Write a function in Matlab that takes as input a size n and a tridiagonal matrix

Question

Write a function in Matlab that takes as input a size n and a tridiagonal matrix given as three vectors: n × 1 vector v representing the main diagonal, (n 1) × 1 vector w representing the upper diagonal, and (n 1) × 1 vector z representing the lower diagonal. Have this function output the LU factorization with the U as two vectors and the L as one vector representing the diagonals. Also output the number of flops used. Use only basic programming.

(a) Write out or print out your function.

(b) Run the case with n = 10, v the vector of 2’s, w and z the vector of 1’s, and b the vector of 1’s. Write down your results for the diagonals of L and U.

(c) Run the case with n = 50 and n = 100 with v the vector of 2’s, w and z the vector of 1’s. Write down your results for the number of flops used

Explanation / Answer

Syntax for diagonals of matrix

X = diag(v,k)
X = diag(v)
v = diag(X,k)
v = diag(X)
Description



X = diag(v,k) when v is a vector of n components, returns a square matrix X of order n+abs(k), with the elements of v on the kth diagonal. k = 0 represents the main diagonal, k > 0 above the main diagonal, and k < 0 below the main diagonal.

X = diag(v) puts v on the main diagonal, same as above with k = 0.

v = diag(X,k) for matrix X, returns a column vector v formed from the elements of the kth diagonal of X.

v = diag(X) returns the main diagonal of X, same as above with k = 0 .
Tips

diag(diag(X)) is a diagonal matrix.

sum(diag(X)) is the trace of X.

diag([]) generates an empty matrix, ([]).

diag(m-by-1,k) generates a matrix of size m+abs(k)-by-m+abs(k).

diag(1-by-n,k) generates a matrix of size n+abs(k)-by-n+abs(k).
Examples

The statement

diag(-m:m)+diag(ones(2*m,1),1)+diag(ones(2*m,1),-1)

produces a tridiagonal matrix of order 2*m+1.

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