Modified Richardson iteration
Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Richardson in his work dated 1910. It is similar to the Jacobi and Gauss–Seidel method.
We seek the solution to a set of linear equations, expressed in matrix terms as
The Richardson iteration is
where is a scalar parameter that has to be chosen such that the sequence converges.
It is easy to see that the method has the correct fixed points, because if it converges, then and has to approximate a solution of .
Subtracting the exact solution , and introducing the notation for the error , we get the equality for the errors
for any vector norm and the corresponding induced matrix norm. Thus, if , the method converges.
Suppose that is symmetric positive definite and that are the eigenvalues of . The error converges to if for all eigenvalues . If, e.g., all eigenvalues are positive, this can be guaranteed if is chosen such that . The optimal choice, minimizing all , is , which gives the simplest Chebyshev iteration. This optimal choice yields a spectral radius of
where is the condition number.
If there are both positive and negative eigenvalues, the method will diverge for any if the initial error has nonzero components in the corresponding eigenvectors.
Equivalence to gradient descent
Define and . Because of the form of A, it is a positive semi-definite matrix, so it has no negative eigenvalues.
A step of gradient descent is
which is equivalent to the Richardson iteration by making .
- Richardson, L.F. (1910). "The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam". Philosophical Transactions of the Royal Society A. 210: 307–357. doi:10.1098/rsta.1911.0009. JSTOR 90994.
- Vyacheslav Ivanovich Lebedev (2002). "Chebyshev iteration method". Springer. Retrieved 2010-05-25. Appeared in Encyclopaedia of Mathematics (2002), Ed. by Michiel Hazewinkel, Kluwer - ISBN 1-4020-0609-8