT FROM AUTHOR]

In this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized condition number converges in distribution to a Fréchet law as the dimension of the matrix increases. [ABSTRACT FROM AUTHOR]

Soft materials such as rubbers, silicones, gels and biological tissues have a nonlinear response to large deformations, a phenomenon which in principle can be captured by hyperelastic models. The suitability of a candidate hyperelastic strain energy function is then determined by comparing its predicted response to the data gleaned from tests and adjusting the material parameters to get a good fit, an exercise which can be deceptive because of nonlinearity. Here we propose to generalise the approach of Rivlin and Saunders (Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 243:251–288, 1951) who, instead of reporting the data as stress against stretch, manipulated these measures to create the 'Mooney plot', where the Mooney-Rivlin model is expected to produce a linear fit. We show that extending this idea to other models and modes of deformation (tension, shear, torsion, etc.) is advantageous, not only (a) for the fitting procedure, but also to (b) delineate trends in the deformation which are not obvious from the raw data (and may be interpreted in terms of micro-, meso-, and macro-structures) and (c) obtain a bounded condition number κ over the whole range of deformation, a robustness which is lacking in other plots and spaces. [ABSTRACT FROM AUTHOR]

We conduct a condition number analysis of a Hybrid High-Order (HHO) scheme for the Poisson problem. We find the condition number of the statically condensed system to be independent of the number of faces in each element, or the relative size between an element and its faces. The dependence of the condition number on the polynomial degree is tracked. Next, we consider HHO schemes on cut background meshes, which are commonly used in unfitted discretisations. It is well known that the linear systems obtained on these meshes can be arbitrarily ill-conditioned due to the presence of sliver-cut and small-cut elements. We show that the condition number arising from HHO schemes on such meshes is not as negatively effected as those arising from conforming methods. We describe how the condition number can be improved by aggregating ill-conditioned elements with their neighbours. [ABSTRACT FROM AUTHOR]

By defining two important terms called basic perturbation vectors and obtaining their linear bounds, we obtain the linear componentwise perturbation bounds for unitary factors and upper triangular factors of the generalized Schur decomposition. The perturbation bounds for the diagonal elements of the upper triangular factors and the generalized invariant subspace are also derived. From the former, we present an upper bound and a condition number of the generalized eigenvalue. Furthermore, with numerical iterative method, the nonlinear componentwise perturbation bounds of the generalized Schur decomposition are also provided. Numerical examples are given to test the obtained bounds. Among them, we compare our upper bound and condition number of the generalized eigenvalue with their counterparts given in the literature. Numerical results show that they are very close to each other but our results don't contain the information on the left and right generalized eigenvectors. [ABSTRACT FROM AUTHOR]

In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond2(V)) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V− 1, by which we prove that Cond 2 (V) = O (n 2) . This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores. [ABSTRACT FROM AUTHOR]

This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem are derived. Computable expressions and upper bounds for these condition numbers are also given to avoid the costly Kronecker product-based operations. The results unify the ones for the TLS problem. For TLSE problems with equilibratory input data, numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition number-based estimates are much tighter. [ABSTRACT FROM AUTHOR]

Broken cells in the finite cell method—especially those with a small volume fraction—lead to a high condition number of the global system of equations. To overcome this problem, in this paper, we apply and adapt an eigenvalue stabilization technique to improve the ill-conditioned matrices of the finite cells and to enhance the robustness for large deformation analysis. In this approach, the modes causing high condition numbers are identified for each cell, based on the eigenvalues of the cell stiffness matrix. Then, those modes are supported directly by adding extra stiffness to the cell stiffness matrix in order to improve the condition number. Furthermore, the same extra stiffness is considered on the right-hand side of the system—which leads to a stabilization scheme that does not modify the solution. The performance of the eigenvalue stabilization technique is demonstrated using different numerical examples. [ABSTRACT FROM AUTHOR]

