Example: 2D Jacobi iteration Stencil code

data dependencies of selected cell in 2d array.

to illustrate formal definition, ll have @ how 2 dimensional jacobi iteration can defined. update function computes arithmetic mean of cell s 4 neighbors. in case set off initial solution of 0. left , right boundary fixed @ 1, while upper , lower boundaries set 0. after sufficient number of iterations, system converges against saddle-shape.

i



=
[
0
,
…
,
99

]

2






s



=

r






s

0





:


z


2


→

r






s

0


(
(
x
,
y
)
)



=


{



1
,


x
<
0




0
,


0
≤
x
<
100




1
,


x
≥
100










s



=
(
(
0
,
−
1
)
,
(
−
1
,
0
)
,
(
1
,
0
)
,
(
0
,
1
)
)




t



:


r


4


→

r





t
(
(

x

1


,

x

2


,

x

3


,

x

4


)
)



=
0.25
⋅
(

x

1


+

x

2


+

x

3


+

x

4


)






{\displaystyle {\begin{aligned}i&=[0,\ldots ,99]^{2}\\s&=\mathbb {r} \\s_{0}&:\mathbb {z} ^{2}\to \mathbb {r} \\s_{0}((x,y))&={\begin{cases}1,&x<0\\0,&0\leq x<100\\1,&x\geq 100\end{cases}}\\s&=((0,-1),(-1,0),(1,0),(0,1))\\t&\colon \mathbb {r} ^{4}\to \mathbb {r} \\t((x_{1},x_{2},x_{3},x_{4}))&=0.25\cdot (x_{1}+x_{2}+x_{3}+x_{4})\end{aligned}}}