On the global convergence of Broyden's method
Authors:
J. J. Moré and J. A. Trangenstein
Journal:
Math. Comp. 30 (1976), 523540
MSC:
Primary 65H10
MathSciNet review:
0418451
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider Broyden's 1965 method for solving nonlinear equations. If the mapping is linear, then a simple modification of this method guarantees global and Qsuperlinear convergence. For nonlinear mappings it is shown that the hybrid strategy for nonlinear equations due to Powell leads to Rsuperlinear convergence provided the search directions form a uniformly linearly independent sequence. We then explore this last concept and its connection with Broyden's method. Finally, we point out how the above results extend to Powell's symmetric version of Broyden's method.
 [1]
C.
G. Broyden, A class of methods for solving
nonlinear simultaneous equations, Math.
Comp. 19 (1965),
577–593. MR 0198670
(33 #6825), http://dx.doi.org/10.1090/S00255718196501986706
 [2]
C.
G. Broyden, The convergence of singlerank
quasiNewton methods, Math. Comp. 24 (1970), 365–382. MR 0279993
(43 #5714), http://dx.doi.org/10.1090/S00255718197002799930
 [3]
C.
G. Broyden, J.
E. Dennis Jr., and Jorge
J. Moré, On the local and superlinear convergence of
quasiNewton methods, J. Inst. Math. Appl. 12 (1973),
223–245. MR 0341853
(49 #6599)
 [4]
J.
E. Dennis Jr. and Jorge
J. Moré, A characterization of superlinear
convergence and its application to quasiNewton methods, Math. Comp. 28 (1974), 549–560. MR 0343581
(49 #8322), http://dx.doi.org/10.1090/S00255718197403435811
 [5]
J. E. DENNIS & J. J. MORÉ, QuasiNewton Methods, Motivation and Theory, Cornell University Computer Science Technical Report TR 74217, 1974.
 [6]
P.
E. Gill and W.
Murray, QuasiNewton methods for unconstrained optimization,
J. Inst. Math. Appl. 9 (1972), 91–108. MR 0300410
(45 #9456)
 [7]
J.
M. Ortega and W.
C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New YorkLondon, 1970. MR 0273810
(42 #8686)
 [8]
M.
J. D. Powell, A hybrid method for nonlinear equations,
Numerical methods for nonlinear algebraic equations (Proc. Conf., Univ.
Essex, Colchester, 1969) Gordon and Breach, London, 1970,
pp. 87–114. MR 0343589
(49 #8330a)
 [9]
M.
J. D. Powell, A Fortran subroutine for solving systems of nonlinear
algebraic equations, Numerical methods for nonlinear algebraic
equations (Proc. Conf., Univ. Essex, Colchester, 1969) Gordon and Breach,
London, 1970, pp. 115–161. MR 0343590
(49 #8330b)
 [10]
M.
J. D. Powell, A new algorithm for unconstrained optimization,
Nonlinear Programming (Proc. Sympos., Univ. of Wisconsin, Madison, Wis.,
1970) Academic Press, New York, 1970, pp. 31–65. MR 0272162
(42 #7043)
 [11]
M. J. D. POWELL, A FORTRAN Subroutine for Unconstrained Minimization, Requiring First Derivatives of the Objective Function, Atomic Energy Research Establishment, Harwell, R.6469, 1970.
 [12]
M.
J. D. Powell, Convergence properties of a class of minimization
algorithms, Nonlinear programming, 2 (Proc. Sympos. Special Interest
Group on Math. Programming, Univ. Wisconsin, Madison, Wis., 1974)
Academic Press, New York, 1974, pp. 1–27. MR 0386270
(52 #7128)
 [13]
H.
Schwetlick, Über die Realisierung und Konvergenz von
Mehrschrittverfahren zur iterativen Lösung nichtlinearer
Gleichungen, Z. Angew. Math. Mech. 54 (1974),
479–493 (German, with English and Russian summaries). MR 0378397
(51 #14565)
 [14]
S. M. THOMAS, Sequential Estimation Techniques for QuasiNewton Algorithms, Cornell University Ph.D. Thesis, 1974.
 [1]
 C. G. BROYDEN, "A class of methods for solving nonlinear simultaneous equations," Math. Comp., v. 19, 1965, pp. 577593. MR 33 #6825. MR 0198670 (33:6825)
 [2]
 C. G. BROYDEN, "The convergence of singlerank quasiNewton methods," Math. Comp., v. 24, 1970, pp. 365382. MR 43 #5714. MR 0279993 (43:5714)
 [3]
 C. G. BROYDEN, J. E. DENNIS & J. J. MORÉ, "On the local and superlinear convergence of quasiNewton methods," J. Inst. Math. Appl., v. 12, 1973, pp. 223245. MR 49 #6599. MR 0341853 (49:6599)
 [4]
 J. E. DENNIS & J. J. MORÉ, "A characterization of superlinear convergence and its application to quasiNewton methods," Math. Comp., v. 28, 1974, pp. 549560. MR 49 #8322. MR 0343581 (49:8322)
 [5]
 J. E. DENNIS & J. J. MORÉ, QuasiNewton Methods, Motivation and Theory, Cornell University Computer Science Technical Report TR 74217, 1974.
 [6]
 P. E. GILL & W. MURRAY, "QuasiNewton methods for unconstrained minimization," J. Inst. Math. Appl., v. 9, 1972, pp. 91108. MR 45 #9456. MR 0300410 (45:9456)
 [7]
 J. M. ORTEGA & W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 42 #8686. MR 0273810 (42:8686)
 [8]
 M. J. D. POWELL, "A hybrid method for nonlinear equations," Numerical Methods for NonLinear Algebraic Equations, Gordon and Breach, London, 1970, pp. 87114. MR 49 #8330a. MR 0343589 (49:8330a)
 [9]
 M. J. D. POWELL, "A FORTRAN subroutine for solving systems of nonlinear algebraic equations," Numerical Methods for NonLinear Algebraic Equations, Gordon and Breach, London, 1970, pp. 115161. MR 49 #8330b. MR 0343590 (49:8330b)
 [10]
 M. J. D. POWELL, "A new algorithm for unconstrained optimization," Nonlinear Programming, Academic Press, New York, 1970, pp. 3165. MR 42 #7043. MR 0272162 (42:7043)
 [11]
 M. J. D. POWELL, A FORTRAN Subroutine for Unconstrained Minimization, Requiring First Derivatives of the Objective Function, Atomic Energy Research Establishment, Harwell, R.6469, 1970.
 [12]
 M. J. D. POWELL, "Convergence properties of a class of minimization algorithms," Nonlinear Programming 2, Academic Press, New York, 1974, pp. 127. MR 0386270 (52:7128)
 [13]
 H. SCHWETLICK, "Über die Realisierung und Konvergenz von Mehrschrittverfahren zur iterativen Lösung nichtlinearer Gleichungen," Z. Angew. Math. Mech., v. 54, 1974, pp. 479493. MR 0378397 (51:14565)
 [14]
 S. M. THOMAS, Sequential Estimation Techniques for QuasiNewton Algorithms, Cornell University Ph.D. Thesis, 1974.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65H10
Retrieve articles in all journals
with MSC:
65H10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197604184512
PII:
S 00255718(1976)04184512
Article copyright:
© Copyright 1976
American Mathematical Society
