**Example 4.** Compare
the "discrete least squares approximation" and "continuous least
squares approximation."

**4 (a).** Use the
function , on
the interval .

**Solution 4 (a).**

**Warning.** We must
proceed with caution when using a least squares method because the
linear system might be ill conditioned, i.e. the solution is highly
sensitive to round off errors in the matrix and
vector. Let us investigate the situation for this
example.

The condition
number of the above system can be determined by
*Mathematica*.

**Fact.** Given
the linear system . If are
input with machine precision then a bound for the error in the
computed solution is
given by

where
is machine epsilon for the
computer. The computed
solution loses
about decimal
digits of accuracy relative to precision of input.

**Caveat.** Although
*Mathematica* uses extended precision sixteen digit numbers,
there is a possibility that the solution might
not have this much accuracy.

**Remark.** When
solving the continuous least squares approximation on the interval
,
the matrix of coefficients is
known to be a Hilbert
matrix which is the classic example of an ill-conditioned
matrix.

We are really done.

**Aside.** We can
calculate the coefficients by
directly setting up the matrix
vector . This
is just for fun !

This is the same as we obtained previously.

(c) John H. Mathews 2005