Modified Richardson iteration

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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 .

Convergence[edit]

Subtracting the exact solution , and introducing the notation for the error , we get the equality for the errors

Thus,

for any vector norm and the corresponding induced matrix norm. Thus, if , the method converges.

Suppose that is diagonalizable and that are the eigenvalues and eigenvectors 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.

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[edit]

Consider minimizing the function . Since this is a convex function, a sufficient condition for optimality is that the gradient is zero () which gives rise to the equation

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 .


See also[edit]

References[edit]