師大 Jacobi Method
a11x(k)1 + a12x(k−1)2 + a13x(k−1)3 + · · · + a1nx(k−1)n = b1
a21x(k−1)1 +a22x(k)2 + a23x(k−1)3 + · · · + a2nx(k−1)n = b2 ... an1x(k−1)1 + an2x(k−1)2 + an3x(k−1)3 + · · · +annx(k)n = bn.
師大 Jacobi Method
If we decompose the coefficient matrix A as A = L + D + U,
whereDis thediagonal part,Lis thestrictly lower triangular part, andU is thestrictly upper triangular part, ofA, and chooseM = D, then we derive the iterative formulation for Jacobi method:
x(k)= −D−1(L + U )x(k−1)+ D−1b.
With this method, the iteration matrixT = −D−1(L + U )and c = D−1b. Each component x(k)i can be computed by
x(k)i =
bi−
i−1
X
j=1
aijx(k−1)j −
n
X
j=i+1
aijx(k−1)j
,
aii.
師大 Jacobi Method
Algorithm (Jacobi Method) For k = 1, 2, . . .
For i = 1, 2, . . . , n x(k)i =
bi−
i−1
X
j=1
aijx(k−1)j −
n
X
j=i+1
aijx(k−1)j
,
aii End for
End for
Only the components of x(k−1) are used to compute x(k).
⇒ x(k)i , i = 1, . . . , n,can be computed in parallel at each iteration k.
