Condition Number Cond Research Articles

On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Abstract Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn ) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn ) for n = 2 kpℓ , where k, ℓ ≥ 0 are integers and p is an odd prime number.

Stability analysis for singularly perturbed differential equations by the upwind difference scheme

AbstractFor solving singularly perturbed differential equations (SPDE), the upwind difference scheme (UDS) and the fitted difference method are chosen. The local refinements of grids are adopted in singular layers with the minimal meshspacing , where h is the maximal meshspacing, and a very small parameter. The traditional condition number in 2‐norm is given by . For the infinitesimal in application, the huge Cond issues a dilemma regarding whether the numerical solutions by the UDS can be trusted. Although the UDS has been used for several decades, such a dilemma has not been clarified yet. The goal of this article is to clarify this dilemma. To this end, we solicit the effective condition number Cond_eff in Li et al. Numer Linear Algebra Appl 15 (2008), 575–690 Effective condition for Numerical Partial Different Equation, 2013, and develop a new actual condition number from the maximum principle. Both of them may offer much smaller bounds of the solution errors caused by perturbation, e.g., rounding, truncation, or discritization errors. We study the Dirichlet problems of SPDE by the UDS in a rectangle. When the Dirichlet boundary condition on the downwind side is homogeneous, we derive . When the entire Dirichlet boundary conditions are homogeneous, the extraordinary bound, , is achieved. Moreover, we derive the actual condition numbers as and for the homogeneous and the nonhomogeneous SPDE, respectively. Note that these bounds do not depend on ε; this is distinct from the traditional Cond. Based on the analysis of this article, the existing dilemma caused by Cond has been removed, to grant a good stability of the UDS for SPDE. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1595–1613, 2014

Multicolinearity Test in Eviews

This program tests multicolinearity with condition number and variance inflation factor (VIF) for given set of explanatory variables. It report high multicolinearity if condition number (cond) be greater than 20(or VIF greater than 10).

Effective condition number for weighted linear least squares problems and applications to the Trefftz method

In [27], the effective condition number Cond_eff is developed for the linear least squares problem. In this paper, we extend the effective condition number for weighted linear least squares problem with both full rank and rank-deficient cases. We apply the effective condition number to the collocation Trefftz method (CTM) [29] for Laplace's equation with a crack singularity, to prove that Cond_eff = O ( L ) and Cond = O ( L 1 / 2 ( 2 ) L ) , where L is the number of singular particular solutions used. The Cond grows exponentially as L increases, but Cond_eff is only O ( L ) . The small effective condition number explains well the high accuracy of the TM solution, but the huge Cond cannot.

Effective Condition Number of Finite Difference Method for Poisson's Equation Involving Boundary Singularities

For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A, the effective condition number Cond_eff is defined in [6, 10] by following Chan and Foulser [2] and Rice [14]. The Cond_eff is smaller, or much smaller, than the traditional condition number Cond. Besides, the simplest condition number Cond_EE is also defined in [6, 10]. This article studies a popular model of Poisson's equation involving the boundary singularities by the finite difference method using the local refinements of grids. The bounds of Cond_EE are derived to display theoretically that the effective condition number is significantly smaller than the Cond. In this article, by exploring local refinement properties, we derive the bounds of effective condition numbers up to O(1) and at least o(h −1/2) for the maximal step size h. They are significant improvements compared with the bound O(h −3/2), which is established in [6, 10]. Therefore, the study of effective condition number in this article reaches a new comprehensive and advanced level.

Effective condition number and its applications

Consider the over-determined system Fx = b where F ∈ Rm × n, m ≥ n and rank (F) = r ≤ n, the effective condition number is defined by Cond_eff = ||b||/σ1||x||, where the singular values of F are given as σmax = σ1 ≥ σ1 ≥...≥ σr > 0 and σr+1 = ... = σn = 0. For the general perturbed system (A + ΔA)(x+ Δx) = b+ Δb involving both ΔA and Δb, the new error bounds pertinent to Cond_eff are derived. Next, we apply the effective condition number to the solutions of Motz's problem by the collocation Trefftz methods (CTM). Motz's problem is the benchmark of singularity problems. We choose the general particular solutions vL = Σk=0L dk(r/Rp)k+1/2 cos(k + 1/2)θ with a radius parameter Rp. The CTM is used to seek the coefficients di by satisfying the boundary conditions only. Based on the new effective condition number, the optimal parameter Rp = 1 is found. which is completely in accordance with the numerical results. However, if based on the traditional condition number Cond, the optimal choice of Rp is misleading. Under the optimal choice Rp = 1, the Cond grows exponentially as L increases, but Cond_eff is only linear. The smaller effective condition number explains well the very accurate solutions obtained. The error analysis in [14, 15] and the stability analysis in this paper grant the CTM to become the most efficient and competent boundary method.

Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace’s equation

Since the stability of the method of fundamental solutions (MFS) is a severe issue, the estimation on the bounds of condition number Cond is important to real application. In this paper, we propose the new approaches for deriving the asymptotes of Cond, and apply them for the Dirichlet problem of Laplace’s equation, to provide the sharp bound of Cond for disk domains. Then the new bound of Cond is derived for bounded simply connected domains with mixed types of boundary conditions. Numerical results are reported for Motz’s problem by adding singular functions. The values of Cond grow exponentially with respect to the number of fundamental solutions used. Note that there seems to exist no stability analysis for the MFS on non-disk (or non-elliptic) domains. Moreover, the expansion coefficients obtained by the MFS are oscillatingly large, to cause the other kind of instability: subtraction cancelation errors in the final harmonic solutions.

Effective condition number for the finite element method using local mesh refinements

This is a continued study but at advanced levels of effective condition number in [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, J. Comput. Appl. Math. 198 (2007) 208–235; Z.C. Li, H.T. Huang, Effective condition number for numerical partial differential equations, Numer. Linear Algebra Appl. 15 (2008) 575–594] for stability analysis. To approximate Poisson's equation with singularity by the finite element method (FEM), the adaptive mesh refinements are an important and popular technique, by which, the FEM solutions with optimal convergence rates can be obtained. The local mesh refinements are essential to FEM for solving complicated problems with singularities, and they have been used for three decades. However, the traditional condition number is given by Cond = O ( h min − 2 ) in Strang and Fix [G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973], where h min is the minimal length of elements. Since h min is infinitesimal near the singular points, Cond is huge. Such a dilemma can be bypassed by small effective condition number. In this paper, the bounds of the simplified effective condition number Cond_EE are derived as O ( 1 ) , O ( h − 1.5 ) or O ( h − 0.5 ) , where h ( < 1 ) is the maximal length of elements. Evidently, Cond_EE is much smaller than Cond. The numerical experiments are carried out, to verify the stability analysis. Small effective condition numbers explain well the satisfactory FEM solutions obtained. This paper provides a stability justification for the adaptive mesh refinements used in FEM. Compared with [Z.C. Li, C.S. Chien, H.T. Huang, Effective condition number for finite difference method, J. Comput. Appl. Math. 198 (2007) 208–235; Z.C. Li, H.T. Huang, Effective condition number for numerical partial differential equations, Numer. Linear Algebra Appl. 15 (2008) 575–594], the analysis in this paper is more difficult and challenging, its proof techniques are new and intriguing, and the results are more important and useful.

Effective condition number of Trefftz methods for biharmonic equations with crack singularities

AbstractThe paper presents the new stability analysis for the collocation Trefftz method (CTM) for biharmonic equations, based on the new effective condition number Cond_eff. The Trefftz method is a special spectral method with the particular solutions as admissible functions, and it has been widely used in engineering. Three crack models in Li et al. (Eng. Anal. Boundary Elements 2004; 28:79–96; Trefftz and Collocation Methods. WIT Publishers: Southampton, Boston, 2008) are considered, and the bounds of Cond_eff and the traditional condition number Cond are derived, to give the polynomial and the exponential growth rates, respectively. The stability analysis explains well the numerical experiments. Hence, the new Cond_eff is more advantageous than Cond. Besides since the bounds of Cond_eff and Cond involve the estimation of the minimal singular value σmin of the discrete matrix F, and since the estimation of σmin is challenging and difficult, the proof for lower bounds of σmin in this paper is important and intriguing. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Study on effective condition number for collocation methods

In this paper, the stability study is further made for three kinds of collocation methods. (I) The general collocation Trefftz methods (GCTMs) using piecewise particular solutions, while in Li et al. [Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems. Eng Anal Boundary Elem 2004;31:163–75] the collocation Trefftz methods (CTMs) using the uniform particular solutions in the entire domain. (II) The method of fundamental solutions (MFSs), i.e., the CTM using the fundamental solution (FS). (III) The collocation method using the radial basis functions (CM-RBFs). In this paper, the new bounds of the traditional condition number Cond and the effective condition number Cond_eff are derived, and comparisons are made to confirm that the effective condition number is a better criterion for stability analysis. The main results are as follows. (1) For Motz's problem by GCTM, Cond _ eff = O ( N ) and Cond = O ( a - N ) ( 0 < a < 1 ) are obtained by a suitable scaling, where N is the number of admissible functions used. (2) For MFS and CM-RBF, both Cond and Cond_eff are exponential with respect to N, but Cond_eff is much smaller than Cond. (3) For MFS and CM-RBF, the expansion coefficients of FS and radial basis functions may be oscillatingly large, to cause a severe subtraction cancellation in the final solutions. Different behaviours of Cond_eff compared to Cond are explored for different collocation methods in this paper, to provide a comprehensive and intrinsic nature of Cond_eff.

