Listed below are several properties that may pertain to various methods for solv

Question

Listed below are several properties that may pertain to various methods for solving systems of linear equation. For each of the properties listed, state whether this quality more accurately describes direct or iterative methods. The entries of the matrix are not altered during the computation. A prior estimate for the solution is helpful. The matrix entries are stored explicitly, using a standard storage scheme such as an array. The work required depends on the conditioning for the problem. Once a given system has been solved, another system with the same matrix but a different right- hand side is easily solve. Acceleration parameters or preconditioners are usually employed. The maximum possible accuracy is relatively easy to obtain. "Black box" software is relatively easy to implement. The matrix can be defined implicitly by its action on an arbitrary vector. A factorization of the matrix is usually performed. The amount of work required can often be determined in advance.

Explanation / Answer

Below are solutions are most of them , please rate if satisified else comment for queries :

a. Iterative Methods : In iterative methods preserve the sparsity of parameter matrix. Parameter matrix is not altered in iteration process and hence because of this iterative methods do not suffer from accumulation of round-off corners

b. Iterative Methods : In iterative methods , a prior estimate is formulated . see below example: Ax = b ,iterative methods for this begins with an approximation to solution, x0, then seek to provide approximations for x1,x2,x3……., that converge to exact solution. This approach appeals more as it can be stopped as soon as the approximations xi have converged to an acceptable precision. With a direct method , this is not possible.

c. Iterative methods : Sparse matrix stores matrix entries in array form explicitly . Iterative methods are more applied to this .Consider numerical analysis which uses sparse matrices , Iterative methods are more common than direct methods in numerical analysis

d. Iterative Methods: Iterative methods depend on conditioning of problem ,Error estimate in the solution decreases with the number of iterations. For well-conditioned problems, this convergence should be quite monotonic. If you are working on problems that are not as well-conditioned, then the convergence will be slower.

f. Iterative Methods : In iterative methods local parameter for accelerating the iteration process is used. Richardson method is first and simplest iterative method that employs above scenario. Preconditioning is also used in iterative methods , preconditioning is defined as convergence of iterative methods depends on spectral properties of the matrix of the linear system and in order to improve these properties one often transforms the linear system by a suitable linear transformation

g Direct method: In direct method , it is easy to obtain maximum possible accuracy where as in iterative method we start with approximation and converge. Consider Centrifugal Separations in Molecular and Cell Biology example.

h Iterative Method: Black box software can be implemented easily in iterative method , consider conjugate gradient solver with inner interactions and variable step preconditioning procedure as an example.

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