Exponential lower bounds for finding Brouwer fixed points
Journal of Complexity
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
On optimality of Krylov's information when solving linear operator equations
Journal of Complexity
Approximating fixed points of weakly contracting mappings
Journal of Complexity
Optimal solution of nonlinear equations
Optimal solution of nonlinear equations
A two-dimensional bisection envelope algorithm for fixed points
Journal of Complexity
Algorithm 825: A deep-cut bisection envelope algorithm for fixed points
ACM Transactions on Mathematical Software (TOMS)
A recursive algorithm for the infinity-norm fixed point problem
Journal of Complexity
A note on two fixed point problems
Journal of Complexity
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We present the PFix algorithm for approximating a fixed point of a function f that has arbitrary dimensionality, is defined on a rectangular domain, and is Lipschitz continuous with respect to the infinity norm with constant 1. PFix has applications in economics, game theory, and the solution of partial differential equations. PFix computes an approximation that satisfies the residual error criterion, and can also compute an approximation satisfying the absolute error criterion when the Lipschitz constant is less than 1. For functions defined on all rectangular domains, the worst-case complexity of PFix has order equal to the logarithm of the reciprocal of the tolerance, raised to the power of the dimension. Dividing this order expression by the factorial of the dimension yields the order of the worst-case bound for the case of the unit hypercube. PFix is a recursive algorithm, in that it uses solutions to a d-dimensional problem to compute a solution to a (d + 1)-dimensional problem. A full analysis of PFix may be found in Shellman and Sikorski [2003b], and a C implementation is available through ACM ToMS.