Jacobi iteration for a three-variable tridiagonal system ($\rho (B)=\sqrt{2}/4$)

Consider the original linear system with a symmetric matrix that is also a tridiagonal matrix:Here $D=4I$ and the Jacobi method readsStarting from $x^{(0)}=0$, the first three vectors areEvery right-hand side uses superscript $k$, including the middle row, even when a new first coordinate has already been computed.

For comparison, a first Gauss-Seidel method sweep in coordinate order $1,2,3$ givesThe new first coordinate enters the middle update, and the new second coordinate enters the final update. Accidentally overwriting old coordinates during a Jacobi method sweep therefore changes the mathematical method.

For the Jacobi iteration for a three-variable tridiagonal system,An orthonormal eigenbasis iswith eigenvalues $q=\sqrt{2}/4$, $0$, 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 coefficientsFor $k\ge 1$,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 $\Vert e^{(0)}{\Vert }_2=\sqrt{14}$ 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.

The curves are computed iterates and exact formulas for this linear system. Since $A=4(I-B)$, the same eigenvectors have eigenvalues $4-\sqrt{2}$, $4$, and $4+\sqrt{2}$ 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.