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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
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.