Let A be a symmetric positive definite n x n matrix with entries a;; that are...

  

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Let A be a symmetric positive definite n x n matrix with entries a;; that are nonzero only if one of the following holds: i = 1, or i = n, or j = 1, or j = n, or i = j. Otherwise, aj = 0. (a) Show that only 5n - 6 of the n elements of A are possibly nonzero. (b) Plot the zero-structure of A. (In MATLAB you can invent such a matrix for n = 20, say, and use spy (A).) (c) Explain why for n = 100,000 using chol (see Section 5.5) to solve Ax = b for a given right-hand-side vector would be problematic. Let A be a symmetric positive definite n x n matrix with entries a;; that are nonzero only if one of the following holds: i = 1, or i = n, or j = 1, or j = n, or i = j. Otherwise, aj = 0. (a) Show that only 5n - 6 of the n elements of A are possibly nonzero. (b) Plot the zero-structure of A. (In MATLAB you can invent such a matrix for n = 20, say, and use spy (A).) (c) Explain why for n = 100,000 using chol (see Section 5.5) to solve Ax = b for a given right-hand-side vector would be problematic.