Solving equations Gauss

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. 

This is a system of equations of the form Ax = B. There are several numeral algorithms to solve this equation including Gauss elimination, Gauss-Jacobi method, Cholesky method, and the LU decomposition method, which are direct methods to solve equations of this type. For a general matrix A, with no special properties such as symmetric, band diagonal, and the like, the LU decomposition is a well-established and frequently used algorithm. 

Four parameters have to be adjusted in order the fit the experimental coordinates of the drop shape the localisation of the drop apex, Y , the radius of curvature R , and the parameter B. A software package called ADSA does the detection of the drop edge coordinates and the fitting of the Gauss-Laplace equation to these data. A suitable algorithm to solve the Gauss Laplace equation is given in Appendix 5F. 

Appendix 5F Numerical Algorithm to Solve the Gauss-Laplace Equation... 

This derivation was first applied by Van Orstrand and Dewey [97], who solved Equation 10.14 for diffusion from a satnrated solntion into pure solvent. Since in the Gauss integral is a number. 

The differentiation of Eq. (10) with respect to each G, gives sets of equations, which are called the normal equations of the linear least-squares problem. These normal equations can be solved by Gauss-Jordan elimination. However, in many cases the normal equations are very close to singular and a zero pivot element may be encountered. In such cases instead of using the normal equations, Eq. (10) can be solved by singular value decomposition. 

In the numerical solution of the Reynolds equation It was assumed that all the variables were distributed quadratlcally in the coordinate directions. Derivatives were thus expressed In terms of three adjacent points, (i,j-l), (i,j) and (l,j+l) In the radial (R) direction and (1-1,j), (l,j) and (1+1,j) in the circumferential (6) direction as shown In Figure 2. The Gauss-Seldel Iterative over-relaxation method was used to solve equation (17) written In finite difference form. 

To solve equation (7) one can eliminate the matrix S(I ) from the equation (again like in the case of atoms and molecules) using Ldwdin s synunetric orthogonalization procedure. To be able to perform a numerical integration procedure (Simpson or preferably Gauss quadrature) for equation (12) we have to solve equation (7) at a number of k points, usually 7-9 k points between 0 and n/a and because... 

As mentioned, Cramer s formula is only suitable for solving equation systems with two or three unknowns the advantage using this solution method is the clear systematics when keying on a pocket calculator. To solve linear equation systems with more unknowns than three, we can, for example, use Gauss elimination. 

The Laplace equation is written for each point and the resulting matrix equations for all points are solved using Gauss elimination. If we set... 

The equation system of eq.(6) can be used to find the input signal (for example a crack) corresponding to a measured output and a known impulse response of a system as well. This way gives a possibility to solve different inverse problems of the non-destructive eddy-current testing. Further developments will be shown the solving of eq.(6) by special numerical operations, like Gauss-Seidel-Method [4]. 

The purpose of this projeet is to gain familiarity with the strengths and limitations of the Gauss-Seidel iterative method (program QGSEID) of solving simultaneous equations. 

If P = I, this is the Gauss-Seidel method. If > I, it is overrelaxation if P < I it is underrelaxation. The value of may be chosen empirically, 0 < P < 2, but it can be selected theoretically tor simple problems hke this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. 

With the aid of effective Gauss method for solving linear equations with such matrices a direct method known as the elimination method has been designed and unveils its potential in solving difference equations,... 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

The four sensitivity equations (Equations 6.56a-d) should be solved simultaneously with the two state equations (Equation 6.52). Integration of these six [=nx(p+1 )=2x(2+1)] equations yields x(t) and G(t) which are used in setting up matrix A and vector b at each iteration of the Gauss-Newton method. 

The above equation represents a set of p nonlinear equations which can be solved to obtain koutput vector around the trajectory xw(t). Kalogerakis and Luus (1983b) showed that when linearization of the output vector is used, the quasilinearization computational algorithm and the Gauss-Newton method yield the same results. 

