Causes Of Ill-conditioning Research Articles

A statistical analysis of the numerical condition of multiple roots of polynomials

Componentwise and normwise condition numbers of an m-tuple root x 0 of a polynomial p( x) that are appropriate for measurement and experimental inaccuracies are derived. These new condition numbers must be compared with the established condition numbers, which are appropriate for quantifying the effect of roundoff errors due to floating point arithmetic. It is shown that the condition numbers that are derived in this paper may be considered average case (as opposed to worst case) because extensive use is made of the expected values of random variables and functions of random variables. Specifically, it is assumed that each coefficient of p( x) is perturbed by an independent zero mean Gaussian random variable, and a measure of the condition of x 0 is defined as the ratio of the expected value of its relative error to the expected value of the relative error in the coefficients of p( x), defined in both the componentwise and normwise forms. It is shown that this distinction between the componentwise and normwise condition estimates is important because they may differ by several orders of magnitude, depending on the coefficients of the polynomial. The cause of ill-conditioning of multiple roots is considered and it is shown that the situations m = 1 and m > 1 must be treated separately. Computational experiments that illustrate the theoretical results are presented.

Study of Ill-conditioning of Linac Stereotactic Irradiation Subspaces using Singular Values Decomposition Analysis

A Linac stereotactic irradiation space is characterized by different angular separations of beams because of the geometry of the stereotactic irradiation. The regions of the stereotactic space characterized by low angular separations are one of the causes of ill-conditioning of the stereotactic irradiation inverse problem. The singular value decomposition (SVD) is a powerful mathematical analysis that permits the measurement of the ill-conditioning of the stereotactic irradiation problem. This study examines the ill-conditioning of the stereotactic irradiation space, provoked by the different angular separations of beams, using the SVD analysis. We subdivided the maximum irradiation space (MIS: (AA)AP

State estimation using augmented blocked matrices

A novel method of blocking the Hachtel formulation for power system state estimation is proposed. The key idea is organization of the Hachtel formulation into a submatrix block structure that conforms to that of the incidence matrix of the network. This leads to the normal first-neighbor topological structure of the blocked matrix, which is processed in the same sparsity-preserving order as the nodal admittance matrix. Explicit inversion of diagonal submatrix blocks overcomes former difficulties with small or zero diagonals and further enhances numerical stability. Equality constraints are enforced exactly instead of as measurements with artificially large weighting factors. Another feature of the method is that it can accommodate a more realistic covariance matrix having nonzero off-diagonal weighting factors corresponding to correlations between the active and reactive power measurements at a bus. In fact, any covariance terms that fall within the submatrix blocks (in locations that are now zeros) can be accommodated with no effects on computational burden. As indicated by its large reduction in the numerical condition number of three representation test problems, the proposed method should be able to overcome the usual causes of ill-conditioning of power system state estimation. As shown by comparisons of its computational requirements with those of other methods, there is no sacrifice in speed to achieve robustness. >

Remote sensing of the middle atmospheric aerosol

Remote sensing of the middle atmospheric aerosol