Determination matrix condition number

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 new subset is selected by first computing the SVD of the least squares matrix A. A rank estimate, f, is obtained by first determining the condition number of the matrix from the singular values and then deciding upon the number of parameters that must be eliminated to reduce k to an acceptable value. The QR scheme with column pivoting is then applied to... 

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... 

Next, we determine the conditions for this matrix to become diagonal (with numbers of norm 1 in the diagonal), which will happen if and only if when p and q fulfill the following relations ... 

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. 

Different symbols have been used for the degree of advancement and the concentrations depending on the conditions. Furthermore vectors have been symbolised by bold letters with arrows, matrices by bold letters, variables as italics. Stoichiometric coefficients and the matrix p are written in Greek symbols. In addition, the concentrations of reactants to be determined and the number of degrees of advancement can be reduced because of the law of conservation of mass and linear dependencies between different steps of the reaction, respectively. Therefore some indices have been introduced to characterise the different variables. The different symbols are summarised in Table 2.2. 

Minimizing of the condition numbers of matrices (5.6) enables choosing optimal input polarizations for measuring the Mueller matrix (5.4). The same for the backscattered Mueller matrix (5.3), with some over determination that canbe used to estimate elements W33 and m. ... 

The direct inversion of Eq. 5 through, for example, the pseudoinverse w = [G G] G w (where indicates estimated value) is not recommended because the square matrix [G G] is generally ill-conditioned, and this produces highly oscillatory solutions with negative peaks. The propagation of errors is determined by 1) the condition number of G G (i.e., the ratio between the largest and the smallest eigenvalue of G G) and 2) the type and amplitude of the noise that contaminates w V). 

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