Practical Considerations
Newton's method is an extremely powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method.
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“Despair, feeding, as it always does, on phantasmagoria, is imperturbably leading literature to the rejection, en masse, of all divine and social laws, towards practical and theoretical evil.”
—Isidore Ducasse, Comte de Lautréamont (1846–1870)