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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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), 962-973
MSC: Primary 65H10
Corrigendum: Math. Comp. 33 (1979), 432-433.
Corrigendum: Math. Comp. 33 (1979), 432-433.
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 $ f(x) = 0$ whose Jacobian matrix $ J(x)$ exists but may be singular. The main idea is to modify small singular values of $ J(x)$ in such a way that the modified Jacobian matrix $ \hat J(x)$ has a continuous pseudoinverse $ {\hat J^ + }(x)$ and that a solution $ {x^\ast}$ of $ f(x) = 0$ may be found by determining an asymptote of the solution to the initial value problem $ x(0) = {x_0},x\prime (t) = - {\hat J^ + }(x)f(x)$. 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 $ J({x^\ast})$ is nonsingular, then $ {x^\ast}$ is unique), algorithms using $ {\hat J^ + }(x)$ may enjoy a larger region of convergence than those that require (an approximation to) $ {J^{ - 1}}(x)$.

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Keywords: Continuation methods, eigenvalues, nonlinear equations, pseudoinverse, singular value decomposition
Article copyright: © Copyright 1977 American Mathematical Society