DerivationLinear systemsintermediate

Derivation: Jacobi iteration matrix

From solving each unknown out of its equation to the matrix form x=−D⁻¹(L+U)x+D⁻¹b.

Solve and arrange in matrices

  1. In each equation ii we solve for xix_i (possible because aii≠0a_{ii}\ne 0):

    xi=1aii(bi−∑j≠iaijxj)x_i=\frac{1}{a_{ii}}\left(b_i-\sum_{j\ne i}a_{ij}x_j\right)

  2. Writing this for all rows and separating the diagonal D from the parts L and U, the matrix form appears:

    x=D−1(b−(L+U)x)=−D−1(L+U)x+D−1bx=D^{-1}\bigl(b-(L+U)x\bigr)=-D^{-1}(L+U)x+D^{-1}b

  3. As an iteration, it is the Jacobi scheme with iteration matrix H_J=−D⁻¹(L+U):

    x(k+1)=−D−1(L+U)x(k)+D−1bx^{(k+1)}=-D^{-1}(L+U)x^{(k)}+D^{-1}b