Question

State the two difference between direct and

iterative methods for solving system of equations​

Answer 1

DIRECT AND ITERATIVE METHOD

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed ininfinite precision arithmetic. Examples includeGaussian elimination, theQRfactorization method for solvingsystems of linear equations, and thesimplex methodoflinear programming.

In contrast to direct methods,iterative methodsare not expected to terminate in a number of steps. Starting from an initial guess, iterative methods form successive approximations thatconvergeto the exact solution only in the limit. Aconvergence criterionis specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples includeNewton’s method, thebisection method, andJacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.

Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g.GMRESand theconjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

In the case of asystem of linear equations, the two main classes of iterative methods are thestationary iterative methods, and the more generalKrylov subspacemethods.

Step-by-step explanation:

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Answer 2

Two differences between direct and iterative methods are :-

1. Direct method deals with a formula to find the answer while iterative methods uses a guess which converge until a approximate value is found using which answer is found upon applying limits.
2. Direct method give precise answer in finite steps while iterative method have infinite successive steps which improve the approximated value.

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