Gauss matrix coefficients

Equations 4.14 and 4.15 are used to evaluate the model response and the sensitivity coefficients that are required for setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

Furthermore, the implementation of the Gauss-Newton method also incorporated the use of the pseudo-inverse method to avoid instabilities caused by the ill-conditioning of matrix A as discussed in Chapter 8. In reservoir simulation this may occur for example when a parameter zone is outside the drainage radius of a well and is therefore not observable from the well data. Most importantly, in order to realize substantial savings in computation time, the sequential computation of the sensitivity coefficients discussed in detail in Section 10.3.1 was implemented. Finally, the numerical integration procedure that was used was a fully implicit one to ensure stability and convergence over a wide range of parameter estimates. 

Program DGC05 implements this solution with a subroutine to evaluate the coefficients of the sleq matrix. It calls subroutine GAUSS to calculate the increments dely, and then it steps forward in time and repeats. 

This set of linear equations can be solved by inspection, or, more formally, by Gauss-Jordan reduction of the augmented coefficient matrix ... 

Because of aliasing, the total number of coefficients obtained should not be greater than N. We have a set of 2c linear equations for the 2c unknown coefficients. A number of standard methods are available for solving a set of linear equations. We used the Gauss-Jordan matrix reduction method. 

Any standard method of matrix inversion, such as the Gauss-Jordan method (N13), may be used to solve the equations. The coefficients in equations 4.11-4.14 may be used without serious error for most ordinary Portland cement clinkers in which the alite composition is not too different from that assumed here. As a byproduct of the calculation described in this section, and using the full compositions of the phases given in Table 1.2, one may calculate a mass balance table (Table 4.3) showing the distributions of all the oxide components among the phases. 

This procedure completes the Gauss elimination. We can carry out the elimination process by writing only the coefficients and the matrix vector in an array as... 

Performs nonlinear regression using the Gauss-Newton estimation method. The jc-data is given as x, while the y-data is given as y. The function, FUN, that is to be fitted must be written as an m-file. It will take three arguments the coefficient values, x, and y (in this order). The function should be written to allow for matrix evaluatitni. The initial guess is specified in bataO. The vector beta contains the estimated values of the coefficients, the vector r contains the residuals, and covb is the estimated covariance matrix for the problem. J is the Jacobian matrix evaluated with the best estimate for the parameters. 

Gaussian elimination is considered the workhorse of computational science for the solution of a system of linear equations. Karl Friedrich Gauss, a great nineteenth-century mathematician, suggested this elimination method as a part of his proof of a particular theorem. Gaussian elimination is a systematic application of elementary row operations to a system of linear equations in order to convert the system to upper triangular form. Once the coefficient matrix is in upper triangular form. 

The matrix algebra used for calculation of the coefficient matrix k is particularly favourable for programming linear regression. If the number of independent variables Xk exceed three or four, it is in many cases better to solve the equation system (3) by Gauss elimination instead of inverting the matrix X X in the traditional way. 

---