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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Fall 2023 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Fall 2023 • Question 4

Solution of a nonlinear fixed point problem

Consider the fixed-point problem \(x = g(x)\) for a continuously differentiable real-valued function \(g\) and assume that it has a solution \(\alpha \) for which \(|g'(\alpha )| < 1\). State and prove a convergence theorem for fixed point iteration \(x^{(k+1)} = g(x^{(k)})\) for this problem. Note: Do not assume more about the function \(g\) than is stated in the problem.

Raw LaTeX

Solution of a nonlinear fixed point problem

Consider the fixed-point problem \(x = g(x)\) for a continuously differentiable real-valued
function \(g\) and assume that it has a solution \(\alpha\) for which \(|g'(\alpha)| < 1\). State and
prove a convergence theorem for fixed point iteration \(x^{(k+1)} = g(x^{(k)})\) for this problem.
Note: Do not assume more about the function \(g\) than is stated in the
problem.

Topic Tags

Fixed-Point Iteration Linear Programming and Optimization Rayleigh Quotient Iteration