Jumat, 21 Agustus 2015
Kamis, 20 Agustus 2015
Jumat, 14 Agustus 2015
Modul praktikum analisi numerik dengan menggunakan matlab Penyelesaian Sistem Persamaan Linier
1. Penyelesaian
Sistem Persamaan Linier
A. Jacobi
clc,clear
A=input('masukkan
matriks berordo 3x3 = ')
B=input('masukkan
matriks berordo 3x1 = ')
[a b]=size(B)
if a==1
B=B'
else
B
end
exp=0.00001
x0=input('masukkan
nilai awal pencarian dalam matriks 3x1 = ')
x(1,1)=x0(1);
x(1,2)=x0(2);x(1,3)=x0(3); nr(1)=norm(A*x'-B);
i=1;
while nr(i)>exp
i=i+1
x(i,1)=(B(1)-A(1,2)*x(i-1,2)-A(1,3)*x(i-1,3))/A(1,1)
x(i,2)=(B(2)-A(2,1)*x(i-1,1)-A(2,3)*x(i-1,3))/A(2,2)
x(i,3)=(B(3)-A(3,1)*x(i-1,1)-A(3,2)*x(i-1,2))/A(3,3)
nr(i)=norm(A*x(i,:)'-B)
end
disp('Hasil komputasi
:')
disp('iterasi x
y z Norm Residu')
disp([[1:i]' x
nr'])
disp('dengan residu
B-Ax = '); disp(B-A*x(i,:)')
disp(['norm residu
||B-Ax||=', num2str(nr(i))])
disp('subtitusi solusi
Ax='); disp(A*x(i,:)')
%a=[3 1 -1;4 7 -3;2 -2
5]
%b=[5 20 10]
%nilai_awal=[0 0 0]
2. Interpolasi
A. Lagrange
clc,clear
n=input('banyak data = ');
dcr=input('data yang dicari = ');
disp ==============================================
for i=1:n
data=i;data
x(i)=input('masukkan nilai Xn,
Xn= ');
y(i)=input('masukkan nilai f(Xn),
f(Xn)= ');
disp
==============================================
end
L=0;
for i=1:n
pii=1;
for
j=1:n
if i ~= j
pii=pii*((dcr-x(j))/(x(i)-x(j)));
end
end
L=L+y(i)*pii;
end
Dengan_interpolasi_Lagrange_didaatkan_Y=L;
Dengan_interpolasi_Lagrange_didaatkan_Y
3. Integral
A. Integral
Traesium
clc,clear
f=inline('3','x')
xa=1; xb=4;
banyak_x=6; %harus ganjil kelipatan 3
x=linspace(xa,xb,banyak_x);
h=x(2)-x(1)
L=0;
for i=1:banyak_x-1
L=L+((h/2)*(f(x(i))+f(x(i+1))))
end
B. Integral
Simson 
clc,clear
f=inline('3','x')
xa=0; xb=3;
banyak_x=9 % Harus Ganjil > 1
x=linspace(xa,xb,banyak_x);
h=x(2)-x(1)
L=0;
for i=1:2:banyak_x-2
L=L+((h/3)*(f(x(i))+4*f(x(i+1))+f(x(i+2))))
end
C. Integral
Simson 
clc,clear
f=inline('3','x')
xa=0; xb=3;
banyak_x=900 % Harus Ganjil kelipatan 3
x=linspace(xa,xb,banyak_x);
h=x(2)-x(1)
L=0;
for i=1:3:banyak_x-3
L=L+((3*h/8)*(f(x(i))+3*(f(x(i+1))+f(x(i+2)))+f(x(i+3))))
end
4. Persamaan
Differensial Biasa
A. Metode
Euler
clc, clear
f=inline('2*x^2','x') %dalam interval [0,3] jarak = 0.1
format long
x0=0;xn=3;h=0.1;
x=x0:h:xn;
size=size(x);
size=size(1,2);
x(1)=0;y(1)=1;
for i=1:size-1
y(i+1)=y(i)+ h*f(x(i));
end
plot(x,y)
B. Metode
Runge Kutta Orde-4
clc,clear
f=inline('sin(x*y)+cos(x+y)','x','y') %dalam interval [0,3], h=0.1
format long
x0=0;xn=3;
h=0.01;
x=x0:h:xn;size=size(x);size=size(1,2);
x(1)=0;
y(1)=1;
for i=1:size-1
k1=h*f(x(i),y(i));
k2=h*f(x(i)+h/2,y(i)+k1/2);
k3=h*f(x(i)+h/2,y(i)+k2/2);
k4=h*f(x(i)+h,y(i)+k3);
y(i+1)=y(i)+(k1+2*(k2+k3)+k4)/6
end
plot(x,y)
Modul praktikum analisi numerik dengan menggunakan matlab Penyelesaian persamaan linier/tak linier
DAFTAR
ISI
1.
