= Error modes of a three-variable Jacobi iteration
{title2=$e^{(k)}=B^ke^{(0)}$}
For the <Jacobi iteration for a three-variable tridiagonal system>,
$$
B=\frac14\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}.
$$
An <orthonormal eigenbasis> is
$$
u_+=\frac12\begin{pmatrix}1\\\sqrt2\\1\end{pmatrix},\qquad u_0=\frac1{\sqrt2}\begin{pmatrix}1\\0\\-1\end{pmatrix},\qquad u_-=\frac12\begin{pmatrix}1\\-\sqrt2\\1\end{pmatrix},
$$
with <eigenvalues> $q=\sqrt2/4$, $0$, and $-q$, respectively. These are error modes: their shapes persist under the <iteration matrix>, while their coefficients change. Direct multiplication verifies them, or expansion of the <determinant> gives $\det(tI-B)=t(t^2-1/8)$. Thus $\rho(B)=q<1$ and convergence holds for every starting <vector>.
With $x^{(0)}=0$, the initial <forward error> has coefficients
$$
e^{(0)}=-x_*=c_+u_++c_0u_0+c_-u_-,\qquad c_+=-2-\sqrt2,\quad c_0=\sqrt2,\quad c_-=-2+\sqrt2.
$$
For $k\geq1$,
$$
e^{(k)}=c_+q^ku_++c_-(-q)^ku_-,\qquad \|e^{(k)}\|_2=2\sqrt3\,q^k.
$$
The zero mode disappears after one sweep; the negative mode alternates in sign. The <Euclidean norm> drops by exactly $q$ per sweep from $k=1$ onward. The first <norm> drop is different because $\|e^{(0)}\|_2=\sqrt{14}$ includes the zero mode. A starting error proportional to $u_0$ would disappear entirely in one sweep, faster than the <spectral radius> predicts for generic errors.
\Image[/jacobi-tridiagonal-error-modes.png]
{title=Computed <Jacobi method> error curves and <eigenvector> modes for the original three-variable <tridiagonal matrix>}
The curves are computed iterates and exact formulas for this <linear system>. Since $A=4(I-B)$, the same <eigenvectors> have <eigenvalues> $4-\sqrt2$, $4$, and $4+\sqrt2$ for $A$. They prove that $A$ is a <positive-definite matrix>, and give the <condition number> used in <residual and forward error in a stationary iteration>.
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