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

${\displaystyle Ax=b.\,}$

The Richardson iteration is

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

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

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

Convergence

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

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

Thus,

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

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

Suppose that ${\displaystyle A}$ is symmetric positive definite and that ${\displaystyle (\lambda _{j})_{j}}$ are the eigenvalues of ${\displaystyle A}$. The error converges to ${\displaystyle 0}$ if ${\displaystyle |1-\omega \lambda _{j}|<1}$ for all eigenvalues ${\displaystyle \lambda _{j}}$. If, e.g., all eigenvalues are positive, this can be guaranteed if ${\displaystyle \omega }$ is chosen such that ${\displaystyle 0<\omega <\omega _{\text{max}}\,,\ \omega _{\text{max}}:=2/\lambda _{\text{max}}(A)}$. The optimal choice, minimizing all ${\displaystyle |1-\omega \lambda _{j}|}$, is ${\displaystyle \omega _{\text{opt}}:=2/(\lambda _{\text{min}}(A)+\lambda _{\text{max}}(A))}$, which gives the simplest Chebyshev iteration. This optimal choice yields a spectral radius of

${\displaystyle \min _{\omega \in (0,\omega _{\text{max}})}\rho (I-\omega A)=\rho (I-\omega _{\text{opt}}A)=1-{\frac {2}{\kappa (A)+1}}\,,}$

where ${\displaystyle \kappa (A)}$ is the condition number.

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

Equivalence to gradient descent

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

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

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

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

which is equivalent to the Richardson iteration by making ${\displaystyle t=\omega }$.

See also

• Richardson extrapolation

References

• 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