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Unformatted text preview: Program 1 Foundations of Computational Math 1 Fall 2011 Due date: via email by 11:59PM on Friday, 21 October 2011 General Task Consider the matrix A ∈ R n × k with linearly independent columns and the transformation H k H k- 1 ··· H 1 A = parenleftbigg R parenrightbigg where H i , 1 ≤ i ≤ k are Householder reflectors and R ∈ R k × k is a nonsingular upper triangular matrix. Implement a code that is capable of performing this transformation and storing the information about the H i and R efficiently in-place in a 2-dimensional n × k array and a small number of additional 1-dimensional arrays of length n or k . Also implement the necessary additional routines to solve the linear least squares problem min x ∈ R k bardbl b − Ax bardbl 2 given b ∈ R n . You must demonstrate your code on multiple examples with a wide range of values of n , k , and b for three situations: 1. n = k , i.e., a square nonsingular matrix A where x min = A- 1 b . 2. n > k and Ax = b for b ∈ R n and b ∈ R ( A ) i.e., a rectangular matrix A with full column rank and a vector b that define a consistent set of overdetermined equations. 3. n > k and b ∈ R n and b negationslash∈ R ( A ) i.e., a rectangular matrix A with full column rank and a vector b that define a linear least squares problem with a nonzero residual....
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- Fall '09
- Linear Algebra, Linear least squares, linearly independent columns, squares problem