Effective condition number for numerical partial differential equations

AbstractIn this paper, the new computational formulas are derived for the effective condition number Cond_eff, and the new error bounds involved in both Cond and Cond_eff are developed. A theoretical analysis is provided to support some conclusions in Banoczi et al. (SIAM J. Sci. Comput. 1998; 20:203–227). For the linear algebraic equations solved by the Gaussian elimination or the QR factorization (QR), the direction of the right‐hand vector is insignificant for the solution errors, but such a conclusion is invalid for the finite difference method for Poisson's equation. The effective condition number is important to the numerical partial differential equations, because the discretization errors are dominant. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Effective condition number for simplified hybrid Trefftz methods

The simplified hybrid Trefftz method was first proposed in Trefftz [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7] in 1926 for solving Laplace's equation, where the harmonic functions are chosen as admissible functions, and their linear combination is sought to satisfy the boundary conditions. The error analysis of the hybrid TM is provided in [Li ZC, Chen YL, Georgiou GG, Xenohontos C. Special boundary approximation methods for Laplace equation problems with boundary singularities—applications to the Motz problem. Int Comput Math Appl 2006;51:115–42; Li ZC, Lu TT, Hu HY, Cheng AH-D. Trefftz and collocation methods. Southampton: WIT Publishers; 2007], but no stability analysis exists so far. Also the simplified hybrid techniques have been applied for the TM to couple with the finite element method (FEM) in our previous study and only the error analysis has been made. Hence, the stability analysis is important for the simplified hybrid TM. In this paper, we will apply the effective condition number Cond _ eff . For the original hybrid TM [Ein Gegenstuck zum Ritz'schen Verfahren. In: Proceedings of the second international congress on applied mechanics, Zurich, 1926. p. 131–7], uniform particular solutions satisfying the governed equation (e.g., the harmonic functions satisfying Laplace's equation) were chosen. In general, piecewise particular solutions can be used for wide application of the hybrid TM, and the interior continuity conditions may be dealt with by hybrid techniques. Their algorithms and error analysis are provided in [Huang HT, Li ZC, Herrera I. Coupling techniques of Trefftz methods. Technical Report, Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan; 2006] without stability analysis. In this paper, two cases of the simplified hybrid TM are considered: Case I: uniform particular solutions used, and Case II: piecewise particular solutions used. It is proved that both Cond _ eff and Cond grow exponentially, with respect to the number of particular solutions used. In Case I, Cond is huge and Cond _ eff is significantly smaller than Cond; but in Case II, Cond is moderately large, and Cond _ eff is significantly smaller than Cond. Hence the ill-conditioning of the simplified hybrid TM for Case II is not severe. Such theoretical results have been validated by the numerical experiments. The study of Cond _ eff in this paper provides a complete and comprehensive knowledge of the simplified hybrid TM.

Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems

For solving the linear algebraic equations Ax = b , the new stability analysis is made based on the effective condition number Cond_eff. The Cond_eff may provide a better upper bound of relative errors of x resulting from the rounding errors of b , than the traditional condition number Cond, which with too large value is, in many times, misleading. In this paper, we apply the effective condition number to the Trefftz methods (TMs) for Poisson's equations with singularities. Two TMs, such as the penalty plus hybrid TM and the Lagrange multiple (i.e., direct) TM, are studied. We focus on the stability analysis of the solutions when the optimal superconvergence is achieved. When solving Motz's problem, the benchmark of singularity problems, by the penalty plus hybrid TM, we have derived that Cond _ eff = O ( N ( 2 ) N ) and Cond = O ( N 2 2 N ) , where N is the number of the singular particular functions used. For solving Motz's problem by the direct TM, we have derived that Cond / Cond _ eff = O ( N 2 N ) . Numerical experiments are provided to verify the stability analysis made. In summary, the two TMs are efficient, but the penalty plus hybrid TM is more recommended, due to simplicity of algorithms without extra-variables and less limitations in applications.

