PENYELESAIAN MENGGUNAKAN MATLABPenyelesaian dengan Jacobi :Soal;

function iterasi_jacobi(A, b, N) %Jacobi(A, b, N) solve iteratively a system of linear equations whereby%A is the coefficient matrix, and b is the right-hand side column vector.%N is the maximum number of iterations.%The method implemented is the Jacobi iterative. %The starting vector is the null vector, but can be adjusted to one's needs.%The iterative form is based on the Jacobi transition/iteration matrix%Tj = inv(D)*(L+U) and the constant vector cj = inv(D)*b.%The output is the solution vector x. %This file follows the algorithmic guidelines given in the book %Numerical Analysis, 7th Ed, by Burden & Faires %Author: Alain G. Kapitho%Date : Dec. 2005%Rev. : Aug. 2007 n = size(A,1);%splitting matrix A into the three matrices L, U and DD = diag(diag(A));L = tril(-A,-1);U = triu(-A,1); %transition matrix and constant vector used for iterationsTj = inv(D)*(L+U);cj = inv(D)*b; tol = 15e-05;k = 1;x = zeros(n,1); %starting vector while k tol || k > N disp('Maximum number of iterations reached without satisfying condition:') disp('||x^(k+1) - x^(k)|| < tol'); disp(tol); disp('Please, examine the sequence of iterates') disp('In case you observe convergence, then increase the maximum number of iterations') disp('In case of divergence, the matrix may not be diagonally dominant') disp(x');end

(disave m-file dengan nama iterasi_jacobi(A, b, N))Sumber:http://m2matlabdb.ma.tum.de/download.jsp?MC_ID=3&MP_ID=407Keterangan:function jacobi(A, b, N)a. A adalah matriks koefisienb. b adalah matriks vector sebelah kanan tanda =c. N adalah jumlah iterasid. Batas error yang digunakan ,tol=0,00015Lalu jalankan program : Pertama kali menginput matriks koefisien (A):>> A=[1 -1 -2 ; 2 -2 1 ; 3 -2 7];>> A

A =

1 -1 -2 2 -2 1 3 -2 7

Kemudian menginput matriks koefisien (b):>> b=[5;1;20];>> b

b =

5 1 20

Kemudian menginput nilai iterasi maksimal (N):>> N=150>> N

N =

150 Memanggil fungsi eksternal untuk menghitung iterasi Jacobi dengan inputan seperti diatas dan menghasilkan data sebagai berikut sebagai berikut:>> iterasi_jacobi(A, b, N)The procedure was successfulCondition ||x^(k+1) - x^(k)|| < tol was met after k iterations 80

x = 29.8000 28.4000 -1.8000

Dan dapat disimpulan bahwa itrasi Jacobi , menghasilkan X1=29.8000 , X2= 28.4000, X3=-1.8000 dengan iterasi ke -80

Dengan menggunakan Gauss Seidel :function gauss_seidel_nizam(A, b, N) %Gauss_seidel(A, b, N) solve iteratively a system of linear equations %whereby A is the coefficient matrix, and b is the right-hand side column vector.%N is the maximum number of iterations.%The method implemented is the Gauss-Seidel iterative. %The starting vector is the null vector, but can be adjusted to one's needs.%The iterative form is based on the Gauss-Seidel transition/iteration matrix%Tg = inv(D-L)*U and the constant vector cg = inv(D-L)*b.%The output is the solution vector x. %This file follows the algorithmic guidelines given in the book %Numerical Analysis, 7th Ed, by Burden & Faires %Author: Alain G. Kapitho%Date : Dec. 2005%Rev. : Aug. 2007 n = size(A,1);%splitting matrix A into the three matrices L, U and DD = diag(diag(A));L = tril(-A,-1);U = triu(-A,1); %transition matrix and constant vector used for iterationsTg = inv(D-L)*U; cg = inv(D-L)*b; tol = 2e-06;k = 1;x = zeros(n,1); %starting vector while k tol || k > N disp('Maximum number of iterations reached without satisfying condition:') disp('||x^(k+1) - x^(k)|| < tol'); disp(tol); disp('Please, examine the sequence of iterates') disp('In case you observe convergence, then increase the maximum number of iterations') disp('In case of divergence, the matrix may not be diagonally dominant') disp(x');end(save m-file dengan nama gauss_seidel_Fandy)Sumber:http://m2matlabdb.ma.tum.de/download.jsp?MC_ID=3&MP_ID=406 A adalah matriks koefisien b adalah matriks vector sebelah kanan tanda = N adalah jumlah iterasi Batas error yang digunakan ,tol=0,000002Eksekusi program: Pertama kali menginput matriks koefisien (A):>> A=[1 -1 -2 ; 2 -2 1 ; 3 -2 7];>> A

A =

1 -1 -2 2 -2 1 3 -2 7 Kemudian menginput matriks koefisien (b):>> b=[5;1;20];>> b

b =

5 1 20 Kemudian menginput nilai iterasi maksimal (N):>> N=150;>> N

N =

150 Terakhir memanggil fungsi eksternal untuk menghitung iterasi Gauss Seidel yang telah disimpan sehingga dihasilkan sebagai berikut:

>> gauss_seidel_nizam(A, b, N)The procedure was successfulCondition ||x^(k+1) - x^(k)|| < tol was met after k iterations 24

x = 29.8000 28.4000 -1.8000Dari hasil diatas bahwa hasil iterasi adalah . Dan hasil di dapatkan pada iterasi ke 24