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Data assimilation algorithms combine prior and observational information, weighted by their respective uncertainties, to obtain the most likely posterior of a dynamical system. In variational data assimilation the posterior is computed by solving a nonlinear least squares problem. Many numerical weather prediction (NWP) centers use full observation error covariance (OEC) weighting matrices, which can slow convergence of the data assimilation procedure. Previous work revealed the importance of the minimum eigenvalue of the OEC matrix for conditioning and convergence of the unpreconditioned data assimilation problem. In this article we examine the use of correlated OEC matrices in the preconditioned data assimilation problem for the first time. We consider the case where there are more state variables than observations, which is typical for applications with sparse measurements, for example, NWP and remote sensing. We find that similarly to the unpreconditioned problem, the minimum eigenvalue of the OEC matrix appears in new bounds on the condition number of the Hessian of the preconditioned objective function. Numerical experiments reveal that the condition number of the Hessian is minimized when the background and observation lengthscales are equal. This contrasts with the unpreconditioned case, where decreasing the observation error lengthscale always improves conditioning. Conjugate gradient experiments show that in this framework the condition number of the Hessian is a good proxy for convergence. Eigenvalue clustering explains cases where convergence is faster than expected. [ABSTRACT FROM AUTHOR]

Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT) matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such p\times q submatrix of the N x N DFT matrix is at least exp (Π/2[min(p, q) pq/N ]), up to algebraic prefactors. That is, fixing the shape parameters (α, β):= (p/N, q/N)\in (0, 1)2, the growth is e\rho N as N, the exponential rate being = Π 2 [min(α, β]. Our proof uses the Kaiser--Bessel transform pair (of which we give a self-contained proof), plus estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kernel eixt, we also prove another lower bound (4/e\Π\alpha)q, up to algebraic prefactors, which is stronger than the above for small and α and β. When combined, the bounds are within a factor of two of the empirical asymptotic rate, uniformly over (0, 1)², and become sharp in certain regions. However, the results are not asymptotic: they apply to essentially all N, p, and q, and with all constants explicit. [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]

Many recent data analysis models are mathematically characterized by a multidimensional total least squares problem with linear equality constraints (TLSE). In this paper, an explicit solution is firstly derived for the multidimensional TLSE problem, as well as the solvability conditions. With applying the perturbation theory of invariant subspace, the multidimensional TLSE problem is proved equivalent to a multidimensional unconstrained weighed total least squares problem in the limit sense. The Kronecker product-based formulae are also given for the normwise, mixed, and componentwise condition numbers of the multidimensional TLSE solution of minimum Frobenius norm, and their computable upper bounds are also provided to reduce the storage and computational cost. All these results are appropriate for the single right-hand-side case and the multidimensional total least squares problem, which are two especial cases of the multidimensional TLSE problem. In numerical experiments, the multidimensional TLSE model is successfully applied to color image deblurring and denoising for the first time, and the numerical results also indicate the effectiveness of the condition numbers. [ABSTRACT FROM AUTHOR]

As known, outliers and multicollinearity in the data set are among the important difficulties in regression models, which badly affect the leastsquares estimators. Under multicollinearity and outliers' existence in the data set, the prediction performance of the least-squares regression method is decreased dramatically. Here, proposing an approximation for the condition number, we suggest a nonlinear mixed-integer programming model to simultaneously control inappropriate effects of the mentioned problems. The model can be effectively solved by popular metaheuristic algorithms. To shed light on importance of our optimization approach, we make some numerical experiments on a classic real data set as well as a simulated data set. [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]

The raise regression was presented as an alternative methodology to estimate models with collinearity, it is to say, when columns of matrix X are nearly linearly dependent. This method consists in transforming matrix X by matrix X that is obtained after raising one of the explanatory variable to move it geometrically away from the span of the remaining variables. In this article, we analyze the behavior of the condition number in the raise regression showing that the effectiveness of the raise method to solve the problem of multicollinearity is limited to the potential reduction of the condition number that presents an asymptote corresponding to the condition number associated to the auxiliary regression of the raised variable. All the contributions are illustrated with an empirical application. [ABSTRACT FROM AUTHOR]

Investigators that seek to employ regression analysis usually encounter the problem of multicollinearity with dependency on two or more explanatory variables. Multicollinearity is associated with unstable estimated coefficients and it results in high variances of the least squares estimators in linear regression models (LRMs). Thus the detection of collinearity is the compulsory first step in regression analysis. Multicollinearity also come out in generalized linear models (GLMs) and has same serious effects on the maximum likelihood estimates. The purposes of this paper are to propose new collinearity diagnostics criteria in GLMs in the context of both the maximum likelihood and ridge estimators, to examine the properties of new collinearity diagnostics via the ridge constant, to exemplify the theoretical results by numerical examples on Poisson, Binomial and Gamma responses. The effects of centering and scaling the information matrix on the sensitivity of the diagnostics in the presence of collinearity are also investigated. [ABSTRACT FROM AUTHOR]

