Solution of matrix equations

Since both L and U are triangular matrices, (1.54) and (1.55) are solved by the simplest backsubstitution except for taking into account the right-hand side interchanges in (1.54). The next module performs these calculations. 

On input the array A contains the decomposed matrix as given by the module M14, and the right-hand side coefficients are placed into the vector X. On output, this vector will store the solution. There is nothing to go wrong in backsubstitution if the previous decomposition was successful, and hence we dropped the error flag. 

Example 1.3.3 Solution of simultaneous linear equations by LU decomposition 

218 IF ER=1 THEN LPRINT COEFFICIENT MATRIX IS SINGULAR 6QTD 23B 

224 LFRINT SOLUTION OF THE SYSTEM OF LINEAR EQUATIONS LPRINT 

---

Solution of matrix equations bv Gauss-Jordan elimination... 

In linear programming problems we will need special solutions of matrix equations with "free" variables set to zero. These are called basic solutions of a matrix equation, where rank(A) is less than the number of variables. 

Multigrid methods with control of the numerical truncation error are very useful for the solution of matrix equations since they start with a small number of grid points, use extrapolation techniques similar to Richardson s and reduce the step size until the result is accurate enough. The method of Bu-lirsch and Stoer (cf. [495, p. 718-725] and [502, p. 288-324]), for instance, consists essentially of three ideas. The calculated values for a given step size h are... 

Listing 12.26. Example of ADI iterative solution of matrix equations for 2D PDE. 

Stechel E B, Walker R B and Light J C 1978 R-matrix solution of coupled equations for inelastic scattering J. Chem. Phys. 69 3518... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

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

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

To obtain the representation of the matrix element <0 (a ) > we, therefore, need to study the solutions of the equation... 

This is possible within the framework of the self-consistent field (SCF) approach to polymer configurations, described more completely elsewhere [18, 19, 51, 52]. Implementation of this method in its full form invariably requires numerical computations which are done in one of two equivalent ways (1) as solutions to diffusion- or Schrodinger-type equations for the polymer configuration subject to the SCF (in which solutions to the continuous-space formulation of the equations are obtained by discretization) or (2) as solutions to matrix equations resulting from a discrete-space formulation of the problem on a lattice. 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

If Caf ) = C is a lower triangle matrix and all the operators Caa are invertible, then the procedure of solving equations (42 ) can be reduced to successive solution of the equations... 

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

In the previous chapter we presented the problem of fitting data when there is more information (in the form of equations relating the several variables involved) available than the minimum amount that will allow for the solution of the equations. We then presented the matrix equations for calculating the least squares solution to this case of overdetermined variables. How did we get from one to the other ... 

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 /... 

Basilevsky, M. V. and Ryaboy, V. M. Two approaches to the calculation of molecular resonance states Solution of scattering equations and matrix diagonalization, J.Comp.Chem., 8 (1987), 683-699... 

The second equation is called the characteristic equation of the matrix A. Once A is known, the vector ut is computed as the unit vector solution of the linear system. It is left to the reader to show that the eigenvalues of a 2 x 2 matrix A can be found as a solution of the equation... 

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... 

---