The n × n factorization A = LU, where L = (li j ) is lower triangular and U = (ui j ) is...

The n × n factorization A = LU, where L = (l_{i j} ) is lower triangular and U = (u_{i j} ) is upper triangular, can be computed directly by the following algorithm (provided zero divisions are not encountered): Specify either l₁₁ or u₁₁ and compute the other such that l₁₁u₁₁ = a₁₁. Compute the first column in L by

and compute the first row in U by

Now suppose that columns 1, 2, . . . , k − 1 have been computed in L and that rows 1, 2, . . . , k − 1 have been computed in U. At the kth step, specify either l_kk or u_kk , and compute the other such that

Compute the kth column in L by

and compute the kth row in U by

This algorithm is continued until all elements of U and L are completely determined. When l_{i i} = 1 (1?i ?n), this procedure is called the Doolittle factorization, and when u _{j j} = 1 (1? j ?n), it is known as the Crout factorization.

Define the test matrix

Using the algorithm above, compute and print factorizations so that the diagonal entries of L and U are of the following forms:

Here the question mark means that the entry is to be computed. Write code to check the results by multiplying L and U together.