Direct secant updates of matrix factorizations
Authors:
J. E. Dennis and Earl S. Marwil
Journal:
Math. Comp. 38 (1982), 459474
MSC:
Primary 65H10
MathSciNet review:
645663
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Abstract: This paper presents a new context for using the sparse Broyden update method to solve systems of nonlinear equations. The setting for this work is that a Newtonlike algorithm is assumed to be available which incorporates a workable strategy for improving poor initial guesses and providing a satisfactory Jacobian matrix approximation whenever required. The total cost of obtaining each Jacobian matrix, or the cost of factoring it to solve for the Newton step, is assumed to be sufficiently high to make it attractive to keep the same Jacobian approximation for several steps. This paper suggests the extremely convenient and apparently effective technique of applying the sparse Broyden update directly to the matrix factors in the iterations between reevaluations in the hope that fewer fresh factorizations will be required. The strategy is shown to be locally and qsuperlinearly convergent, and some encouraging numerical results are presented.
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 K. W. Brodlie, A. R. Gourlay & J. Greenstadt, "Rankone and ranktwo corrections to positive definite matrices expressed in product form," J. Inst. Math. Appl., v. 11, 1973, pp. 7382. MR 0331770 (48:10102)
 [2]
 C. G. Broyden, "The convergence of an algorithm for solving sparse nonlinear systems," Math. Comp., v. 25, 1971, pp. 285294. MR 0297122 (45:6180)
 [3]
 A. K. Cline, C. B. Moler, G. W. Stewart & J. H. Wilkinson, "An estimate for the condition number of a matrix," SIAM J. Numer. Anal., v. 16, 1979, pp. 368375. MR 526498 (80g:65048)
 [4]
 A. R. Curtis, M. J. D. Powell & J. K. Reid, "On the estimation of sparse Jacobian matrices," J. Inst. Math. Appl., v. 13, 1974, pp 117119.
 [5]
 J. E. Dennis, Jr., "A brief introduction to quasiNewton methods" in Numerical Analysis (Edited by G. Golub and J. Oliger), Proc. Sympos. Appl. Math., vol. 22, Amer. Math. Soc., Providence, R. I., 1978. MR 533049 (80d:65003)
 [6]
 J. E. Dennis, Jr. & J. J. Moré, "A characterization of superlinear convergence and its application to quasiNewton methods," Math. Comp., v. 28, 1974, pp. 549560. MR 0343581 (49:8322)
 [7]
 J. E. Dennis, Jr & J. J. Moré, "QuasiNewton methods, motivation and theory," SIAM Rev., v. 19, 1977, pp. 4689. MR 0445812 (56:4146)
 [8]
 J. E. Dennis, Jr. & H. F. Walker, "Convergence theorems for leastchange secant update methods," SIAM J. Numer. Anal., v. 18, 1981, pp 949987. MR 638993 (83a:65052a)
 [9]
 J. J. Dongarra, J. R. Bunch, C. B. Moler & G. W. Stewart, LINPACK User's Guide, SIAM, Philadelphia, Pa., 1979.
 [10]
 I. S. Duff, "A survey of sparse matrix research," Proc. IEEE, v. 65, 1977, pp. 500535.
 [11]
 A. M. Erisman & J. K. Reid, "Monitoring the stability of the triangular factorization of a sparse matrix," Numer. Math., v. 22, 1974, pp. 183186. MR 0345395 (49:10131)
 [12]
 P. E. Gill, G. Golub, W. Murray & M. A. Saunders, "Methods for modifying matrix factorizations," Math. Comp., v. 28, 1974, pp. 505535. MR 0343558 (49:8299)
 [13]
 A. Hindmarsh, private communication, 1978.
 [14]
 A. Lucia, Thesis, Dept. of Chem. Engr., University of Connecticut, Storrs, 1980.
 [15]
 J. J. Moré, B. Garbow & K. Hillstrom, User Guide for MINPACK1, Report ANL8074, Argonne National Laboratories, 1980.
 [16]
 J. M. Ortega & W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
 [17]
 L. K. Schubert, "Modification of a quasiNewton method for nonlinear equations with a sparse Jacobian," Math. Comp., v. 24, 1970, pp. 2730. MR 0258276 (41:2923)
 [18]
 C. Van Loan, private communication, 1978.
 [19]
 J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 0184422 (32:1894)
 [20]
 M. Wheeler & T. Potempa, private communication, 1980.
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DOI:
http://dx.doi.org/10.1090/S00255718198206456638
PII:
S 00255718(1982)06456638
Article copyright:
© Copyright 1982 American Mathematical Society
