Jacobi method explained

In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration 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.

Description

Let

Ax=b

be a square system of n linear equations, where:

A = \begin a_ & a_ & \cdots & a_ \\ a_ & a_ & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\a_ & a_ & \cdots & a_ \end, \qquad \mathbf = \begin x_ \\ x_2 \\ \vdots \\ x_n \end, \qquad \mathbf = \begin b_ \\ b_2 \\ \vdots \\ b_n \end.

When

and

are known, and

is unknown, we can use the Jacobi method to approximate

. The vector

x⁽⁰⁾

denotes our initial guess for

(often

	(0)
x
	i=0

for

i=1,2,...,n

). We denote

x^(k)

as the k-th approximation or iteration of

, and

x^(k+1)

is the next (or k+1) iteration of

Matrix-based formula

Then A can be decomposed into a diagonal component D, a lower triangular part L and an upper triangular part U: $A=D+L+U \qquad \text \qquad D = \begin a_ & 0 & \cdots & 0 \\ 0 & a_ & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & a_ \end \text L+U = \begin 0 & a_ & \cdots & a_ \\ a_ & 0 & \cdots & a_ \\ \vdots & \vdots & \ddots & \vdots \\ a_ & a_ & \cdots & 0 \end.$ The solution is then obtained iteratively via

x^(k+1)=D^-1(b-(L+U)x^(k)).

Element-based formula

The element-based formula for each row

is thus:

x^_i = \frac \left(b_i -\sum_a_x^_j\right),\quad i=1,2,\ldots,n.

The computation of

	(k+1)
x
	i

requires each element in

x^(k)

except itself. Unlike the Gauss–Seidel method, we can't overwrite

	(k)
x
	i

with

	(k+1)
x
	i

, as that value will be needed by the rest of the computation. The minimum amount of storage is two vectors of size n.

Algorithm

Input:, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion Output: 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 increment k end

Convergence

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

\rho(D^-1(L+U))<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:

\left|a_ii\right|>\sum_j{\left|a_ij\right|}.

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, $A =\begin 29 & 2 & 1\\ 2 & 6 & 1\\ 1 & 1 & \frac\end\quad \Rightarrow \quadD^ (L+U) =\begin 0 & \frac & \frac\\ \frac & 0 & \frac\\ 5 & 5 & 0\end\quad \Rightarrow \quad\rho(D^(L+U)) \approx 1.0661 \,.$

Examples

Example question

A linear system of the form

Ax=b

with initial estimate

x⁽⁰⁾

is given by

A= \begin{bmatrix} 2&1\\ 5&7\\ \end{bmatrix}, b= \begin{bmatrix} 11\\ 13\\ \end{bmatrix} and x⁽⁰⁾= \begin{bmatrix} 1\\ 1\\ \end{bmatrix}.

We use the equation

x^(k+1)=D^-1(b-(L+U)x^(k))

, described above, to estimate

. First, we rewrite the equation in a more convenient form

D^-1(b-(L+U)x^(k))=Tx^(k)+C

, where

T=-D^-1(L+U)

and

C=D^-1b

. From the known values

D^= \begin 1/2 & 0 \\ 0 & 1/7 \\ \end, \ L= \begin 0 & 0 \\ 5 & 0 \\ \end\quad \text \quad U = \begin 0 & 1 \\ 0 & 0 \\ \end .

we determine

T=-D^-1(L+U)

T= \begin 1/2 & 0 \\ 0 & 1/7 \\ \end\left\ = \begin 0 & -1/2 \\ -5/7 & 0 \\ \end .

Further,

is found as

C = \begin 1/2 & 0 \\ 0 & 1/7 \\ \end \begin 11 \\ 13 \\ \end = \begin 11/2 \\ 13/7 \\ \end.

With

and

calculated, we estimate

x⁽¹⁾=Tx⁽⁰⁾+C

x^= \begin 0 & -1/2 \\ -5/7 & 0 \\ \end \begin 1 \\ 1 \\ \end + \begin 11/2 \\ 13/7 \\ \end = \begin 5.0 \\ 8/7 \\ \end \approx \begin 5 \\ 1.143 \\ \end .

The next iteration yields

x^= \begin 0 & -1/2 \\ -5/7 & 0 \\ \end\begin 5.0 \\ 8/7 \\ \end + \begin 11/2 \\ 13/7 \\ \end = \begin 69/14 \\ -12/7 \\ \end \approx \begin 4.929 \\ -1.714 \\ \end .

This process is repeated until convergence (i.e., until

\|Ax⁽ⁿ⁾-b\|

is small). The solution after 25 iterations is

x=\begin{bmatrix} 7.111\\ -3.222 \end{bmatrix} .

Example question 2

Suppose we are given the following linear system:

\begin{align} 10x₁-x₂+2x₃&=6,\\ -x₁+11x₂-x₃+3x₄&=25,\\ 2x_1-x₂₊10x₃-x₄&=-11,\\ 3x₂-x₃+8x₄&=15. \end{align}

If we choose as the initial approximation, then the first approximate solution is given by $\beginx_1 & = (6 + 0 - (2 * 0)) / 10 = 0.6, \\x_2 & = (25 + 0 + 0 - (3 * 0)) / 11 = 25/11 = 2.2727, \\x_3 & = (-11 - (2 * 0) + 0 + 0) / 10 = -1.1,\\ x_4 & = (15 - (3 * 0) + 0) / 8 = 1.875.\end$ 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.

x₁	x₂	x₃	x₄
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 .

Python example

import numpy as np

ITERATION_LIMIT = 1000

initialize the matrix

A = np.array(10., -1., 2., 0., [-1., 11., -1., 3.], [2., -1., 10., -1.], [0.0, 3., -1., 8.]])

initialize the RHS vector

b = np.array([6., 25., -11., 15.])

prints the system

print("System:")for i in range(A.shape[0]): row = [f"{A[i, j]}*x" for j in range(A.shape[1])] print(f' = ')print

x = np.zeros_like(b)for it_count in range(ITERATION_LIMIT): if it_count != 0: print(f"Iteration : ") x_new = np.zeros_like(x)

for i in range(A.shape[0]): s1 = np.dot(A[i, :i], x[:i]) s2 = np.dot(A[i, i + 1:], x[i + 1:]) x_new[i] = (b[i] - s1 - s2) / A[i, i] if x_new[i]

x_new[i-1]: break

if np.allclose(x, x_new, atol=1e-10, rtol=0.): break

x = x_new

print("Solution: ")print(x)error = np.dot(A, x) - bprint("Error:")print(error)

Weighted Jacobi method

The weighted Jacobi iteration uses a parameter

\omega

to compute the iteration as

x^(k+1)=\omegaD^-1(b-(L+U)x^(k))+\left(1-\omega\right)x^(k)

with

\omega=2/3

being the usual choice.^[1] From the relation

L+U=A-D

, this may also be expressed as

x^(k+1)=\omegaD^-1b+\left(I-\omegaD^-1A\right)x^(k)

Convergence in the symmetric positive definite case

In case that the system matrix

is of symmetric positive-definite type one can show convergence.

Let

C=C_\omega=I-\omegaD^-1A

be the iteration matrix.Then, convergence is guaranteed for

\rho(C_\omega)<1 \Longleftrightarrow 0<\omega<

	2
	λ_max(D^-1A)

where

λ_max

is the maximal eigenvalue.

The spectral radius can be minimized for a particular choice of

\omega=\omega_opt

as follows

\min_\omega \rho (C_\omega) = \rho (C_) = 1-\frac \quad \text \quad \omega_\text := \frac \,,