Matrix Gauss factorization

Recently (Buzzi-Ferraris, 2011a) proposed a novel Gauss factorization that allows the factorization of an underdimensioned matrix in a stable way (see Section 8.2). 

If QR or LQ factorization is adopted, it is possible to use the methods described by Buzzi-Ferraris and Manenti (2010a), which are also less efficient than the ones used to update a symmetric matrix. On the other hand, if a Gauss factorization is adopted the problem is more complex and some stability problems may arise with the algorithm. 

The m equations (8.3) with the n — m conditions (8.12) make the system square and, consequently, solvable through an appropriate algorithm (i.e.. Gauss factorization) when the resulting matrix is nonsingular. As special but important case, the values cj are all zero. Also to obtain a solution of the underdimensioned linear system that can exploit the Gauss factorization, the variables x are separated into m dependent variables, x, and into n — m independent variables, x , to which a numerical value is assigned. 

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 second defect of LQ factorization is important for sparse matrices. In fact, dense matrices require double the computational effort of Gauss factorization. If the matrix is sparse, this gap may become larger and a dramatic filling of the factorized matrix may occur. The advantages of LQ factorization are a stable solution of an underdimensioned system (if the system is standardized), and the easy and safe removal of all linear combinations among equations. 

The majority of the programs that exploit Gauss factorization select the pivot without any column swap, but with row swaps only, when needed. It is effective only if the matrix is square and relatively well conditioned. 

When E is large and sparse, LQ factorization can be computationally onerous and lead to the dangerous matrix filling as well in this case a stable Gauss factorization is required to calculate the null space of the matrix (see Chapter 8). 

In Chapter 8, we showed that in the case of linear constraints with the matrix A m < n it is possible to obtain the null space of the matrix using a Gauss factorization with good stability features. 

The main difference with projection methods is in the use of the null space of the active constraints obtained with Gauss factorization in spite of the projection matrix. 

LV Factorization of a Matrix To eveiy m X n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and a.nm X n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessaiy in carrying out the Gauss elimination to... 

The Gauss-Newton method arises when the second order terms on the right hand side of Equation 5.20 are ignored. As seen, the Hessian matrix used in Equation 5.11 contains only first derivatives of the model equations f(x,k). Leaving out the second derivative containing terms may be justified by the fact that these terms contain the residuals e, as factors. These residuals are expected to be small quantities. 

According to Scales (1985) the best way to solve Equation 5.12b is by performing a Cholesky factorization of the Hessian matrix. One may also perform a Gauss-Jordan elimination method (Press et al., 1992). An excellent user-oriented presentation of solution methods is provided by Lawson and Hanson (1974). We prefer to perform an eigenvalue decomposition as discussed in Chapter 8. 

One method for finding A is called Gauss-Jordan elimination, which is a method of solving simultaneous linear algebraic equations. It consists of a set of operations to be applied to Eq. (9.51). In order maintain a valid equation, these operations must be applied to both sides of the equation. The first operation is applied to the matrix A and to the matrix E on the right-hand side of the equation, but not to the unknown matrix A . This is analogous to the fact that if you have an equation ax = c, you would multiply a and c by some factor, but not multiply... 

Usually, the Jacobian matrix of the system (7.38) is nonsymmetric. Thus, it is neither possible to solve the linear system by means of the Cholesky algorithm nor to halve memory allocation. The most efficient methods (Gauss or PLR variant) adopted for Jacobian factorization require twice as much time and memory allocation as the Cholesky algorithm. 

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. 

The quantity o> in (4.11) is called the relaxation factor. We observe that, for CO = 1, this iterative method reduces to the Gauss-Seidel iterative method of (4.8)-(4.8 ). For reasons of brevity, we shall say that a matrix G, which is cyclic of index 2, is consistently ordered [52] if it is the form of (4.10). With the concept of a consistent ordering. Young [52] established the following general relationship between the eigenvalues A of the successive overrelaxation matrix... 

Suppose the matrix A is invertible. Then if there were no row interchanges in carrying out the above Gauss ehmination procedure, we have the LU factorization of the matrix A ... 

The Newton-Raphson technique Is used since It offers better convergence chan the Gauss-Seldel scheme. There ace, however some limitations to this technique, since the matrix Inversion procedure requires a CPU time approximately proportional to 0(N ) and Che storage Is proportional to N (where N Is Che total number of nodes In the computational region). These factors combine to make Che approach unsuitable for extension to point contact problems where Che number of nodes Is large. 

Free radical Generation Matrix or solvent T K V kMo/s g-factor Splitting parameters gauss References for g and a Further references... 

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