Brent’s method

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Brent s method (Press et al. 1992) is used to find the root of Eq. 3.38 (Vr). Some terms in the latter equation have Vr raised to the fifth power, which means that there could be as many as five roots to such an equation. This root-finding method does not require a derivative as the following Newton-Raphson method does, but it does require a lower and upper bracket within which the true solution must he. In this particular case with... 

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Brent s and Brown s methods are variations of Newton s method that improve convergence. The calculation of the elements in in Eq. (L.ll) and the solving of the linear equations are intermingled. Each row of Jt is obtained as needed using the latest information available. Then one more step in the solution of the linear equations is executed. Brown s method is an extension of Gaussian elimination Brent s method is an extension of QR factorization. Computer codes are generally implemented by using numerical approximations for the partial derivatives in J. ... 

Obtaining the surface resistance R is to search R which minimizes Eqn. (5). The general optimization method, such as Brent s method [5], can be applied for searching R. 

Powell and Brent s methods are both an elaborated version of Smith s method (1962), which is analyzed briefly here for the two-dimensional case. 

The multidimensional case is more complex and in this extension only, is the shining theory of Powell and Brent s methods. 

Elh is by given L and H, a one-dimensional function of 9. Any onedimensional minimization routine may be therefore suitable for solving this problem. For example, the Brent s method in one-dimension as described in Ref. 17 was found to be quite suitable for this purpose [20]. 

Brent, A.C., Hietkamp, S., 2003. Comparative evaluation of bfe cycle impact assessment methods with a South African case study. International Journal of Life Cycle Assessment 8, 27-38. 

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