# An investigation of the solution of least squares problems using the QR factorisation

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## Abstract

Experimental data inevitably contains error. These experimental observations are often compared to theoretical predictions by writing as a least squares problem, i.e. minimising the sum of squares between the experimental data and theoretical predictions. These least squares problems are often solved using a QR-factorisation of a known matrix, which uses the Gram-Schmidt method to write the columns of this matrix as a linear sum of orthonormal vectors. This method, when used in practice, can exhibit numerical instabilities, where the (inevitable) numerical errors due to fixed precision calculations on a computer are magnified, and may swamp the calculation. Instead, a modified Gram-Schmidt method is used for the QR-factorisation. This modified Gram-Schmidt factorisation avoids numerical instabilities, but is less computationally efficient. The first aim of this project is to investigate the relative computational efficiencies of the two methods for QR-factorisation. The second aim is to use the QR-factorisation to identify: (i) what parameters can be recovered from experimental data; and (ii) whether the data can automatically be classified as "good" or "bad".