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Solving Overdetermined Systems with the QR Decomposition

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Solving Overdetermined Systems with the QR Decomposition

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Deploying code to production can be filled with uncertainty. Reduce the risks, and deploy earlier and more often. Download this free guide to learn more. Brought to you in partnership with Rollbar.

A system of linear equations is considered overdetermined if there are more equations than unknowns. In practice, we have a system Ax=b where A is a m by n matrix and b is a m dimensional vector b but m is greater than n. In this case, the vector b cannot be expressed as a linear combination of the columns of A. Hence, we can't find x so that satisfies the problem Ax=b (except in specific cases) but it is possible to determine x so that Ax is as close to b as possible. So we wish to find x which minimizes the following error




Considering the QR decomposition of A we have that Ax=b becomes




multiplying by Q^T we obtain




and since Q^T is orthogonal (this means that Q^T*Q=I) we have




Now, this is a well defined system, R is an upper triangular matrix and Q^T*b is a vector. More precisely b is the orthogonal projection of b onto the range of A. And,




The function linalg.lstsq() provided by numpy returns the least-squares solution to a linear system equation and is able to solve overdetermined systems. Let's compare the solutions of linalg.lstsq() with the ones computed using the QR decomposition:
from numpy import *

# generating a random overdetermined system
A = random.rand(5,3)
b = random.rand(5,1) 

x_lstsq = linalg.lstsq(A,b)[0] # computing the numpy solution

Q,R = linalg.qr(A) # qr decomposition of A
Qb = dot(Q.T,b) # computing Q^T*b (project b onto the range of A)
x_qr = linalg.solve(R,Qb) # solving R*x = Q^T*b

# comparing the solutions
print 'qr solution'
print x_qr
print 'lstqs solution'
print x_lstsq

This is the output of the script above:

qr solution [[ 0.08704059] [-0.10106932] [ 0.56961487]]
lstqs solution [[ 0.08704059] [-0.10106932] [ 0.56961487]]

As we can see, the solutions are the same.

Deploying code to production can be filled with uncertainty. Reduce the risks, and deploy earlier and more often. Download this free guide to learn more. Brought to you in partnership with Rollbar.

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