Let $x^{(k)}$ and $b$ be vectors and $L,\ D, \ U$ be lower, diagonal, upper triangular matrices.
We have:
$x^{(k)}=(I+D^{-1}L)^{-1}(D^{-1}b-D^{-1}Ux^{(k-1)})$ (1)
How does the following follow from (1):
$x^{(k)}=-(D+L)^{-1}Ux^{(k-1)}+(D+L)^{-1}b$
Let $x^{(k)}$ and $b$ be vectors and $L,\ D, \ U$ be lower, diagonal, upper triangular matrices.
We have:
$x^{(k)}=(I+D^{-1}L)^{-1}(D^{-1}b-D^{-1}Ux^{(k-1)})$ (1)
How does the following follow from (1):
$x^{(k)}=-(D+L)^{-1}Ux^{(k-1)}+(D+L)^{-1}b$
With Cocopuffs input:
$(I+D^{-1}L)^{-1}(D^{-1}b-D^{-1}Ux^{(k-1)})=(I+D^{-1}L)^{-1}D^{-1}(b-Ux^{(k-1)})=(D(I+D^{-1}L))^{-1}(b-Ux^{(k-1)})=-(D+L)^{-1}Ux^{(k-1)}+(D+L)^{-1}b$