Solution of Linear Equations

We will first address the question of when the set of linear equations in Equation (2.71) has a solution. Homogeneous equations always have a solution. This is also evident from the fact that x = 0 is always a solution. Whether nontrivial solutions also exist depends on the rank of the matrix 

If r is the rank of the matrix A, then there are (n r) linearly independent solutions obtained 

Equation (2.71) has a unique solution when its homogeneous version has only the zero solution. Since the latter occurs only when r = n, we conclude that, for the case ofm = n, i.e., the number of equations is the same as the number of unknowns, a unique solution occurs regardless of b when A is nonsingular. When m n and r = n, a. unique solution exists for Ax = b as long as the ranks of matrices A and [A b] are the same, where the augmented matrix [A b] is defined by 

It should be noted that the calculation of an inverse matrix requires considerable computational time, and hence various methods have been proposed for the solution of a set of linear equations that take advantage of the structure of the matrix A. The methods may be divided into two classes, namely, direct and iterative methods. 

Iterative methods are sometimes used due to ease of computer coding and lesser computational storage requirements. The Jacobi method is the simplest iterative method but has slower convergence in comparison with the Gauss-Seidel method. In the Gauss-Seidel method, the (A -tl)th iteration of the value of the unknown x, is given by 

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These Uj may be solved for by the methods under Numerical Solution of Linear Equations and Associated Problems and substituted into Eq. (3-78) to yield an approximate solution for Eq. (3-77). 

The principal use of the inverse matrix is in solution of linear equations or the application of transformations. If... 

Westlake, J. R. (1968) A handbook of numerical matrix inversion and solution of linear equations (Wiley). 

Gaussian elimination is a very efficient method for solving n equations in n unknowns, and this algorithm is readily available in many software packages. For solution of linear equations, this method is preferred computationally over the use of the matrix inverse. For hand calculations, Cramer s rule is also popular. 

A. 1 Definitions / A.2 Basic Matrix Operations / A.3 Linear Independence and Row Operations / A.4 Solution of Linear Equations / A. 5 Eigenvalues, Eigenvectors / References /... 

Let us estimate the correspondent left eigenvector f (a vector row). The eigenvalue is known, hence it is easy to do just by solution of linear equations. This system of —1 equations is ... 

EX15 1.5 Solution of linear equations with tridiagonal matrix M17... 

REM EK. 1.5. SOLUTION OF LINEAR EQUATIONS KITH TRIDIAGONAL MATRIX 104 REH MERGE N17... 

Solution of linear equations. A set of linear equations is represented by ax = b. The solution x can be obtained in Mathematica by use of LinearSolve[ , b. Matrix a can be square or rectangular. 

Despite these potential difficulties, efforts to attack this problem have been undertaken and some progress has been made. The nonlinear equations are generally attacked by methods (e.g. Newton-Raphson) which require periodic solution of linear equations. 

Beckman FS (1960) The solution of linear equations by conjugate gradient method In Ralston A, Wilf HS (eds) (1960) Mathematical methods for digital computers 1, chap 4. Wiley, New York, p 62-67... 

The spectral Lanczos decomposition method is designed for effective calculation of the matrix functions. This method has found a useful application in the solution of linear equations in electromagnetic modeling (Druskin and Knizhnerman, 1994 Druskin et ah, 1999)). Let us consider, for example, the matrix equation (12.35). The formal solution of this equation has the form... 

This matrix formulation may be used in the iterative procedure by replacing the inner cycle with the solution of linear equation system of eq.(51) (Coitino et al., 1995a). However, this approach could be too cumbersome a more interesting application is the direct minimization of the free energy functional. We need to make a digression here. 

Therefore, m solutions of linear equations (with a perturbation-dependent vector like W on the right-hand side) can replace the 0 m ) solutions for Note that the situation, though similar, is not completely analogous to the case of the (2n -I- 1) rule. 

Although the full Navier Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u Vu were identically equal to zero or else appeared only in an equation for the crossstream pressure gradient, which was decoupled from the primary linear flow equation, as in the ID analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs. 

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