Following recent interest by the community, the scaling of the minimal singular value of a Vandermonde matrix with nodes forming clusters on the length scale of Rayleigh distance on the complex unit circle is studied. Using approximation theoretic properties of exponential sums, we show that the decay is only single exponential in the size of the largest cluster, and the bound holds for arbitrary small minimal separation distance. We also obtain a generalization of well-known bounds on the smallest eigenvalue of the generalized prolate matrix in the multi-cluster geometry. Finally, the results are extended to the entire spectrum. [ABSTRACT FROM AUTHOR]

BACKGROUND: The limited-angle reconstruction problem is of both theoretical and practical importance. Due to the severe ill-posedness of the problem, it is very challenging to get a valid reconstructed result from the known small limited-angle projection data. The theoretical ill-posedness leads the normal equation AT Ax = AT b of the linear system derived by discretizing the Radon transform to be severely ill-posed, which is quantified as the large condition number of AT A. OBJECTIVE: To develop and test a new valid algorithm for improving the limited-angle image reconstruction with the known appropriately small angle range from [ 0 , π 3 ] ∼ [ 0 , π 2 ]. METHODS: We propose a reweighted method of improving the condition number of AT Ax = AT b and the corresponding preconditioned Landweber iteration scheme. The weight means multiplying AT Ax = AT b by a matrix related to AT A, and the weighting process is repeated multiple times. In the experiment, the condition number of the coefficient matrix in the reweighted linear system decreases monotonically to 1 as the weighting times approaches infinity. RESULTS: The numerical experiments showed that the proposed algorithm is significantly superior to other iterative algorithms (Landweber, Cimmino, NWL-a and AEDS) and can reconstruct a valid image from the known appropriately small angle range. CONCLUSIONS: The proposed algorithm is effective for the limited-angle reconstruction problem with the known appropriately small angle range. [ABSTRACT FROM AUTHOR]

This paper presents theoretical results on estimation the condition number with application in image restoration problems.

• Nonuniform E-field is restored by measuring E-FISH signals along probing laser. • Convergence of inversion algorism related to measuring system parameter is given. • Accuracy of inversion algorism is validated via experimental investigations. • Robustness of inversion algorism in resisting measurement noise is confirmed. The electric field-induced second-harmonic generation (E-FISH) method can offer high spatiotemporal resolution in electric field measurement, but faces challenge in reconstructing the inhomogeneous electric field along the probing laser. This paper presents an inversion method to reconstruct the electric field profile by using E-FISH signals measured from multiple points along the probing laser. The relationship between the condition number of the inversion coefficient matrix and the parameters of E-FISH measurement system is clarified. The inversion method is validated through experimental investigations conducted in three different electrode configurations, namely the plate-to-plate gap, the rod-to-rod gap, and the rod-plate gap. With the inhomogeneity coefficient varies from 1 to 10.7, the maximum deviation between the measured electric field and the finite element simulation is less than 6%. This method does not rely on priori assumptions and electrode configurations, and exhibits good robustness in resisting the effect of measurement noise on the inversion accuracy. [ABSTRACT FROM AUTHOR]

The objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.

The previous research on the controllability of complex networks is qualitative issue which can only identify whether the network is controllable or not, and it is difficult to assess the controlled ability of network. In this paper, a quantitative measurement index of network controllability is introduced. And the index is used to study the controllability of the IEEE power network system. The relationship between controllability and network parameters (including the average degree, the clustering coefficient, the average path length) of power network is analyzed by comparing with the ER random network, WS small world network, BA scale-free network and CM network. The results show that the power networks have a small average path length and a higher clustering coefficient, which belong to the small world network. And, the controllability of power networks is larger than the corresponding model networks, indicating that power networks are relatively easier to be controlled.

