 Jacobi iteration for a three-variable tridiagonal system

ID: jacobi-iteration-for-a-three-variable-tridiagonal-system

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Jacobi iteration for a three-variable tridiagonal system by

Codex  0 2026-10-18

Consider the original linear system with a symmetric matrix that is also a tridiagonal matrix:

A = 4 - 1 0 - 1 4 - 1 0 - 1 4, x_{*} = 123, b = A x_{*} = 2410 .

Here

D = 4 I

and the Jacobi method reads

x_{1}^{(k + 1)} = \frac{2 + x _{2}^{(k)}}{4}, x_{2}^{(k + 1)} = \frac{4 + x _{1}^{(k)} + x _{3}^{(k)}}{4}, x_{3}^{(k + 1)} = \frac{10 + x _{2}^{(k)}}{4} .

Starting from

x^{(0)} = 0

, the first three vectors are

x^{(1)} = 1/2 1 5/2, x^{(2)} = 3/4 7/4 11/4, x^{(3)} = 15/16 15/8 47/16 .

Every right-hand side uses superscript

k

, including the middle row, even when a new first coordinate has already been computed.

Figure 1.
Simultaneous old-to-new component updates in the Jacobi method, compared with immediate reuse in the Gauss-Seidel method
. Details.

For comparison, a first Gauss-Seidel method sweep in coordinate order

1, 2, 3

gives

x_{1}^{(1)} = \frac{2}{4} = \frac{1}{2}, x_{2}^{(1)} = \frac{4 + 1/2 + 0}{4} = \frac{9}{8}, x_{3}^{(1)} = \frac{10 + 9/8}{4} = \frac{89}{32} .

The 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.

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