Condition number, definition

The condition number is always greater than one and it represents the maximum amplification of the errors in the right hand side in the solution vector. The condition number is also equal to the square root of the ratio of the largest to the smallest singular value of A. In parameter estimation applications. A is a positive definite symmetric matrix and hence, the cond ) is also equal to the ratio of the largest to the smallest eigenvalue of A, i.e.,... 

The condition number of the Hessian matrix of the objective function is an important measure of difficulty in unconstrained optimization. By definition, the smallest a condition number can be is 1.0. A condition number of 105 is moderately large, 109 is large, and 1014 is extremely large. Recall that, if Newton s method is used to minimize a function/, the Newton search direction s is found by solving the linear equations... 

Accurate definition of certain process parameters, such as severity of process conditions, number of recycles, process blocks and equipment, multiphase streams, and unusual separations No matter how much time and effort are spent preparing estimates, there is a chance of errors occurring due to Engineering errors and omissions Cost and labor rate changes Construction problems Estimating inaccuracies Miscellaneous unforeseens ... 

There are several sub-groups of MWR methods, according to the particular choices for the weighting function Wi z) employed. The performance of the resulting MWR is to a certain extent tied to the properties of the resulting coefficient matrix. To enable an efficient solution process it is desired that the coefficient matrix is symmetric, positive definite and characterized by a small condition number. At the same time the work needed to assemble the coefficient vaiues shouid be minimized. 

The collocation methods can be shown to give rise to symmetric, positive definite coefficient matrices that is characterized with a acceptable condition number for diffusion dominated problems or other higher order even derivative terms. For convection dominated problems the collocation method produces non-symmetric coefficient matrices that are not positive definite and characterized with a large condition number. This method is thus frequently employed in reactor engineering solving problems containing second order derivatives of smooth functions. A t3q)ical example is the pellet equations in heterogeneous dispersion models. 

Preconditioning is a technique which improves the condition number of a matrix and thereby increases the convergence rate of Krylov subspace methods. Thus, if the preconditioner A4 is a symmetric, positive definite matrix, the original problem Ax = b can be solved indirectly by solving M Ax = M h. The the residual can then be written as ... 

The latter method is often used simply because the conditions numbers are smaller. The user should be aware how a software package computes a condition number. For instance, SAS uses Eq. (2.49). For this book Eq. (2.48) will be used as the definition of the condition number. Condition numbers range from 1, which indicates perfect stability, to infinity, which indicates perfect... 

This definition holds for exactly determined systems where the number of rows in X is equal to the number of columns, that is, n=p (cf. Eq. (6.41)). In the case of overdetermined systems, with n>p, the condition number is obtained from... 

As one of the by-products of computing the SVD of A, the number ai/or can be computed. That number equals cond2(A) = IIAII2 A II2 for nonsingular matrices A and extends both the definition of the condition number of a matrix and its applications to error analysis to the case of singular matrices. 

Now the question arises How should the parameter 0 be chosen When looking for a minimizer, Davidon proposed to select the parameter 0 that minimizes among all positive definite matrices the ratio of the largest to the smallest eigenvalue condition number) of the matrix... 

Under the assumption of theorem 12 the procedure IV converges if Mj is a sequence of symmetric, positive definite matrices vith uniformly bounded condition numbers. 

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