Condition number

Forward Analysis In this type of analysis, we are interested in the propagation of initial perturbations Sxq along the flow of (1), i.e., in the growth of the perturbations 5x t xo) = (xo -h Sxq) — xq. The condition number K,(t) may be defined as the worst case error propagation factor (cf. textbook [4]), so that, in first order perturbation analysis and with a suitable norm j ... 

The degree of conditioning of a matrix is detennined by the "condition number defined as (Fox and Mayei s, 1977)... 

A matrix with a large condition number is commonly referred to as ill-conditioned and particularly vulnerable to round-off errors. Special techniques. 

The higher the Condition Number, the more ill-conditioned the % matrix is... 

LoglO of Condition Number gives approx, number of decimal % places... 

The conversion from state-space to transfer function has produced some smaii erroneous numerator terms, which can be negiected. These errors reiate to the condition of A, and wiii increase as the condition number increases. 

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The matrix A is known as the preconditioner and has to be chosen such that the condition number of the transformed linear system is smaller than that of the original system. 

The ratio of the largest to the smallest eigenvalue of the Hessian matrix at the minimum is defined as the condition number. For most algorithms the larger the condition number, the larger the limit in Equation 5.5 and the more difficult it is for the minimization to converge (Scales, 1985). 

A measure of the degree of ill-conditioning of a nonsingular square matrix is through the condition number which is defined as... 

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.,... 

Generally speaking, for condition numbers less than 10 the parameter estimation problem is well-posed. For condition numbers greater than 1010 the problem is relatively ill-conditioned whereas for condition numbers 10 ° or greater the problem is very ill-conditioned and we may encounter computer overflow problems. 

The condition number of a matrix A is intimately connected with the sensitivity of the solution of the linear system of equations A x = b. When solving this equation, the error in the solution can be magnified by an amount as large as cortd A) times the norm of the error in A and b due to the presence of the error in the data. 

If however, matrix A is reasonably well-conditioned at the optimum, A could easily be ill-conditioned when the parameters are away from their optimal values. This is quite often the case in parameter estimation and it is particularly true for highly nonlinear systems. In such cases, we would like to have the means to move the parameters estimates from the initial guess to the optimum even if the condition number of matrix A is excessively high for these initial iterations. 

Again, we can determine the condition number and X,nin of matrix Anew using any eigenvalue decomposition routine that computes the eigenvalues of a real symmetric matrix and use the conditions (xN+0 that correspond to a maximum of... 

Step 3. Perform an eigenvalue decomposition of matrix Anew to determine its condition number, determinant and Xn,n. 

The condition number of matrix Anew can be used to indicate which of the optimization criteria (volume or shape) is more appropriate. In this example the... 

In this problem it is very difficult to obtain convergence to the global optimum as the condition number of matrix A at the above local optimum is 3xl018. Even if this was the global optimum, a small change in the data would result in widely different parameter estimates since this parameter estimation problem appears to be fairly ill-conditioned. 

The LS objective function was found to be 0.7604x10"9. This value is almost three orders of magnitude smaller than the one found earlier at a local optimum. The estimated parameter values were At=22.672, A2=132.4, A3=585320, Ej=l3899, E2=2439.6 and E3=13506 where parameters A, and E were estimated back from Ai and E. With this reparameterization we were able to lessen the ill-conditioning of the problem since the condition number of matrix A was now 5.6x108. 

You are asked to verify the calculations of Watts (1994) using the Gauss-Newton method. You are also asked to determine by how much the condition number of matrix A is improved when the centered formulation is used. 

There is more to looking at K. We can, for example, make use of its singular value and condition number, which should be deferred to a second course in control. 

The desired independence between the variables of the different analytical signals corresponds directly with the selectivity of the analytical system (Kaiser [1972] Danzer [2001], and Sect. 7.3). In case of multivariate calibration, the selectivity is characterized by means of the condition number... 

Equation (6.79) is valid for exactly determined systems (m = n). In case of overdetermined systems, m > n, the condition number is given by... 

If systems are well-conditioned the selectivity is expressed by condition numbers close to 1. 

Selectivity. In general, selectivity of analytical multicomponent systems can be expressed qualitatively (Vessman et al. [2001]) and estimated quantitatively according to a statement of Kaiser [1972] and advanced models (Danzer [2001]). In multivariate calibration, selectivity is mostly quantified by the condition number see Eqs. (6.80)-(6.82). Unfortunately, the condition number does not consider the concentrations of the species and gives therefore only an aid to orientation of maximum expectable analytical errors. Inclusion of the concentrations of calibration standards into selectivity models makes it possible to derive multivariate limits of detection. 

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... 

These equations become harder and harder to solve numerically as V2/ becomes more ill-conditioned. When its condition number exceeds 1014, there will be few if any correct digits in the computed solution using double precision arithmetic (see Luenberger, 1984). 

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