The harmonic balance method has emerged as an efficient and accurate approach for computing periodic, as well as almost periodic, solutions to nonlinear ordinary differential equations. The accuracy of the harmonic balance method can however be negatively impacted by aliasing. Aliasing occurs because Fourier coefficients of nonlinear terms in the governing equations are approximated by a discrete Fourier transform (DFT). Understanding how aliasing occurs when the DFT is applied is therefore essential in improving the accuracy of the harmonic balance method. In this work, a new operator that describe the fold-back, i.e. aliasing, of unresolved frequencies onto the resolved ones is developed. The norm of this operator is then used as a metric for investigating how the time sampling should be performed to minimize aliasing. It is found that a time sampling which minimizes the condition number of the DFT matrix is the best choice in this regard, both for single and multiple frequency problems. These findings are also verified for the Duffing oscillator. Finally, a strategy for oversampling multiple frequency harmonic balance computations is developed and tested. [ABSTRACT FROM AUTHOR]

This paper considers the mixed least squares-scaled total least squares (MLSSTLS) problem which unifies the mixed least squares-total least squares (MTLS) problem and the scaled total least squares (STLS) problem. Firstly, we present the explicit expression of the MLSSTLS solution under some conditions. Then, the perturbation analysis and condition numbers of the MLSSTLS solution are obtained. These results can reduce to some corresponding published results of the MTLS problem and the STLS problem, respectively. Finally, numerical experiments are given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper presents a quantum algorithm to decide whether a Boolean equation system F has a solution and to compute one if F does have solutions with any given success probability. The runtime complexity of the algorithm is polynomial in the size of F and the condition number of certain Macaulay matrix associated with F . As a consequence, the authors give a polynomial-time quantum algorithm for solving Boolean equation systems if their condition numbers are polynomial in the size of F . The authors apply the proposed quantum algorithm to the cryptanalysis of several important cryptosystems: The stream cipher Trivum, the block cipher AES, the hash function SHA-3/Keccak, the multivariate public key cryptosystems, and show that they are secure under quantum algebraic attack only if the corresponding condition numbers are large. This leads to a new criterion for designing such cryptosystems which are safe against the attack of quantum computers: The corresponding condition number. [ABSTRACT FROM AUTHOR]

We study the equivalence between the ring learning with errors and polynomial learning with errors problems for cyclotomic number fields, namely: we prove that both problems are equivalent via a polynomial noise increase as long as the number of distinct primes dividing the conductor is kept constant. We refine our bound in the case where the conductor is divisible by at most three primes and we give an asymptotic subexponential formula for the condition number of the attached Vandermonde matrix valid for arbitrary degree. [ABSTRACT FROM AUTHOR]

Heavily damped quadratic eigenvalue problem (QEP) is a special class of QEP, which has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize it to produce a matrix pencil. We investigate upper bounds for the conditioning of eigenvalues of linearizations of four common forms relative to that of the quadratic and compare them with the previous studies. Based on the analysis of upper bounds, we introduce applying tropical scaling for the linearizations to reduce the bounds and the condition number ratios. Furthermore, we establish upper bounds for the condition number ratios with tropical scaling and make a comparison with the unscaled bounds. Several numerical experiments are performed to illustrate our results. [ABSTRACT FROM AUTHOR]

For the computation of the generalized singular value decomposition (GSVD) of a large matrix pair (A,B) of full column rank, the GSVD is commonly formulated as two mathematically equivalent generalized eigenvalue problems, so that a generalized eigensolver can be applied to one of them and the desired GSVD components are then recovered from the computed generalized eigenpairs. Our concern in this paper is, in finite precision arithmetic, which generalized eigenvalue formulation is numerically preferable to compute the desired GSVD components more accurately. We make a detailed perturbation analysis on the two formulations and show how to make a suitable choice between them. Numerical experiments illustrate the results obtained. [ABSTRACT FROM AUTHOR]

We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies n p ≥ log n + k (n) with k (n) → ∞ as n → ∞ , then the adjacency matrix is invertible with probability approaching one (n is the number of vertices in the two former cases and the same for each part in the latter case). For n p ≤ log n - k (n) these matrices are invertible with probability approaching zero, as n → ∞ . In the intermediate region, when n p = log n + k (n) , for a bounded sequence k (n) ∈ R , the event Ω 0 that the adjacency matrix has a zero row or a column and its complement both have a non-vanishing probability. For such choices of p our results show that conditioned on the event Ω 0 c the matrices are again invertible with probability tending to one. This shows that the primary reason for the non-invertibility of such matrices is the existence of a zero row or a column. We further derive a bound on the (modified) condition number of these matrices on Ω 0 c , with a large probability, establishing von Neumann's prediction about the condition number up to a factor of n o (1) . [ABSTRACT FROM AUTHOR]

