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Material Type: Assignment; Class: Numerical Mathematics I; Subject: Mathematics; University: University of Tennessee - Knoxville; Term: Fall 2007;
Typology: Assignments
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Math 571 – Fall 2007 – Homework # Due Thursday, Sept 20
You can use any written resources, but please work alone.
A =
1 1 + δ 1 1 1 1 + δ 1 1 1
.
(a) Use Householder QR Factorization to reduce A to an upper triangular matrix. (Algorithm 10.1) (b) Let b = (2, 2 , 2 − δ)T^. Compute Q∗b by Algorithm 10.2. (c) Use the results from (a) and (b) to find the solution x to Ax = b.
minx ||Ax − b||^22
where
A =
and b =
using the QR factorization of A (either Gram-Schmidt or Householder).
(a) Determine values for c and s (representing cos(θ) and sin(θ)) such that c^2 + s^2 = 1 and ( c −s s c
) ( x 1 x 2
( α 0
) .
(b) Given a vector x ∈ Cm, describe how you could use such 2 × 2 rotation matrices to form a m × m rotation matrix R such that Rx = αe 1. (c) Determine the flop count for constructing the rotation matrix R and the flop count for applying it to another vector y. Hint: you should keep R as a set of 2 × 2 matrices rather than constructing a large m × m matrix explicitly.