Penyelesaian persamaan linier/tak linier
A.
Metode Biseksi (Bagi Dua) .................................................. 1
B.
Metode Regula Falsi ............................................................ 1
C.
Metode Newton Rhapson .................................................... 2
D.
Metode Secant ....................................................................... 3
2.
Penyelesaian Sistem Persamaan Linier
A.
Jacobi ....................................................................................... 3
3.
Interpolasi
A. Lagrange
................................................................................. 4
4.
Integral
A. Integral
Trapesium ............................................................... 5
B.
Integral Simson ............................................................... 5
............................................................... 5
C. Integral
Simson ............................................................... 6
............................................................... 6
5.
Persamaan Differensial Biasa
A.
Meode Euler .......................................................................... 6
B.
Metode Runge Kutta Orde IV ............................................ 6
6.
Lampiran (Dasar-dasar Matlab)...............................................
8
MODUL
PRAKTIKUM METODE NUMERIK
1. Penyelesaian
Persamaan Linier/Tak Linier
A. Metode
Biseksi (Bagi Dua)
clc, clear
e1=1/100000000000;
x1=-4;x2=3;xt=(x1+x2)/2;
iterasi=100;
y=inline('x^3','x')
ezplot(y)
axis ([-5 5 -5 5])
grid on
for i=1:iterasi
if
abs(y(x1))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=x1; nilaiX, break
elseif abs(y(x2))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=x2; nilaiX, break
elseif abs(y(xt))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=xt; nilaiX, break
elseif y(x1)*y(x2)<0
xt=(x1+x2)/2;
if y(x1)*y(xt)<0
x1=x1; x2=xt;
elseif y(x2)*y(xt)<0
x1=xt; x2=x2;
end
else
disp 'tidak ada akar dalam rentang ini', break
end
end
B. Metode
Regula Falsi
clc, clear
e1=0.001;
x1=-2;x2=3;
iterasi=100;
y=inline('x^3','x')
ezplot(y)
axis ([-5 5 -5 5]);grid on
for i=1:iterasi
xc=x2-((y(x2)*(x2-x1))/(y(x2)-y(x1)))
if
abs(y(x1))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=x1; nilaiX, break
elseif abs(y(x2))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=x2; nilaiX, break
elseif abs(y(xc))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=xc; nilaiX, break
elseif y(x1)*y(x2)<0
if y(x1)*y(xc)<0
x1=x1; x2=xc;
elseif y(x2)*y(xc)<0
x1=xc; x2=x2;
end
else
disp 'tidak ada akar dalam rentang ini'
break
end
end
C. Metode
Newton Rhapson
clc, clear
e1=0.00000000000000001;
x=3;
iterasi=100
y=inline('x^3','x')
dy=diff('x^3','x');
dy=inline(dy)
ezplot(y)
axis ([-5 5 -5 5])
grid on
for i=1:iterasi
x=x-y(x)/dy(x)
if
abs(y(x))<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=x; nilaiX, break
end
end
D. Metode
Secant
clc, clear
e1=0.001;
x1=3; x2=2;
iterasi=100
y=inline('x^3','x')
ezplot(y)
axis ([-5 5 -5 5]); grid on
for i=1:iterasi
x3=x2-(y(x2)*((x2-x1)/(y(x2)-y(x1))))
if
abs(x3-x2)<e1
Pada_iterasi_ke=i;Pada_iterasi_ke, nilaiX=x3; nilaiX, break
end
x1=x2; x2=x3;
end
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