Consider the LU decomposition of an upper Hessenberg (no, it’s not a place in Germany) matrix,…

Consider the
LU decomposition of an upper Hessenberg (no, it’s not a place in Germany)
matrix, defined on the facing page, assuming that no pivoting is needed: A =
LU.

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(a) Provide
an efficient algorithm for this LU decomposition (do not worry about questions
of memory access and vectorization).

(b) What is
the sparsity structure of the resulting matrix L (i.e., where are its
nonzeros)?

(c) How many
operations (to a leading order) does it take to solve a linear system Ax = b,
where A is upper Hessenberg?

(d) Suppose
now that partial pivoting is applied. What are the sparsity patterns of the
factors of A?