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3.8.3 Line Search Method
A powerful technique often used to improve the convergence rate of Newton based methods is the line search method. In this method, the direction of the displacement vector is considered as optimal, but the magnitude is controlled by a parameter : The value of is usually chosen so that the total potential energy at the end of the iteration is minimized in the direction of . This is equivalent to the requirement that the residual force at the end of the iteration is orthogonal to : However, in practice it is sufficient to obtain a value of such that, where typically a value of is used. Under normal conditions the value automatically satisfies equation (3.8.3-3) and therefore few extra operations are involved. However, when this is not the case, a more suitable value for needs to be obtained. For this reason it is convenient to approximate as a quadratic in : which yields a value for as If , the square root is positive and a first improved value for is obtained: If the can be obtained by using the value that minimizes the quadratic function, that is, . This procedure is now repeated with replaced by until equation (3.8.3-3) is satisfied.