We have already considered Gaussian elimination (and back substitution) as examples of direct solution methods. Our most powerful supercomputers are needed for weather forecasting even if we used iterative methods. For large agricultural farms which use remote sensing for weather monitoring and forecast, it would be useless to tell them the exact solution to a storm a long time after the storm has passed and devastated the crop production because you tried to solve the equations involved exactly, and from the time the storm was on the horizon it took you such a long time to find the exact solution, assuming that such an exact solution even exist in the 1st place. You could imagine why weather forecasting is very strict on time. However, our current exact methods are prohibitively expensive in terms of time taken, so iterative method is the best we have.
These algorithms start with an initial guess at the solution ( \(\pmb_k\) is still very much an approximation rather than an exact answer, you could imagine why weather forecasting, which uses iterative methods, is still very much an approximation rather than an exact answer. In an attempt to further reduce this cost iterative algorithms were devised. In this case the \(n^3\) operations required of a direct algorithm such as Gaussian elimination is completely prohibitive. However, as pointed out previously, \(n\) could be billions for hard-core applications such as in numerical weather forecasting. ignoring the round off related issues) in a finite number of operations. \(n\) here refers to the number of unknowns or equations, or sometimes termed the degrees of freedom of the problem.Īn advantage of direct methods is that they provide the exact solution (assuming exact arithmetic, i.e. For large \(n\) Gaussian elimination will clearly be more efficient.
The computational cost of this method (in terms of arithmetic operations required also termed complexity) scales with \((n+1)!\), whereas the Gaussian elimination method (which is basically the substitution method) scales with \(n^3\). This transformed the equations making up the linear system into equivalent ones with the aim of eliminating unknowns from some of the equations and hence allowing for easy solution through back (or forward) substitution.Ĭramer’s rule gives an explicit formula for the inverse of a matrix, or for the solution of a linear matrix system. These are termed direct methods and iterative (or indirect) methods.ĭirect methods perform operations on the linear equations (the matrix system), e.g. Two types/families of methods exist to solve matrix systems.
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