Modifying singular values: existence of solutions to sytems of nonlinear equations having a possibly singular Jacobian matrix
Author:
David Gay
Journal:
Math. Comp. 31 (1977), 962973
MSC:
Primary 65H10
Corrigendum:
Math. Comp. 33 (1979), 432433.
Corrigendum:
Math. Comp. 33 (1979), 432433.
MathSciNet review:
0443325
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Abstract: We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations whose Jacobian matrix exists but may be singular. The main idea is to modify small singular values of in such a way that the modified Jacobian matrix has a continuous pseudoinverse and that a solution of may be found by determining an asymptote of the solution to the initial value problem . We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and is nonsingular, then is unique), algorithms using may enjoy a larger region of convergence than those that require (an approximation to) .
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197704433251
PII:
S 00255718(1977)04433251
Keywords:
Continuation methods,
eigenvalues,
nonlinear equations,
pseudoinverse,
singular value decomposition
Article copyright:
© Copyright 1977 American Mathematical Society