High dimensional error covariance matrices and their inverses are used to weight the contribution of observation and background information in data assimilation procedures. As observation error covariance matrices are often obtained by sampling methods, estimates are often degenerate or ill-conditioned, making it impossible to invert an observation error covariance matrix without the use of techniques to reduce its condition number. In this paper, we present new theory for two existing methods that can be used to ‘recondition’ any covariance matrix: ridge regression and the minimum eigenvalue method. We compare these methods with multiplicative variance inflation, which cannot alter the condition number of a matrix, but is often used to account for neglected correlation information. We investigate the impact of reconditioning on variances and correlations of a general covariance matrix in both a theoretical and practical setting. Improved theoretical understanding provides guidance to users regarding method selection, and choice of target condition number. The new theory shows that, for the same target condition number, both methods increase variances compared to the original matrix, with larger increases for ridge regression than the minimum eigenvalue method. We prove that the ridge regression method strictly decreases the absolute value of off-diagonal correlations. Theoretical comparison of the impact of reconditioning and multiplicative variance inflation on the data assimilation objective function shows that variance inflation alters information across all scales uniformly, whereas reconditioning has a larger effect on scales corresponding to smaller eigenvalues. We then consider two examples: a general correlation function, and an observation error covariance matrix arising from interchannel correlations. The minimum eigenvalue method results in smaller overall changes to the correlation matrix than ridge regression but can increase off-diagonal correlations. Data assimilation experiments reveal that reconditioning corrects spurious noise in the analysis but underestimates the true signal compared to multiplicative variance inflation.

In geodesy and geophysics, many large-scale over-determined linear equations need to be solved which are often ill-conditioned. When the conjugate gradient method is used, their ill-conditioning effects to the solutions must be overcome, which is studied in this paper. Through the regularization ideas, the conjugate gradient method is improved, and the regularization iterative solution based on controlling condition number is put forward. Firstly by constructing the interference source vector, a new equation is derived with ill-condition diminished greatly, which has the same solution to the original normal equation. Then the new equation is solved by conjugate gradient method. Finally the effectiveness of the new method is verified by some numerical experiments of airborne gravity downward to the earth surface. In the numerical experiments the new method is compared with LS, CG and Tikhonov methods, and its accuracy is the highest.

The paper generalizes the concept of “solving a system of linear algebraic equations in order to formulate a unified approach to the analysis of incompatible, indefinite and unstable systems”. Examples of unstable systems of linear algebraic equations are considered, which solutions depend on small changes in the numerical coefficients in the equations. The reasons for the instability of linear systems and the regularization algorithm for finding the solution of any system of linear algebraic equations are discussed. As the author notes, the Tikhonov regulatory algorithm is the most popular and practically convenient for solving unstable SLAES.

The error propagation of Capon’s minimum variance estimator resulting from measurement errors and position errors is derived within a linear approximation. It turns out, that Capon’s estimator provides the same error propagation as the conventionally used least square fit method. The shape matrix which describes the location depence of the measurement positions is the key parameter for the error propagation, since the condition number of the shape matrix determines how the errors are amplified. Furthermore, the error resulting from a finite number of data samples is derived by regarding Capon’s estimator as a special case of the maximum likelihood estimator.

Gyroscope biases may drift in long-term storage, and the drifts become the most important factors that affect the azimuth alignment accuracy of the inertial navigation systems (INSs). In this article, a three-position in- field calibration method is proposed, providing a completely new way to solve the gyroscope bias estimation problem without the turntable available. Based on the analysis of the azimuth errors of attitude tracking, the attitude tracking information at the current position is used to extract the azimuth error of self-alignment at the previous position. The observation equations are derived, with the input arguments of the differences between the attitude tracking results and the self-alignment results at the last two positions, and the solution vectors of the gyroscope biases. Condition number is used to analyze the ill-conditioned problem of the equations. Simulation and experiment results show that the gyroscope biases can be effectively calibrated by designing the rotation angles reasonably. [ABSTRACT FROM AUTHOR]

In this paper, we develop a family of high order cut discontinuous Galerkin (DG) methods for hyperbolic conservation laws in one space dimension. Ghost penalty stabilization is used to stabilize the scheme for small cut elements. Analysis shows that our proposed methods have similar stability and accuracy properties as the standard DG methods on a regular mesh. We also prove that the cut DG method with piecewise constants in space is total variation diminishing. We use the strong stability preserving Runge--Kutta method for time discretization and the time step is independent on the size of cut element. Numerical examples demonstrate the cut DG methods are high order accurate for smooth problems and perform well for discontinuous problems. [ABSTRACT FROM AUTHOR]

