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(28) |

(29) |

**x**which minimizes the weighted error

*E*of (29) is the

**x**which satisfies the simultaneous equations

(30) |

*w*is equivalent to choosing it equal to unity.

A case of common interest
is where some equations should be solved exactly.
Such equations are called constraint equations.
Constraint equations often
arise out of theoretical considerations so they may,
in principle,
not have any error.
The rest of the equations often involve some measurement.
Since the measurement can often be made many times,
it is easy to get a lot more
equations than unknowns.
Since measurement always involves error,
we then use the method of least squares
to minimize the average error.
In order to be certain that the constraint equations are solved exactly,
one could use the trick of applying very large weight factors
to the constraint equations.
A problem is that
``very large'' is not well defined.
A weight equal 10^{10}
might not be large enough
to guarantee the constraint equation is satisfied
with sufficient accuracy.
On the other hand, 10^{10} might lead to
disastrous round-off when solving
the simultaneous equations in a computer
with eight-digit accuracy.
The best approach is to analyze the situation
theoretically for .

An example of a constraint equation
is that the sum of the *x*_{i} equals *M*.
Another constraint would be *x _{1}* =

*x*. Arranged in a matrix, these two constraint equations are

_{2}(31) |

*k*constraint equations as

(32) |

**Bx**

**0**. The rows of are just like some extra rows for

**B**. The resulting equation for

**x**is

(33) |

*w*

_{i}to equal and we will let tend to zero. Also let

(34) | ||

(35) |

(36) |

(37) | ||

(38) | ||

*m*equations in

*m*unknowns. It will automatically be satisfied if the

*k*equations in (32) are satisfied. Equation (38) appears to involve the

*m*unknowns in plus

*m*more unknowns in . In fact, we do not need ;the

*k*unknowns

(39) |

Arranging (38) and (32)
together and dropping superscripts,
we get a square matrix in *m* + *k* unknowns.

(40) |

Equation (40) is now a simultaneous set for the unknowns and . It might also be thought of as the solution to the problem of minimizing the quadratic form

(41) |