Model fitting by least squares |

The compute time for a rectangular matrix is slightly more pessimistic. It is the product of the number of data points times the number of model points squared which is also the cost of computing the matrix from . Because the number of data points generally exceeds the number of model points by a substantial factor (to allow averaging of noises), it leaves us with significantly fewer than 4,000 points in model space.

A square image packed into a 4,096-point vector is a array. The computer power for linear algebra to give us solutions that fit in a image is thus proportional to , 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 roughly 1,000 1,000
which is a lot bigger than .
Life offers us many occasions in which final images exceed the 4,000
points of a array.
To make linear algebra (and inverse theory) relevant to such applications,
we investigate special techniques.
A numerical technique known as the
``**conjugate-direction method**''
works well for all values of 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 and 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.

- Sign convention
- Method of random directions and steepest descent
- Why steepest descent is so slow
- Null space and iterative methods
- The magical property of the conjugate direction method
- Conjugate-direction theory for programmers
- Routine for one step of conjugate-direction descent
- A basic solver program
- Fitting success and solver success
- Roundoff
- Test case: solving some simultaneous equations

Model fitting by least squares |

2014-12-01