Least Squares Problems
The origin of the least squares data-fitting problem is the need of a notion of “generalized solutions” for a linear system Ax = b that has no solution in the classical sense (that is, b does not belong to the range of A). The idea is then to look for a vector x such that Ax is “the closest possible” to b. Several norms are at hand to measure the distance between Ax and b, but the simplest choice (which corresponds to the denomination “least squares”) is the Euclidean vector norm. In other words, a least squares problem amounts to finding the solution (possibly nonunique) x ∈ ℝ p to the following minimization problem:
KeywordsOrthogonal Projection Normal Equation Triangular Matrix Orthogonal Matrix Operation Count
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