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

 A x = b.\,

The Richardson iteration is

 
x^{(k+1)}  = x^{(k)} + \omega \left( b - A x^{(k)} \right),

where \omega is a scalar parameter that has to be chosen such that the sequence x^{(k)} converges.

It is easy to see that the method has the correct fixed points, because if it converges, then x^{(k+1)} \approx x^{(k)} and x^{(k)} has to approximate a solution of A x = b.

Convergence[edit]

Subtracting the exact solution x, and introducing the notation for the error e^{(k)} = x^{(k)}-x, we get the equality for the errors

 
e^{(k+1)}  = e^{(k)} - \omega A e^{(k)} = (I-\omega A) e^{(k)}.

Thus,

 
\|e^{(k+1)}\| = \|(I-\omega A) e^{(k)}\|\leq  \|I-\omega A\| \|e^{(k)}\|,

for any vector norm and the corresponding induced matrix norm. Thus, if \|I-\omega A\|<1, the method converges.

Suppose that A is diagonalizable and that (\lambda_j, v_j) are the eigenvalues and eigenvectors of A. The error converges to 0 if | 1 - \omega \lambda_j |< 1 for all eigenvalues \lambda_j. If, e.g., all eigenvalues are positive, this can be guaranteed if \omega is chosen such that 0 < \omega < 2/\lambda_{max}(A). The optimal choice, minimizing all | 1 - \omega \lambda_j |, is \omega = 2/(\lambda_{min}(A)+\lambda_{max}(A)), which gives the simplest Chebyshev iteration.

If there are both positive and negative eigenvalues, the method will diverge for any \omega if the initial error e^{(0)} has nonzero components in the corresponding eigenvectors.

Equivalence to gradient descent[edit]

Consider minimizing the function F(x) = \frac{1}{2}\|\tilde{A}x-b\|_2^2. Since this is a convex function, a sufficient condition for optimality is that the gradient is zero (\nabla F(x) = 0) which gives rise to the equation

\tilde{A}^T\tilde{A}x = \tilde{A}^T\tilde{b}.

Define A=\tilde{A}^T\tilde{A} and b=\tilde{A}^T\tilde{b}. 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

 x^{(k+1)} = x^{(k)} - t \nabla F(x^{(k)}) = x^{(k)} - t( Ax^{(k)} - b )

which is equivalent to the Richardson iteration by making t=\omega.


See also[edit]

References[edit]