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Local convergence
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<p>In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, an <a href="/facts/Iterative_method/uKMAJhEh">iterative method</a> is called locally convergent if the successive <a href="/facts/Approximation/LvYHsXzq">approximations</a> produced by the method are guaranteed to <a href="/facts/Limit_of_a_sequence/Ktge2RwL">converge</a> to a solution when the initial approximation is already close enough to the solution. Iterative methods for <a href="/facts/Nonlinear_equation/4aYItALC">nonlinear equations</a> and their systems, such as <a href="/facts/Newton%27s_method/VlRhI2FL">Newton's method</a> are usually only locally convergent. </p><p>An iterative method that converges for an arbitrary initial approximation is called globally convergent. Iterative methods for systems of <a href="/facts/Linear_equations/JfCgALwf">linear equations</a> are usually globally convergent. </p>