:StuRat, good point about the extraneous solution.
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Extraneous solutions are not too difficult to deal with because they just require checking all solutions for validity.
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An over-determined situation eliminates the possibility of ambiguous and / or extraneous solutions that can occur when only the minimum required number of measurements are available.
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In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero.
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In that case, A'is certainly " not " injective, so applying its transformation to the equation AX = B creates an extraneous solution.
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The method of Lagrange multipliers also eliminates the dependence on other variables cleanly and furthermore, is strictly true, while you potentially introduce extraneous solutions with the extra parameters.
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If some function is applied to both sides of an equation, the resulting equation has the solutions of the initial equation among its solutions, but may have further solutions called extraneous solutions.
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In mathematics, an "'extraneous solution "'( or "'spurious solution "') is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the original problem.
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Raising both sides to the exponent of 2 ( which means applying the function f ( s ) = s ^ 2 to both sides of the equation ) changes the equation to x ^ 2 = 1, which not only has the previous solution but also introduces the extraneous solution, x =-1.
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This procedure produces extraneous solutions, but when we have found the correct ones by numerical means we can also write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable & mdash; an algebraic solution of the general quintic.
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