We consider a measurement constrained supervised learning problem, that is, (i) full sample of the predictors are given; (ii) the response observations are unavailable and expensive to measure. Thus, it is ideal to select a subsample of predictor observations, measure the corresponding responses, and then fit the supervised learning model on the subsample of the predictors and responses. However, model fitting is a trial and error process, and a postulated model for the data could be misspecified. Our empirical studies demonstrate that most of the existing subsampling methods have unsatisfactory performances when the models are misspecified. In this paper, we develop a novel subsampling method, called "LowCon," which outperforms the competing methods when the working linear model is misspecified. Our method uses orthogonal Latin hypercube designs to achieve a robust estimation. We show that the proposed design-based estimator approximately minimizes the so-called worst-case bias with respect to many possible misspecification terms. Both the simulated and real-data analyses demonstrate the proposed estimator is more robust than several subsample least-squares estimators obtained by state-of-the-art subsampling methods. for this article are available online. [ABSTRACT FROM AUTHOR]

Tight frames are similar to orthogonal bases, except that the frame coefficients are not unique, but they are stable in calculations and numerical algorithms. Not all frames are tight frames, but some have the ability to become tight frames. These frames are called scalable frames. In this article, we extend this good property of frames to G-frames. For this purpose, we define the scalable G-frame based on the diagonal operators, and obtain a preconditioner for its analysis operator by block diagonal operator. We also provide the necessary and sufficient conditions for the scalability of the G-frames based on the frames induced by the G-frames. [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]

Perturbation analysis of the equality constrained quadratic programming is considered. We present two different perturbation bounds to explore underlying factors for affecting the conditioning of equality constrained quadratic programming, and propose the condition numbers to give sharp forward error bounds. To improve the computational efficiency of condition numbers, some new compact forms and tight upper bounds of the condition numbers are introduced. Numerical examples are given to illustrate our theoretical results. As a special case of equality constrained quadratic programming, the rigorous perturbation analysis of Markowitz mean–variance model is also studied, which can be used to give a formal characterization of the roles of condition number and the smallest eigenvalue of the covariance matrix in bounding the forward errors. With respect to condition number and the smallest eigenvalue of the covariance matrix, numerical performances of two different covariance matrix estimators on optimal portfolio selection are also presented through simulations. [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]

Following a recent attempt by Waziri et al. [2019] to find an appropriate choice for the nonnegative parameter of the Hager–Zhang conjugate gradient method, we have proposed two adaptive options for the Hager–Zhang nonnegative parameter by analyzing the search direction matrix. We also used the proposed parameters with the projection technique to solve convex constraint monotone equations. Furthermore, the global convergence of the methods is proved using some proper assumptions. Finally, the efficacy of the proposed methods is demonstrated using a number of numerical examples. [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, the contingency is ranked for the system under a single transmission line outage condition. The ranking is carried with the static model of Quadrature Booster (QB) in the system. QB is incorporated by modifying the Jacobian matrix of the Newton-Raphson load flow technique. The contingencies are ranked based on the Condition Number of the Jacobian. The ranking is also carried for the enhanced system loading condition. IEEE-30 bus system is used for the proposed methodology and MATLAB environment is considered for the simulation purpose. [ABSTRACT FROM AUTHOR]

In this paper we analyze the nearly optimal block diagonal scalings of the rows of one factor and the columns of the other factor in the triangular form of the SR decomposition. The result is a block generalization of the result of the van der Sluis about the almost optimal diagonal scalings of the general rectangular matrices. [ABSTRACT FROM AUTHOR]

This paper is devoted to perturbation analysis of the total least squares problem with linear equality constraint (TLSE). Different magnitudes of perturbations in input data are taken into account, and the proposed results deliver better estimates of forward errors of the solution than those based on already well-known condition numbers in several numerical tests. Relations between the TLSE solution/residual and the ones from the equality constrained least squares problem (LSE) are revealed. Various condition numbers and perturbation results for the LSE problem in literature can be recovered from newly derived results. [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]

Given a family of linear constraints and a linear objective function one can consider whether to apply a Linear Programming (LP) algorithm or use a Linear Superiorization (LinSup) algorithm on this data. In the LP methodology one aims at finding a point that fulfills the constraints and has the minimal value of the objective function over these constraints. The Linear Superiorization approach considers the same data as linear programming problems but instead of attempting to solve those with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward feasible points with reduced (not necessarily minimal) objective function values. Previous studies compared LP and LinSup in terms of their respective outputs and the resources they use. In this paper we investigate these two approaches in terms of their sensitivity to condition numbers of the system of linear constraints. Condition numbers are a measure for the impact of deviations in the input data on the output of a problem and, in particular, they describe the factor of error propagation when given wrong or erroneous data. Therefore, the ability of LP and LinSup to cope with increased condition numbers, thus with ill-posed problems, is an important matter to consider which was not studied until now. We investigate experimentally the advantages and disadvantages of both LP and LinSup on examplary problems of linear programming with multiple condition numbers and different problem dimensions.