# What is the **contraction factor and convergence factor** of a iteration method?

For any iteration method from A=M-N, e.g.,

$$Mx_{k+1}=Nx_{k}+b,\quad k=0,1,...$$

we know that it converges iff $$\rho(M^{-1}N)<1$$. And when it converges, there exists a concept called asymtotic convergence rate $$R = -\ln(\rho M^{-1}N)$$. So, we usually call $$\rho$$ the convergence factor.

I also heard that a concept called contraction factor of the iteration method. What is the contraction factor?

These concepts are related. Let $$A = M - N$$ and consider the iteration $$M x_{k+1} = N x_k + b.$$ We can write this as a mapping $$\Phi : \mathbb{R}^n \to \mathbb{R}^n$$ defined by $$\Phi(y) = M^{-1}(N y + b),$$ and so $$x_{k+1} = \Phi(x_k)$$. $$\Phi$$ is called a contraction mapping if there exists a constant $$0 \leq L < 1$$ (called the Lipschitz constant, or contraction factor) such that $$\| \Phi(y) - \Phi(z) \| \leq L \| y - z \|.$$ Intuitively speaking, this means that the mapping $$\Phi$$ takes any two vectors and moves them closer together. Note that \begin{align} \Phi(y) - \Phi(z) &= M^{-1}(Ny + b) - M^{-1}(Nz + b) \\ &= M^{-1}N(y - z), \tag{\ast} \end{align} and so we have that $$L = \|M^{-1} N\|$$.
Now, let's relate this to the convergence of the iterative method. Let $$x$$ be the solution to $$Ax = b$$. Notice that $$x$$ is a fixed point of this iteration: \begin{align} \Phi(x) &= M^{-1}(Nx + b)\\ &= M^{-1}((M-A)x + b) \\ &= (I - M^{-1}A)x + M^{-1} b \\ &= x. \end{align}
Combining this fact with equation ($$\ast$$), we have \begin{align} M^{-1}N(x_{k} - x) &= \Phi(x_k) - \Phi(x) \\ &= x_{k+1} - x, \end{align} and therefore, defining the error of the method at step $$k$$ to be $$e_k = x_{k}-x$$, we have the following recurrence for the errors: $$e_{k+1} = M^{-1}N e_k. \tag{\ast\ast}$$ So, we certainly have that $$\|e_{k+1}\| \leq \| M^{-1} N \| \| e_k \| = L \| e_k \|,$$ relating the convergence of the iteration to the contraction factor of the mapping.
In your question, you mention the spectral radius of the operator $$M^{-1}N$$. The condition $$\rho(M^{-1}N)$$ indeed is necessary and sufficient for convergence of the iteration, which can be seen by applying the power sequence theorem for the spectral radius to equation ($$\ast\ast$$).