Broyden’s Method - Case Study - Mathematics Assignment Help

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Broyden’s Method

A careful study of Newton Raphson method as reported earlier, revealed that a major disadvantage of the method has is the computation of the Jacobian matrix and its inverse at each iterative step, hence the only way to avoid this problem will be to employ the use of Quasi-Newton methods. Quasi-Newton methods are methods which approximate the Jacobian matrix or its inverse with another matrix (i.e. . A very popular Quasi-Newton method is the Broyden’s method which was proposed by Charles Broyden in 1965 (Broyden, 1965). The iterative procedure involved in Broyden’s method is similar to that used in Newton Raphson method. The only difference is that an approximated matrix is used instead of .

According to Charles Broyden, (1965), there are two methods that can be used to find the approximate solution for nonlinear systems of equations as reported in Al-Towaiq, et al., (2017). The first method gives an approximate matrix for using the following assumption; must satisfy the secant equation where . However, Broyden’s method involves the computation of and not , this brings our attention to the next theorem.
 

THEOREM: (Sherman-Morrison Formula) If is a nonsingular matrix and are vectors, then is nonsingular provided that and The theorem above is a matrix inverse formula (Deng, 2011). It allows to be computed directly using , rather than computing and then its inverse at each iteration. Hence using the theorem and setting , , as well as using as defined above we have that

Hence we get From the assumption above, Broyden’s method is defined as where is computed using equation (14).

Below is the Broyden’s method algorithm:

STEP 1: Let be the initial vector given.
STEP 2: Calculate
STEP 3: In this step we compute . Because we do not have enough information to compute directly, Broyden’s method permits us to let , which implies that .
STEP 4: Calculate

STEP 5: Calculate
STEP 6: Take and and calculate . Next take the first two iterations of and calculate .
STEP 7: Calculate
STEP 8: Compute
STEP 9: Take that we found in step 8, and calculate
STEP 10: Repeat the process until it converges at , i.e. when . This will indicate that we have reached the solution of the system.

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