[SOLVED] Math1426 homework 2- numerical analysis

1. Consider the linear equation system Ax = b with
and .
.
(a) What is the solution x of Ax = b?
(b) Let x + δx denote the solution of the perturbed equation (A + δA)(x + δx) = b for ∥δA∥∞ ≤ 0.01. Show that the relative error is bounded by
.
2. Consider the symmetric and positive definite tri-diagonal matrix
.
(a) Sketch a computer code that computes a Cholesky factorization, A = LDL⊺. Make sure that the computational complexity of your code is at most O(n).
(b) What is the computational complexity of computing a QR-factorization of the abovetri-diagonal matrix? Would you recommend using such a factorization? 3. Let the vectors a1 ∈ R2 and a2 ∈ R2 be given by
and .
We consider the standard scalar product in R2, ⟨x,y⟩ = x1y1 + x2y2 for all x,y ∈ R2.
(a) Explain how to use the Gram-Schmidt algorithm to find an orthonormal basis of thevector space V = span(a1,a2).
(b) Let A = ( a1 a2 ) ∈ R2 be a matrix with columns a1 and a2. Find an orthogonal matrix Q ∈ R2×2 and an upper triangular matrix R ∈ R2×2 such that A = QR.
(c) Also explain how to use Householder’s method for computing a QR-factorization of thematrix A.
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