Matrix condition numbers

Thus, there are infinite (weD- and ill-) matrix condition numbers for the same linear system that depend on the system formulation. Conversely, there is one single standard form for each linear system and a unique system conditioning. 

The matrix condition number of a system in its standard form is called the system conditioning or system condition number. 

The matrix condition number (traditional approach without weighing the right-hand side terms) is 242 hence, the system should be considered well conditioned. The system conditioning is small too and is 5.4. If we solve this system with a traditional Gauss factorization without passing through the standard form, the selected pivot for the first column is 10 since it is... 

The solution of this linear system is completely wrong if it is not preventively written in its standard form the matrix condition number is 1467, while the system conditioning is 37.8. The incorrect solution is obtained with all the factorizations, not just the Gauss one. 

Table B.17 Matrix Analysis cond Matrix condition number,...

A formal measure of conditioning is the matrix condition number, defined as ... 

The main advantage of EPC is that it uses an analytical expression of the system matrix condition number to determine the primary tuning parameter. Considering a SISO plant and an = 3 as an example, the weighted matrix A Aajprox.+W becomes... 

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... 

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.,... 

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. 

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... 

Euclidean norm and condition number of a square matrix... 

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