Consider the problem of computing the covariance matrix where x i = ( i 1 , i 2 ) T . The question addressed is, what is the fastest way to compute this matrix. In doing this, the x i ’s should be...

Consider the problem of computing the covariance matrix

where
x

_i

= (

_i1,


_i2)
^T

. The question addressed is, what is the fastest way to compute this matrix. In doing this, the
x

_i
’s should be randomly generated, and this should be done before answering the following questions. Also, you should answer the questions taking = 2,000,
 = 20,000, and
 = 200,000. Finally, you should take advantage of any mathematical simplifications available (e.g., symmetry), to reduce the computational time.

(a) How long does it take to compute
B
using the summation formula? This means that each 2 × 2 matrix
x

_i
(x

_i
)
^T

is computed, and then added into the sum.

(b) How long does it take to compute
B
by first forming the matrix
X, and then using (9.33)? You should use MATLAB’s matrix multiply command to compute the product.

(c) How long does it take to compute
B
by first forming the vectors
c
^∗
₁
and
c
^∗
₂, and then using (9.34)? You should use MATLAB’s dot multiply command to compute the dot products.

(d) How long does it take to compute
B
by first forming the matrix
X, and then using MATLAB’s cov command?

(e) Explain the differences, or non-differences, in the computing times. Also, comment if any of the methods failed, and why that happened.