In the context of solving nonlinear equations, the term "affine invariance" was introduced to describe the fact that when a function F: Rn → Rn is transformed to G = AF ,where A is an invertible matrix, then the equation F(x) = 0 has the same solutions as G(x) = 0, and the Newton iterates Xk+1 = Xk-F'(Xk)-1F(Xk) remain unchanged when F is replaced by G. The idea was that this property of Newton's method should be reflected in its convergence analysis and practical implementation, not only on aesthetic grounds but also because the resulting algorithms would likely be less sensitive to scaling, conditioning, and other numerical issues.
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Ypma, Tjalling J., "Review of: Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, by P. Deuflhard" (2005). Mathematics. Paper 94.