Several approaches including optimization based methods were developed for mesh quality improvement using only node movement, keeping intact the element connectivity. In this research, a socio-inspired optimization approach referred to as cohort intelligence (CI) was investigated for mesh smoothing. Minimization of summation of condition numbers of all elements was the final aim. The geometrical boundaries of the object defined the surface and edge constraints for movement of external nodes. Movement of internal nodes was completely governed by variations of CI algorithm, viz. roulette wheel, follow best, follow better, alienation and random selection, follow worst and follow itself. The approach was demonstrated with pentagonal prism, hexagonal prism and hexagonal prism with hole. The performance of follow best and roulette wheel variations of CI algorithm was observed to be satisfactory as compared to other variations of the algorithm. [ABSTRACT FROM AUTHOR]

We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are "off the grid," pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated. [ABSTRACT FROM AUTHOR]

Interacting processes are available in Industries. Control of such interacting processes is a challenging problem. Various control schemes like multiloop control, decentralised control with decoupler and centralised control are available for interacting processes. Multiloop decentralised control is the simplest among all the control because of its simplicity and easy adaptation. Decentralised control involves splitting of MIMO systems into n number of SISO systems and design of controller for the SISO system. But the interaction effects cannot be eliminated and involves decoupler to nullify the interaction effects in traditional approach. Online tuning of the optimal parameters is done by putting the process under servo regulatory condition and by varying the set point of tanks simultaneously leading to a controller which avoids decoupler. In this article, the servo and regulatory conditions for the system is analysed by using fractional order PI controller. Performance shows that for MIMO process the proposed FOPI controller is best suited for control applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations. We apply the Crank–Nicolson technique to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. We transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. We prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners. [ABSTRACT FROM AUTHOR]

More lightweight structure and higher control bandwidth are highly desirable in next-generation motion stages, satisfying the continuously increasing requirements in throughput and accuracy. However, these lead to more severe flexible deformation, causing that the estimation accuracy of the generalized displacements of a point of interest (POI) cannot be guaranteed under the rigid-body assumption. In this paper, a method for estimating the generalized displacement of the POI using critical modes is proposed. This method can realize a more accurate estimation under the limited measurement points. In this method, since the number of measurement points is limited, the selection criterion of the critical modes is firstly proposed for the over-actuator system; then, with regard to the estimation accuracy, the influences of the measurement layout and the residual modes on the estimation matrix are analyzed mathematically, and a performance measure is proposed for evaluating this method from the perspective of system control. In the verification section, the validity of the estimation method is demonstrated through numerical simulation and an experiment on a representative but straightforward case using a plate structure. [ABSTRACT FROM AUTHOR]

We mainly consider the stability and reconstruction of convolution random sampling in multiply generated shift-invariant subspaces V p (Φ) = ∑ k ∈ Z d c (k) T Φ (· - k) : (c (k)) k ∈ Z d ∈ (ℓ p (Z d)) r of L p (R d) , 1 < p < ∞ , where Φ = (ϕ 1 , ϕ 2 , … , ϕ r) T with ϕ i ∈ L p (R d) and c = (c 1 , c 2 , … , c r) T with c i ∈ ℓ p (Z d) , i = 1 , 2 , … , r . The sampling set { x j } j ∈ N is randomly chosen with a general probability distribution over a bounded cube C K and the samples are the form of convolution { f ∗ ψ (x j) } j ∈ N of the signal f. Under some proper conditions for the generator Φ , convolution function ψ and probability density function ρ , we first approximate V p (Φ) by a finite dimensional subspace V N p (Φ) = ∑ i = 1 r ∑ | k | ≤ N c i (k) ϕ i (· - k) : c i ∈ ℓ p ([ - N , N ] d). Then we show that the sampling stability holds with high probability for all functions in certain compact subsets V K p (Φ) = f ∈ V p (Φ) : ∫ C K | f ∗ ψ (x) | p d x ≥ (1 - δ) ∫ R d | f ∗ ψ (x) | p d x of V p (Φ) when the sampling size is large enough. Finally, we prove that the stability is related to the properties of the random matrix generated by { ϕ i ∗ ψ } 1 ≤ i ≤ r and give a reconstruction algorithm for the convolution random sampling of functions in V N p (Φ) . [ABSTRACT FROM AUTHOR]

