Definition:Gauss-Seidel Method/Mistake

Source Work

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.)

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.):

Gauss-Seidel method

Mistake

Now, writing $\mathbf x_n$ for the column vector of values of $x_1, x_2, x_3$ after the $n$th iteration, and

$\mathbf L = \begin {pmatrix} a_{1 1} & 0 & 0 \\ a_{2 1} & a_{2 3} & 0 \\ a_{3 1} & a_{3 2} & a_{3 3} \end {pmatrix}, \mathbf U = \begin {pmatrix} 0 & a_{1 2} & a_{1 3} \\ 0 & 0 & a_{2 3} \\ 0 & 0 & 0 \end {pmatrix}$

the iterative relationship is

$\mathbf x_{n + 1} = \mathbf L^{-1} \paren {\mathbf b - \mathbf U \mathbf x_n}$

Correction

The definition of $\mathbf L$ is wrong.

It should be:

$\mathbf L = \begin {pmatrix} a_{1 1} & 0 & 0 \\ a_{2 1} & \color {red} {a_{2 2} } & 0 \\ a_{3 1} & a_{3 2} & a_{3 3} \end {pmatrix}$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gauss-Seidel method
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gauss-Seidel method