Jacobi method

In numerical linear algebra, the Jacobi method is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is 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 method is named after Carl Gustav Jacob Jacobi.

DescriptionEdit

Let

 

be a square system of n linear equations, where:

 

Then A can be decomposed into a diagonal component D, a lower triangular part L and an upper triangular part U:

 

The solution is then obtained iteratively via

 

where   is the kth approximation or iteration of   and   is the next or k + 1 iteration of  . The element-based formula is thus:

 

The computation of   requires each element in x(k) except itself. Unlike the Gauss–Seidel method, we can't overwrite   with  , as that value will be needed by the rest of the computation. The minimum amount of storage is two vectors of size n.

AlgorithmEdit

Input: initial guess   to the solution, (diagonal dominant) matrix  , right-hand side vector  , convergence criterion
Output: solution when convergence is reached
Comments: pseudocode based on the element-based formula above

 
while convergence not reached do
    for i := 1 step until n do
         
        for j := 1 step until n do
            if j ≠ i then
                 
            end
        end
         
    end
     
end

ConvergenceEdit

The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1:

 

A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms:

 

The Jacobi method sometimes converges even if these conditions are not satisfied.

Note that the Jacobi method does not converge for every symmetric positive-definite matrix. For example

 

ExamplesEdit

Example 1Edit

A linear system of the form   with initial estimate   is given by

 

We use the equation  , described above, to estimate  . First, we rewrite the equation in a more convenient form  , where   and  . From the known values

 

we determine   as

 

Further,   is found as

 

With   and   calculated, we estimate   as  :

 

The next iteration yields

 

This process is repeated until convergence (i.e., until   is small). The solution after 25 iterations is

 

Example 2Edit

Suppose we are given the following linear system:

 

If we choose (0, 0, 0, 0) as the initial approximation, then the first approximate solution is given by

 

Using the approximations obtained, the iterative procedure is repeated until the desired accuracy has been reached. The following are the approximated solutions after five iterations.

       
0.6 2.27272 -1.1 1.875
1.04727 1.7159 -0.80522 0.88522
0.93263 2.05330 -1.0493 1.13088
1.01519 1.95369 -0.9681 0.97384
0.98899 2.0114 -1.0102 1.02135

The exact solution of the system is (1, 2, −1, 1).

Example 3 using Python and NumPyEdit

The following numerical procedure simply iterates to produce the solution vector.

def jacobi(A, b, x_init, epsilon=1e-10, max_iterations=500):
    D = np.diag(np.diag(A))
    LU = A - D
    x = x_init
    D_inv = np.diag(1 / np.diag(D))
    for i in range(max_iterations):
        x_new = np.dot(D_inv, b - np.dot(LU, x))
        if np.linalg.norm(x_new - x) < epsilon:
            return x_new
        x = x_new
    return x

# problem data
A = np.array([
    [5, 2, 1, 1],
    [2, 6, 2, 1],
    [1, 2, 7, 1],
    [1, 1, 2, 8]
])
b = np.array([29, 31, 26, 19])

# you can choose any starting vector
x_init = np.zeros(len(b))
x = jacobi(A, b, x_init)

print("x:", x)
print("computed b:", np.dot(A, x))
print("real b:", b)

Produces the output:

x: [3.99275362 2.95410628 2.16183575 0.96618357]
computed b: [29. 31. 26. 19.]
real b: [29 31 26 19]

Weighted Jacobi methodEdit

The weighted Jacobi iteration uses a parameter   to compute the iteration as

 

with   being the usual choice.[1] From the relation  , this may also be expressed as

 .

Convergence in the symmetric positive definite caseEdit

In case that the system matrix   is of symmetric positive-definite type one can show convergence.

Let   be the iteration matrix. Then, convergence is guaranteed for

 

where   is the maximal eigenvalue.

The spectral radius can be minimized for a particular choice of   as follows

 

where   is the matrix condition number.

See alsoEdit

ReferencesEdit

  1. ^ Saad, Yousef (2003). Iterative Methods for Sparse Linear Systems (2nd ed.). SIAM. p. 414. ISBN 0898715342.

External linksEdit