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Numerical Linear Algebra with Applications by William Ford

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Chapter 17

Implementing the QR Decomposition

Abstract

After reviewing the reduced QR decomposition done using Gram-Schmidt, this chapter develops two efficient methods for computing the QR decomposition, using Givens rotations and Householder reflections. Givens rotations are defined, and the use of a rotation to zero out a particular entry in a vector is developed. This is followed by showing how to use Givens rotations to zero out multiple entries in a vector. If J(i,j,c,s) is a Givens rotation and A is a matrix, the product J(i,j,c,s)*A can be performed by modifying only two rows of A. These ideas are then applied to zeroing out entries in a column of a matrix. Due to possibility of overflow, the Givens parameters c and s must be computed ...

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