The Jacobi Method

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 n equations in the linear system of equations

in isolation. If, in the i- th equation

solve for the value of xi ,while assuming the other entries of x remain fixed. This gives

2

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

3

where the matrices D, -L, and -U represent the diagonal, strictly lower triangular, and strictly upper triangular parts of A , respectively.

Two assumptions made on the Jacobi Method:

1. The system given by

Has a unique solution.

2. The coefficient matrix has no zeros on its main diagonal are nonzeros.

The main idea of Jacobi:

To begin, solve the 1st equation for x1, the 2nd equation for x2 and so on to obtain the rewritten equations:

Then make an initial guess of the solution x(0)={x1(0), x2(0),…..xn(0)}. Substitute these values into the right-hand side of the rewritten equations to obtain the first approximation. This accomplishes one iteration.

In the same way, the second approximation is computed by substituting the first approximation’s values into the right-hand side of the rewritten equations.

By repeated iterations, we form a sequence of approximations.

The Jacobi Method summary:

Algorithm for Jacobi method:

Example: Apply the Jacobi method to solve

Continue iterations until two successive approximations are identical when rounding to three significant digits.

Solution:

To begin, rewrite the system