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Concept
Jacobi method
Summary
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.
Let be a square system of n linear equations, where:
When and are known, and is unknown, we can use the Jacobi method to approximate . The vector denotes our initial guess for (often for ). We denote as the k-th approximation or iteration of , and is the next (or k+1) iteration of .
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
The element-based formula for each row is thus:The computation of requires each element in 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.
Input: initial guess x(0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion
Output: solution when convergence is reached
Comments: pseudocode based on the element-based formula above
k = 0
while convergence not reached do
for i := 1 step until n do
σ = 0
for j := 1 step until n do
if j ≠ i then
σ = σ + aij xj(k)
end
end
xi(k+1) = (bi − σ) / aii
end
increment k
end
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.
Official source
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