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Although the method of least squares can be used to solve an "overdetermined" system of equations; I do NOT believe it is the usual/elementary case. The predictor matrix is assumed to be invertible (non-singular) which usually explained as "just determined" with the same number of equations as unknowns. "Over-determined" and "under-determined matrices" (systems of equations) are treated as special cases.
See for example the Wikipedia article, System of linear equations
"In general, the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns: Usually, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system. Usually, a system with the same number of equations and unknowns has a single unique solution. Usually, a system with more equations than unknowns has no solution. Such a system is also known as an overdetermined system."
The least squares "solution" is a minimum of a parabola; mathematically it has a unique "closed form" solution (one can use calculus to work out a specific algebraic formula that will yield an exact solution).
An analogy, a screwdriver CAN be used as a chisel, but it would be a mistake to begin a Wikipedia article on "screwdrivers" by stating that "a screwdriver is a tool for chiseling wood" that would ignore the primary use of screwdrivers to turn screws to attach or detach two pieces of wood, metal or plastic.
Yes, least squares can be used to solve overdetermined systems of equations, just as a screwdriver can be used to chisel wood, but that is not initial intended use of either one. Jim.Callahan,Orlando (talk) 01:19, 7 February 2016 (UTC)
The force constant in Hooke's Law is typically defined by rather than as here, where it has been replaced by . Perhaps this is just to make it look simpler for the purposes of this article? — Preceding unsigned comment added by 22.214.171.124 (talk) 06:32, 17 August 2016 (UTC)
Nonlinear least squares: notation
In the equation immediately under the paragraph that begins "The Jacobian J is a function of...", the meaning of k changes silently in the middle of the equation, which is horribly confusing to someone less than completely comfortable with the topic. If you know what's going on, then you realize that the superscript k used early in the equation refers to the iteration, but the subscript k in the summation later in that equation is an index over the parameter number. If you aren't so confident, then you wonder why you suddenly are summing over iteration numbers. The confusion is aggravated in the immediately following paragraph and equation, where the text reads
- (...) solved for :
but does not actually appear in the accompanying equation; it is only implied by the unindexed subscript j in that equation, and k again is (silently) used in the new sense, as the summation index over parameter number.
I think the easiest solution would be either to change all the k superscript indices referring to iteration to some other symbol, e.g. a capital K, as this would distinguish notationally between the two meanings and retain the use of subscript k as a summation index, OR to insert the word superscript into the first paragraph of this section to indicate that only superscripts refer to iteration numbers. I think that latter is closest to standard usage, so I've made that change. BSVulturis (talk) 19:22, 11 April 2017 (UTC)