Effective condition number for finite difference method

For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A , from elliptic equations, the traditional condition number in the 2-norm is defined by Cond . = λ 1 / λ n , where λ 1 and λ n are the maximal and minimal eigenvalues of the matrix A , respectively. The condition number is used to provide the bounds of the relative errors from the perturbation of both A and b . Such a Cond. can only be reached by the worst situation of all rounding errors and all b . For the given b , the true relative errors may be smaller, or even much smaller than the Cond., which is called the effective condition number in Chan and Foulser [Effectively well-conditioned linear systems, SIAM J. Sci. Statist. Comput. 9 (1988) 963–969] and Christiansen and Hansen [The effective condition number applied to error analysis of certain boundary collocation methods, J. Comput. Appl. Math. 54(1) (1994) 15–36]. In this paper, we propose the new computational formulas for effective condition number Cond_eff, and define the new simplified effective condition number Cond_E. For the latter, we only need the eigenvector corresponding to the minimal eigenvalue of A , which can be easily obtained by the inverse power method. In this paper, we also apply the effective condition number for the finite difference method for Poisson's equation. The difference grids are not supposed to be quasiuniform. Under a non-orthogonality assumption, the effective condition number is proven to be O ( 1 ) for the homogeneous boundary conditions. Such a result is extraordinary, compared with the traditional Cond. = O ( h min - 2 ) , where h min is the minimal meshspacing of the difference grids used. For the non-homogeneous Neumann and Dirichlet boundary conditions, the effective condition number is proven to be O ( h - 1 / 2 ) and O ( h - 1 / 2 h min - 1 ) , respectively, where h is the maximal meshspacing of the difference grids. Numerical experiments are carried out to verify the analysis made.

Pivoting strategies leading to small bounds of the errors for certain linear systems

For the solution of a linear system Ax=b using Gaussian elimination, some new properties of scaled partial pivoting strategies for a strictly monotone vector norm ∥·∥ are obtained. Given a nonsingular matrix A, we have proved that, if there exists a permutation matrix P such that PA=LU and |PA|=|L| |U| (where |B| denotes the matrix of absolute values of the entries of B), then P is associated with the scaled partial pivoting strategy for ∥·∥. This result is applied to extend a well-known small componentwise relative backward error for the Gaussian elimination of certain matrices to larger classes of matrices. On the other hand, given a matrix A=LU it is shown that, if there exists an optimal pivoting strategy in order to diminish the Skeel condition number Cond(U) of the resulting upper triangular matrix U, then it coincides with the scaled partial pivoting for ∥·∥. Furthermore, the class of matrices for which no row exchanges is an optimal pivoting strategy to diminish Cond(U) is characterized.

On the condition of a matrix arising in the numerical inversion of the Laplace transform

Bellman, Kalaba, and Lockett recently proposed a numerical method for inverting the Laplace transform. The method consists in first reducing the infinite interval of integration to a finite one by a preliminary substitution of variables, and then employing an n n -point Gauss-Legendre quadrature formula to reduce the inversion problem (approximately) to that of solving a system of n n linear algebraic equations. Luke suggests the possibility of using Gauss-Jacobi quadrature (with parameters α \alpha and β \beta ) in place of Gauss-Legendre quadrature, and in particular raises the question whether a judicious choice of the parameters α \alpha , β \beta may have a beneficial influence on the condition of the linear system of equations. The object of this note is to investigate the condition number cond ( n , α , β ) (n,\alpha ,\beta ) of this system as a function of n n , α \alpha , and β \beta . It is found that cond ( n , α , β ) (n,\alpha ,\beta ) is usually larger than cond ( n , β , α ) (n,\beta ,\alpha ) if β &gt; α \beta &gt; \alpha , at least asymptotically as n → ∞ n \to \infty . Lower bounds for cond ( n , α , β ) (n,\alpha ,\beta ) are obtained together with their asymptotic behavior as n → ∞ n \to \infty . Sharper bounds are derived in the special cases n n , n n odd, and α = β = ± 1 2 \alpha = \beta = \pm \frac {1} {2} , n n arbitrary. There is also a short table of cond ( n , α , β ) (n,\alpha ,\beta ) for α \alpha , β = − .8 ( .2 ) 0 , .5 , 1 , 2 , 4 , 8 , 16 , β ≦ α \beta = - .8(.2)0,.5,1,2,4,8,16,\beta \leqq \alpha , and n = 5 , 10 , 20 , 40 n = 5,10,20,40 . The general conclusion is that cond ( n , α , β ) (n,\alpha ,\beta ) grows at a rate which is something like a constant times ( 3 + √ 8 ) n {(3 + \surd 8)^n} , where the constant depends on α \alpha and β \beta , varies relatively slowly as a function of α \alpha , β \beta , and appears to be smallest near α = β = − 1 \alpha = \beta = - 1 . For quadrature rules with equidistant points the condition grows like ( 2 √ 2 / 3 π ) 8 n (2\surd 2/3\pi ){8^n} .