The computing times for three matrix algorithms are given in Table 8.10, which is adapted from Table 4.13. The assumption is that the computing time
can be modeled using the function
The goal of this exercise is to use regression to find
and
. Note that you do not need to know anything about how the matrix methods work to do this exercise.
(a) Using the results from Exercise 8.6(a),(b), fit the model function to the LU times, and then plot the model function and data on the same axis.
(b) Show that to minimize the least square error
one gets that
(c) It would seem reasonable to expect that the computing time is a reflection of the flops used to calculate the factorization. If so, then
should be a positive integer. Using the resulting from part (b), and assuming
is a positive integer, find
and
. Also, using these values, plot the model function and data on the same axis. Explain how you find
, and also comment on how the assumption that
is an integer affects how well the model function fits the data.
(d) Redo (a) and (c) for the QR times.
(e) Redo (a) and (c) for the SVD times.
Exercise 8.6(a),(b)
(a) Writing
=
1
v2
, and then taking the log of this equation, show that the transformed model function can be written as
(
) =
1
+
2
, where
1
= log
1
and
2
=
2. Also, show that the transformed data points (
i
,
i) are
i
= log
i
and
i
= log
i
.
(b) Continuing from part (a), using the least squares error
(
1,
2) =
(
1
+
2
i
−
i
)2, and the common log, show that
1
= 10
V1
and
2
=
2, where
1
and
2
are given in (8.31).