In
numerical linear algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...
, the Jacobi method is an iterative algorithm for determining the solutions of a
strictly diagonally dominant system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in t ...
. 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
:
be a square system of ''n'' linear equations, where:
Then ''A'' can be decomposed into a
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ� ...
component ''D'', a lower triangular part ''L'' and an upper triangular part ''U'':
:
The solution is then obtained iteratively via
:
where
is the ''k''th 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''.
Algorithm
Input: , (diagonal dominant) matrix ''A'', right-hand side vector ''b'', convergence criterion
Output:
Comments:
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:
:
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
:
Examples
Example 1
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 2
Suppose we are given the following linear system:
:
If we choose 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.
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 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. I ...
1., 11., -1., 3.
., -1., 10., -1.
.0, 3., -1., 8.)
# initialize the RHS vector
b = np.array( ., 25., -11., 15.
# prints the system
print("System:")
for i in range(A.shape :
row = *x".format(A[i,_j_j_+_1)_for_j_in_range(A.shape[1.html" ;"title=",_j.html" ;"title="*x".format(A[i, j">*x".format(A[i, j j + 1) for j in range(A.shape[1">,_j.html" ;"title="*x".format(A[i, j">*x".format(A[i, j j + 1) 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("Iteration : ".format(it_count, x))
x_new = np.zeros_like(x)
for i in range(A.shape :
s1 = np.dot(A , :i x i
s2 = np.dot(A , i + 1: x + 1:
x_new = (b - s1 - s2) / A, i
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
if x_new x_new -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) - b
print("Error:")
print(error)
Weighted Jacobi method
The weighted Jacobi iteration uses a parameter
to compute the iteration as
:
with
being the usual choice.
From the relation
, this may also be expressed as
:
.
Convergence in the symmetric positive definite case
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 also
*
Gauss–Seidel method
*
Successive over-relaxation In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging ...
*
Iterative method § Linear systems
*
Gaussian Belief Propagation
*
Matrix splitting
References
External links
*
*
Jacobi Method from www.math-linux.com
{{Authority control
Numerical linear algebra
Articles with example pseudocode
Relaxation (iterative methods)
Articles with example Python (programming language) code