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What is one to do when one has fewer equations
than unknowns: give up?
Certainly not,
just apply the principle of simplicity.
Let us find the simplest solution
which satisfies all the equations.
This situation often arises.
Suppose,
after having made a finite number of measurements
one is trying to determine a continuous function,
for example,
the mass density as a function of depth in the earth.
Then,
in a computer would be represented
by sampled at *N* depths Then merely taking *N* large,
one has more unknowns than equations.

One measure of simplicity is that the unknown
function *x*_{i} has minimum wiggliness.
In other words minimize

| |
(42) |

subject to satisfying exactly the observation
or constraint equations

| |
(43) |

Another more popular measure of simplicity
(which does not imply an ordering of the variables *x*_{i})
is the minimization of

| |
(44) |

If we set out to minimize (44) without any constraints,
*x* would satisfy the simultaneous equations

| |
(45) |

By inspection one sees the obvious result
that *x*_{i} = 0.
Now let us include two constraint equations and,
for definiteness,
take three unknowns.
The method of the previous section gives

| |
(46) |

Equation (46) has a size equal
to the number of variables plus the number of constraints.
It may be solved numerically
or it may be reduced to a matrix
whose size is given by the number of constraints.
Let us split up (46) into two equations:

| |
(47) |

and
| |
(48) |

We abbreviate these equations by
and ,Premultiply (47) by **G**,

insert (48)
solve for
put back into (47)
Written out in full this is
| |
(49) |

This is the final result,
a minimum wiggliness solution which exactly satisfies an underdetermined
set called the constraint equations.

## EXERCISES:

- If wiggliness is defined by (42) instead of (44),
what form does (49) take?
- Given the mass and moment of inertia of the earth,
calculate mass density as a function of depth
utilizing the principle of minimum wiggliness (49).
What criticism do you have of this procedure?
(HINT: An elegant solution uses integrals instead
of infinite sums.)
- Use the techniques of this section on (40)
to reduce the size of the matrices to be inverted.

** Next:** HOUSEHOLDER TRANSFORMATIONS AND GOLUB'S
** Up:** Data modeling by least
** Previous:** WEIGHTS AND CONSTRAINTS
Stanford Exploration Project

10/30/1997