Solving linear equations

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,... 

Although this is a conceptually convenient way to solve for x, it is not necessarily the most efficient method for doing so. We shall return to the matter of solving linear equations in Section A.4. 

Solving linear equations Ax = b is done in MATLAB via the backslash A b command. For example, let us consider a system of 4 linear equations in 4 unknowns... 

A number of standard computer programs easily handle problems of this type such as spreadsheet packages, Matlab, Mathcad, Polymath, and so on as well as symbolic manipulators such as Mathematica, Maple, Derive, etc. Most statistic packages and equation solvers will also solve linear equations and have a simple user interface. 

Methods for Solving Linear Equations with Symmetric Positive Definite Matrices. 

It must be remembered that linear algebraic equations of the form (6.31) are approximate forms of the original non-linear discretized equations. The overall iterative procedure of repeatedly solving linearized equations to obtain solutions of non-linear equations is susceptible to divergence, especially when these are coupled with equations of other variables. In order to control the magnitude of change during each iteration, an under-relaxation parameter, is introduced ... 

One can solve equations in Maple using the solve and fsolve commands. The solve command is used to solve linear equations in symbolic form and the fsolve command is used to solve linear and nonlinear equations numerically. For example,... 

The results obtained in (10) are employed in solving linear equations. 

Linear algebra provides a notation for concisely representing linear algebraic equations and it provides a set of operations by whidi those equations can be solved. Linear equations arise in many common situations, such as taking inventories by material balances. As a particular example, consider a closed system initially loaded with N total moles of three liquid components 1, 2, and 3. That initial mixture had mole fractions z, Z2, and Z3. When equilibrium is attained, the system is found to have divided into three phases a, P, and y. The mole fractions for phase a are ze,, those in phase p are x,, and those in phase y are 1/ . These mole fractions are related to the original overall mole fractions z,- by material balances ... 

Any row can be interchanged with another row. This process is called pivoting. The main purpose of this operation is to create a new matrix that has dominant diagonal terms, which is important in solving linear equations. 

The elimination procedure described in the last sections forms a process, commonly called Gauss elimination. It is the backbone of the direct methods, and is the most useful in solving linear equations. Scaling and pivoting are essential in the Gauss elimination process. 

Obtaining the matrix inverse using the Gauss-Jordan method provides a compact way of solving linear equations. For a given problem,... 

This example is already a little tedious to solve by hand and it is a good idea to resort to computer methods especially for more complex structures. I (O Keeffe, 1990a) find it easiest to program the computer to read off all equations from the connectivity matrix and to use an algorithm that solves linear equations that include redundancies. It should be noted that one never has an over-determined set of equations. 

This method was originally developed to solve linear equation systems Ax = b. Starting with the assumption that the exact solution x is a minimum of... 

A convenient method for solving linear equations is to apply the Laplace or Fourier transform. In the transfer function (transmittance), the dynamic properties are encoded. The function is defined as the quotient of the Laplace transforms of the output and input functions. The spectrum transmittance is defined as the quotient of the Fourier transforms of the output and input functions under zero initial conditions. 

It can be seen that all parameters Xand d can be obtained by reference to values that are already known at the nodes. These are in fact node (or time-to-maturity) dependent constants. In other words, we have a system of linear equations from node 1 to N. Simultaneous linear equations can be solved by substitution. This method of solving linear equation can be applied to larger sets of linear equations, although we require increased processing power. 

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. 

Instability is a nonlinear phenomenon. However, the dynamic behavior of nuclear reactors can be assumed to be linear for small perturbations around steady-state conditions. This allows the reactor stability to be studied and the threshold of instability in nuclear reactors to be predicted by using a linear model and solving linearized equations. Linear stability analyses in the frequency domain have been... 

Listing 4.3 Example of solving linear equations with Gauss elimination program. 

The next seetion will diseuss general numerical operations performed using matrix notation and diseuss the development of a set of eoded routines for matrix manipulation. These approaehes builds upon the solution teehniques developed for solving linear equations in Section 4.1. 

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