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KRYLOV SUBSPACE ITERATIVE METHODS

The solution time for simultaneous linear equations grows cubically with the number of unknowns. There are three regimes for solution; which one is applicable depends on the number of unknowns m. For m three or less, we use analytical methods. We also sometimes use analytical methods on matrices of size when the matrix contains enough zeros. Today in year 2001, a deskside workstation, working an hour solves about a set of simultaneous equations. A square image packed into a 4096 point vector is a array. The computer power for linear algebra to give us solutions that fit in a image is thus proportional to k6, which means that even though computer power grows rapidly, imaging resolution using exact numerical methods'' hardly grows at all from our current practical limit.

The retina in our eyes captures an image of size about which is a lot bigger than .Life offers us many occasions where final images exceed the 4000 points of a array. To make linear algebra (and inverse theory) relevant to such problems, we investigate special techniques. A numerical technique known as the conjugate-direction method'' works well for all values of m and is our subject here. As with most simultaneous equation solvers, an exact answer (assuming exact arithmetic) is attained in a finite number of steps. And if n and m are too large to allow enough iterations, the iterative methods can be interrupted at any stage, the partial result often proving useful. Whether or not a partial result actually is useful is the subject of much research; naturally, the results vary from one application to the next.