Example: (Programming)\

Avec(u)=vec(b)

\

[[4,-1,0,-1,0,0,0,0,0,4,-1,0,-1,0,0,0,0\ 0,-1,4,0,0,-1,0,0,0\ -1,0,0,4,-1,0,-1,0,0\ 0,-1,0,-1,4,-1,0,-1,0\ 0,0,-1,0,-1,4,0,0,-1\ 0,0,0,-1,0,0,4,-1,0\ 0,0,0,0,-1,0,-1,4,-1\ 0,0,0,0,0,-1,0,-1,4]]\ 0\ 0\ 0\ 0\ 0\ -1\ 0\ -1[[(u_(1,1))]]\ u_(2,1)\ u_(3,1)\ u_(1,2)\ u_(2,2)\ u_(3,2)\ u_(1,3)\ u_(2,3)\ u_(3,3)=[[0]]\ 0\ 25\ 0\ 0\ 50\ 25\ 50\ 50\ u^(0)=(1,1,1,dots,1)^(T)

\

Avec(u)=vec(b)

\

[[83,11,-4],[7,52,13],[3,8,29]][[u_(1)],[u_(2)],[u_(3)]]=[[95],[104],[71]]

\

u^(0)=(0,0,0)^(T)

\ Additional notes\ a) Jacobi:\

x_(i)^((r+1))=(b_(i))/(a_(ii))-\\\\sum_({(:[j

]

=

[

1]),(j!=i):})^n (a_(ij))/(a_(ii))x_(j)^((r))

\ b) Gauss-Seidel:\

x_(i)^((r+1))=(b_(i))/(a_(ii))-\\\\sum_(j=1)^(i-1) (a_(ij))/(a_(ii))x_(j)^((r+1))-\\\\sum_(j=i+1)^n (a_(ij))/(a_(ii))x_(j)^((r))

\ c) S.O.R:\

x_(i)^((r+1))=\\\\omega (GS)+(1-\\\\omega )x_(i)^((r))

Example: (Programming) 1) Au=b 410100000141010000014001000100410100010141010001014001000100410000010141000001014u1,1u2,1u3,1u1,2u2,2u3,2u1,3u2,3u3,3=00250050255050 u0=(1,1,1,,1)T 2) Au=b 83731152841329u1u2u3=9510471 u0=(0,0,0)T Additional notes a) Jacobi: xi(r+1)=aiibij=1j=inaiiaijxj(r) b) Gauss-Seidel: xi(r+1)=aiibij=1i1aiiaijxj(r+1)j=i+1naiiaijxj(r) c) S.O.R: xi(r+1)=(GS)+(1)xi(r) Example: (Programming) 1) Au=b 410100000141010000014001000100410100010141010001014001000100410000010141000001014u1,1u2,1u3,1u1,2u2,2u3,2u1,3u2,3u3,3=00250050255050 u0=(1,1,1,,1)T 2) Au=b 83731152841329u1u2u3=9510471 u0=(0,0,0)T Additional notes a) Jacobi: xi(r+1)=aiibij=1j=inaiiaijxj(r) b) Gauss-Seidel: xi(r+1)=aiibij=1i1aiiaijxj(r+1)j=i+1naiiaijxj(r) c) S.O.R: xi(r+1)=(GS)+(1)xi(r)