In order to improve the convergence characteristics and robustness of the Gauss-Newton method, Levenberg in 1944 and later Marquardt (1963) proposed to modify the normal equations by adding a small positive number, y2, to the diagonal elements of A. Namely, at each iteration the increment in the parameter vector is obtained by solving the following equation... 

The above unconstrained estimation problem can be solved by a small modification of the Gauss-Newton method. Let us assume that we have an estimate kw of the parameters at the j iteration. Linearization of the model equation and the constraint around kw yields,... 

Equations 10.15 to 10.17 define a set of (nxp) partial differential equations for the sensitivity coefficients that need to be solved at each iteration of the Gauss-Newton method together with the n PDEs for the state variables. 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

The solution of Equation 10.28 is obtained in one step by performing a simple matrix multiplication since the inverse of the matrix on the left hand side of Equation 10.28 is already available from the integration of the state equations. Equation 10.28 is solved for r=l,...,p and thus the whole sensitivity matrix G(tr,) is obtained as [gi(tHt), g2(t,+1),- - , gP(t,+i)]. The computational savings that are realized by the above procedure are substantial, especially when the number of unknown parameters is large (Tan and Kalogerakis, 1991). With this modification the computational requirements of the Gauss-Newton method for PDE models become reasonable and hence, the estimation method becomes implementable. 

When the Gauss-Newton method is used to estimate the unknown parameters, we linearize the model equations and at each iteration we solve the corresponding linear least squares problem. As a result, the estimated parameter values have linear least squares properties. Namely, the parameter estimates are normally distributed, unbiased (i.e., (k )=k) and their covariance matrix is given by... 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

The more interesting problems tend to be neither steady state nor linear, and the reverse Euler method can be applied to coupled systems of ordinary differential equations. As it happens, the application requires solving a system of linear algebraic equations, and so subroutine GAUSS can be put to work at once to solve a linear system that evolves in time. The solution of nonlinear systems will be taken up in the next chapter. 

GAUSS Subroutine GAUSS solves a system of simultaneous linear algebraic equations by Gaussian elimination and back substitution. The number of equations (equal to the number of unknowns) is NROW. The coefficients are in array SLEQ(NR0W,NR0W+1), where the last column is the constants. 

Program DGC04 solves the time-dependent problem. Subroutine EQUATIONS evaluates the coefficients of the unknown delys in the manner just outlined, and then subroutine GAUSS solves for the values of dely. Subroutine STEPPER steps forward in time by incrementing x and y. Subroutine SPECS sets the values of the parameters of the problem, converting units where necessary, and PRINTER writes the results to a file for plotting. 

The application of the reverse Euler method of solution to a system of coupled differential equations yields a system of coupled algebraic equations that can be solved by the method of Gaussian elimination and back substitution. In this chapter I demonstrated the solution of simultaneous algebraic equations by means of this method and showed how the solution of algebraic equations can be used to solve the related differential equations. In the process, I presented subroutine GAUSS, the computational engine of all of the programs discussed in the chapters that follow. 

Then this system of simultaneous linear algebraic equations can be solved using the subroutine GAUSS developed in Section 3.3. Because I have dropped the nonlinear term, I must always use a delx sufficiently small to ensure that all the dely values are indeed much smaller than the y values. 

I presented a group of subroutines—CORE, CHECKSTEP, STEPPER, SLOPER, GAUSS, and SWAPPER—that can be used to solve diverse theoretical problems in Earth system science. Together these subroutines can solve systems of coupled ordinary differential equations, systems that arise in the mathematical description of the history of environmental properties. The systems to be solved are described by subroutines EQUATIONS and SPECS. The systems need not be linear, as linearization is handled automatically by subroutine SLOPER. Subroutine CHECKSTEP ensures that the time steps are small enough to permit the linear approximation. Subroutine PRINTER simply preserves during the calculation whatever values will be needed for subsequent study. 

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

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