[M] Consider the heat plate in the following figure (refer to Exercise 33 in Section 1.1...

[M] Consider the heat plate in the following figure (refer to Exercise 33 in Section 1.1).

The solution to the steady-state heat flow problem for this plate is approximated by the solution to the equation Ax = b, where b = (5, 15, 0, 10, 0, 10, 20, 30) and

The missing entries in A are zeros. The nonzero entries of A lie within a band along the main diagonal. Such band matrices occur in a variety of applications and often are extremely large (with thousands of rows and columns but relatively narrow bands).

a. Use the method in Example 2 to construct an LU factorization of A, and note that both factors are band matrices (with two nonzero diagonals below or above the main diagonal). Compute LU – A to check your work.

b. Use the LU factorization to solve Ax = b.

c. Obtain A^–1 and note that A^–1 is a dense matrix with no band structure. When A is large, L and U can be stored in much less space than A^–1. This fact is another reason for preferring the LU factorization of A to A^–1 itself.

Exercise 33: Write a system of four equations whose solution gives estimates for the temperatures T₁,…, T₄