Broyden's method

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In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in k variables. It was originally described by C. G. Broyden in 1965.[1]

Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden's method is to compute the whole Jacobian at most only at the first iteration, and to do rank-one updates at other iterations.

In 1979 Gay proved that when Broyden's method is applied to a linear system of size n × n, it terminates in 2 n steps,[2] although like all quasi-Newton methods, it may not converge for nonlinear systems.

Description of the method

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Solving single-variable nonlinear equation

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In the secant method, we replace the first derivative f at xn with the finite-difference approximation:

 

and proceed similar to Newton's method:

 

where n is the iteration index.

Solving a system of nonlinear equations

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Consider a system of k nonlinear equations in   unknowns

 

where f is a vector-valued function of vector x

 
 

For such problems, Broyden gives a variation of the one-dimensional Newton's method, replacing the derivative with an approximate Jacobian J. The approximate Jacobian matrix is determined iteratively based on the secant equation, a finite-difference approximation:

 

where n is the iteration index. For clarity, define

 
 
 

so the above may be rewritten as

 

The above equation is underdetermined when k is greater than one. Broyden suggested using the most recent estimate of the Jacobian matrix, Jn−1, and then improving upon it by requiring that the new form is a solution to the most recent secant equation, and that there is minimal modification to Jn−1:

 

This minimizes the Frobenius norm

 

One then updates the variables using the approximate Jacobian, what is called a quasi-Newton approach.

 

If   this is the full Newton step; commonly a line search or trust region method is used to control  . The initial Jacobian can be taken as a diagonal, unit matrix, although more common is to scale it based upon the first step.[3] Broyden also suggested using the Sherman–Morrison formula[4] to directly update the inverse of the approximate Jacobian matrix:

 

This first method is commonly known as the "good Broyden's method."

A similar technique can be derived by using a slightly different modification to Jn−1. This yields a second method, the so-called "bad Broyden's method":

 

This minimizes a different Frobenius norm

 

In his original paper Broyden could not get the bad method to work, but there are cases where it does[5] for which several explanations have been proposed.[6][7] Many other quasi-Newton schemes have been suggested in optimization such as the BFGS, where one seeks a maximum or minimum by finding zeros of the first derivatives (zeros of the gradient in multiple dimensions). The Jacobian of the gradient is called the Hessian and is symmetric, adding further constraints to its approximation.

The Broyden Class of Methods

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In addition to the two methods described above, Broyden defined a wider class of related methods.[1]: 578  In general, methods in the Broyden class are given in the form[8]: 150    where     and   and   for each  . The choice of   determines the method.

Other methods in the Broyden class have been introduced by other authors.

  • The Davidon–Fletcher–Powell (DFP) method, which is the only member of this class being published before the two methods defined by Broyden.[1]: 582  For the DFP method,  .[8]: 150 
  • Anderson's iterative method, which uses a least squares approach to the Jacobian.[9]
  • Schubert's or sparse Broyden algorithm – a modification for sparse Jacobian matrices.[10]
  • The Pulay approach, often used in density functional theory.[11][12]
  • A limited memory method by Srivastava for the root finding problem which only uses a few recent iterations.[13]
  • Klement (2014) – uses fewer iterations to solve some systems.[14][15]
  • Multisecant methods for density functional theory problems[7][16]

See also

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References

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  1. ^ a b c Broyden, C. G. (1965). "A Class of Methods for Solving Nonlinear Simultaneous Equations". Mathematics of Computation. 19 (92). American Mathematical Society: 577–593. doi:10.1090/S0025-5718-1965-0198670-6. JSTOR 2003941.
  2. ^ Gay, D. M. (1979). "Some convergence properties of Broyden's method". SIAM Journal on Numerical Analysis. 16 (4). SIAM: 623–630. doi:10.1137/0716047.
  3. ^ Shanno, D. F.; Phua, Kang -Hoh (1978). "Matrix conditioning and nonlinear optimization". Mathematical Programming. 14 (1): 149–160. doi:10.1007/BF01588962. ISSN 0025-5610.
  4. ^ Sherman, Jack; Morrison, Winifred J. (1950). "Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix". The Annals of Mathematical Statistics. 21 (1): 124–127. doi:10.1214/aoms/1177729893. ISSN 0003-4851.
  5. ^ Kvaalen, Eric (1991). "A faster Broyden method". BIT Numerical Mathematics. 31 (2). SIAM: 369–372. doi:10.1007/BF01931297.
  6. ^ Martı́nez, José Mario (2000). "Practical quasi-Newton methods for solving nonlinear systems". Journal of Computational and Applied Mathematics. 124 (1–2): 97–121. doi:10.1016/s0377-0427(00)00434-9. ISSN 0377-0427.
  7. ^ a b Marks, L. D.; Luke, D. R. (2008). "Robust mixing for ab initio quantum mechanical calculations". Physical Review B. 78 (7). doi:10.1103/physrevb.78.075114. ISSN 1098-0121.
  8. ^ a b Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer New York. doi:10.1007/978-0-387-40065-5. ISBN 978-0-387-30303-1.
  9. ^ Anderson, Donald G. (1965). "Iterative Procedures for Nonlinear Integral Equations". Journal of the ACM. 12 (4): 547–560. doi:10.1145/321296.321305. ISSN 0004-5411.
  10. ^ Schubert, L. K. (1970). "Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian". Mathematics of Computation. 24 (109): 27–30. doi:10.1090/S0025-5718-1970-0258276-9. ISSN 0025-5718.
  11. ^ Pulay, Péter (1980). "Convergence acceleration of iterative sequences. the case of scf iteration". Chemical Physics Letters. 73 (2): 393–398. doi:10.1016/0009-2614(80)80396-4.
  12. ^ Kresse, G.; Furthmüller, J. (1996). "Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set". Physical Review B. 54 (16): 11169–11186. doi:10.1103/PhysRevB.54.11169. ISSN 0163-1829.
  13. ^ Srivastava, G P (1984). "Broyden's method for self-consistent field convergence acceleration". Journal of Physics A: Mathematical and General. 17 (6): L317–L321. doi:10.1088/0305-4470/17/6/002. ISSN 0305-4470.
  14. ^ Klement, Jan (2014). "On Using Quasi-Newton Algorithms of the Broyden Class for Model-to-Test Correlation". Journal of Aerospace Technology and Management. 6 (4): 407–414. doi:10.5028/jatm.v6i4.373. ISSN 2175-9146.
  15. ^ "Broyden class methods – File Exchange – MATLAB Central". www.mathworks.com. Retrieved 2016-02-04.
  16. ^ Woods, N D; Payne, M C; Hasnip, P J (2019). "Computing the self-consistent field in Kohn–Sham density functional theory". Journal of Physics: Condensed Matter. 31 (45): 453001. doi:10.1088/1361-648X/ab31c0. ISSN 0953-8984.

Further reading

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