We propose a new approach to the theory of conditioning for numerical analysis problems for which both classical and stochastic perturbation theories fail to predict the observed accuracy of computed solutions. To motivate our ideas, we present examples of problems that are discontinuous at a given input and even have infinite stochastic condition number, but where the solution is still computed to machine precision without relying on structured algorithms. Stimulated by the failure of classical and stochastic perturbation theory in capturing such phenomena, we define and analyse a weak worst-case and a weak stochastic condition number. This new theory is a more powerful predictor of the accuracy of computations than existing tools, especially when the worst-case and the expected sensitivity of a problem to perturbations of the input is not finite. We apply our analysis to the computation of simple eigenvalues of matrix polynomials, including the more difficult case of singular matrix polynomials. In addition, we show how the weak condition numbers can be estimated in practice. [ABSTRACT FROM AUTHOR]

Heat transfer coefficient (HTC) is a useful concept to characterize the heat exchange performance of rock fracture. However, existing study has preliminarily found that some HTC formulas would lead to abnormal values, which claims a systematical evaluation of existing formulas. In this paper, 214 test records are employed to evaluate 8 existing formulas from literature. It is found that, for all these experimental data, Bai's formula (Formula D) developed in 2017 shows fairly good numerical stability, while the rest show numerical oscillations and anomalies in different degrees. Moreover, the numerical oscillations and anomalies would be exacerbated with the increase of flow rates, which leads to poor applicability. Error propagation theory in numerical analysis is used to analyze the mechanism of the numerical oscillations and anomalies which can be effectively explained with the condition number and the relative error of the corresponding formula. [ABSTRACT FROM AUTHOR]

By reducing the systems of singular integral equations (SIE) to two types, we carry out a numerical analysis of the problem of mathematical physics about interaction of stationary plane strain waves with a rigid inclusion (cavity with a clamped contour) located in an infinite isotropic elastic medium. The problem is solved using the systems of SIEs of the 1st and 2nd kinds, where the latter has an indefinable index. The conditionality of the models is analyzed using cluster high-precision computational schemes. [ABSTRACT FROM AUTHOR]

We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium. [ABSTRACT FROM AUTHOR]

We introduce an effective iterative method for solving rectangular linear systems, based on gradients along with the steepest descent optimization. We show that the proposed method is applicable with any initial vectors as long as the coefficient matrix is of full column rank. Convergence analysis produces error estimates and the asymptotic convergence rate of the algorithm, which is governed by the term 1 − κ − 2 , where κ is the condition number of the coefficient matrix. Moreover, we apply the proposed method to a sparse linear system arising from a discretization of the one-dimensional Poisson equation. Numerical simulations illustrate the capability and effectiveness of the proposed method in comparison to the well-known and recent methods. [ABSTRACT FROM AUTHOR]

It is known that the convergence rate of the traditional iteration methods like the conjugate gradient method depends on the condition number of the stiffness matrix. Moreover the construction of fast solvers like multigrid and domain decomposition methods also need to estimate the condition number of the stiffness matrix. The main purpose of this paper is to give a rigorous condition number estimate of the stiffness matrix resulting from the linear and bilinear immersed finite element approximations of the high-contrast interface problem. It is shown that the condition number is C ρ h - 2 , where ρ is the jump of the discontinuous coefficients, h is the mesh size, and the constant C is independent of ρ and the location of the interface on the triangulation. Numerical results are also given to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

The problem of computing the roots of a particular sequence of sparse polynomials pn(t) is considered. Each instance pn(t) incorporates only the n + 1 monomial terms t , t 2 , t 4 , ... , t 2 n associated with the binomial coefficients of order n and alternating signs. It is shown that pn(t) has (in addition to the obvious roots t = 0 and 1) precisely n − 1 simple roots on the interval (0,1) with no roots greater than 1, and a recursion relating pn(t) and pn+ 1(t) is used to show that they possess interlaced roots. Closed–form expressions for the Bernstein coefficients of pn(t) on [0,1] are derived and employed to compute the roots in double–precision arithmetic. Despite the severe variation of the graph of pn(t), and tight clustering of roots near t = 1, it is shown that for n ≤ 10, the roots on (0,1) are remarkably well–conditioned and can be computed to high accuracy using both the power and Bernstein forms. [ABSTRACT FROM AUTHOR]

We study the average condition number for polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average. [ABSTRACT FROM AUTHOR]