The rule for matrix multiplicationhowever, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second i. Any matrix can be multiplied element-wise by a scalar from its associated field.
In addition to this we will have either Dirichlet, von Neumann or mixed boundary conditions to specify the boundary values of ij.
The system of linear equations described by in combination with the boundary conditions may be solved in a variety of ways. The structure of the system ensures A is relatively sparse, consisting of a tridiagonal core with one nonzero diagonal above and another below this. These nonzero diagonals are offset by either m or n from the leading diagonal.
Provided pivoting if required is conducted in such a way that it does not place any nonzero elements outside this band then solution by Gauss Elimination or LU Decomposition will only produce nonzero elements inside this band, substantially reducing the storage and computational requirements see section 4.
Careful choice of the order of the matrix elements i. Because of the wide spread need to solve Laplace's and related equations, specialised solvers have been developed for this problem.
This process may be performed iteratively to reduce an n dimensional finite difference approximation to Laplace's equation to a tridiagonal system of equations with n-1 applications. The main drawback of this method is that the boundary conditions must be able to be cast into the block tridiagonal format.
These iterative methods are often referred to as relaxation methods as an initial guess at the solution is allowed to slowly relax towards the true solution, reducing the errors as it does so. There are a variety of approaches with differing complexity and speed. We shall introduce these methods before looking at the basic mathematics behind them.
The Jacobi Iteration is the simplest approach. To find the solution for a two-dimensional Laplace equation simply: Initialise Fij to some initial guess. Apply the boundary conditions.
If solution does not satisfy tolerance, repeat from step 2. Higher order approximations may be obtained by simply employing a stencil which utilises more points.
While very simple and cheap per iteration, the Jacobi Iteration is very slow to converge, especially for larger grids. Corrections to errors in the estimate Fij diffuse only slowly from the boundaries taking O max m,n iterations to diffuse across the entire mesh.
The advantages of this are: Faster convergence although still relatively slow. On the other hand, the method is less amenable to vectorisation as, for a given iteration, the new estimate of one mesh point is dependent on the new estimates for those already scanned.
If we consider the mesh points as a chess board, then the white squares would be updated on the first pass and the black squares on the second pass. The advantages No interdependence of the solution updates within a single pass aids vectorisation. Faster convergence at low wave numbers.
It has been found that the errors in the solution obtained by any of the three preceding methods decrease only slowly and often decrease in a monotonic manner. The optimal value of s will depend on the problem being solved and may vary as the iteration process converges.
Typically, however, a value of around 1. In some special cases it is possible to determine an optimal value analytically. Multigrid methods try to improve the rate of convergence by considering the problem of a hierarchy of grids.
The larger wave length errors in the solution are dissipated on a coarser grid while the shorter wave length errors are dissipated on a finer grid. Typically the relaxation steps will be performed using Successive Over Relaxtion with Red-Black ordering and some relaxation coefficient s.
The hierarchy of grids is normally chosen to differ in dimensions by a factor of 2 in each direction.Solving a system of equations using a matrix is a great method, especially for larger systems (with more variables and more equations).
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