The Jacobi method is a method of solving a matrix equation on a matrix that has no zeros along its main diagonal (Bronshtein and
Semendyayev 1997, p. 892). Each diagonal element is solved for, and an approximate
value plugged in. The process is then iterated until it converges. This algorithm
is a stripped-down version of the Jacobi transformation
method of matrix diagonalization.
The Jacobi method is easily derived by examining each of the equations in the linear
system of equations
in isolation. If, in the th
solve for the value of
while assuming the other entries of remain fixed. This gives
which is the Jacobi method.
In this method, the order in which the equations are examined is irrelevant, since the Jacobi method treats them independently. The definition of the Jacobi method
can be expressed with matrices as