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Penyelesaian persamaan linier simultan Matriks 3x3 dengan menggunakan Metode Jacoby Persamaan : 4x1 - 2x2 + 2x = 2 4x1 - 5x2 + 2x = -4 -2x1 + 4x2 + 4 = 10 Solusi : A=
x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 =
4 4 -2
-2 -5 4
2 2 4
2 -4 10
4 4 -2
x1 x1 x1
2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2
4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4
x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1
-2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2
B=
2 -4 10
x2 x2 x2
2 2 4
x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2
2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2
x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3
= = = = = = = = = = = = = = = = = = = = = = = = =
0.000 0.000 0.000 NILAI ERROR 0.500 0.800 2.500 NILAI ERROR -0.350 2.200 1.950 NILAI ERROR 0.625 1.300 0.125 NILAI ERROR 1.088 1.350 1.513 NILAI ERROR 0.419 2.275 1.694 NILAI ERROR 0.791 1.813 0.434 NILAI ERROR 1.189 1.606 1.083 NILAI ERROR 0.762
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x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 =
-4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10 2
4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4 4 -2 4
x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1 x1
-5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2
x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2
2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2 2 4 2
x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
2.184 1.488 NILAI ERROR 0.848 2.005 0.696 NILAI ERROR 1.154 1.757 0.919 NILAI ERROR 0.919 2.091 1.320 NILAI ERROR 0.885 2.063 0.868 NILAI ERROR 1.097 1.856 0.880 NILAI ERROR 0.988 2.030 1.193 NILAI ERROR 0.918 2.068 0.964 NILAI ERROR 1.052 1.920 0.892 NILAI ERROR 1.014 1.998 1.105 NILAI ERROR 0.946 2.054 1.009 NILAI ERROR 1.022 1.961 0.919 NILAI ERROR 1.021
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x2 = x3 =
-4 10
4 -2
x1 x1
-5 4
x2 x2
2 4
x3 x3
= =
x1 = x2 = x3 =
2 -4 10
4 4 -2
x1 x1 x1
-2 -5 4
x2 x2 x2
2 2 4
x3 x3 x3
= = =
x1 = x2 = x3 =
2 -4 10
4 4 -2
x1 x1 x1
-2 -5 4
x2 x2 x2
2 2 4
x3 x3 x3
= = =
Verifikasi menggunakan excel : A-1 = 0.47 -0.27 -0.10 0.33 -0.33 0.00 -0.10 0.20 0.20 x = A-1.b
1 2 1
1.986 1.050 NILAI ERROR RELATIF 0.968 2.037 1.025 NILAI ERROR RELATIF 1.006 1.984 0.947
an Metode Jacoby
ERROR RELATIF
2.67
ERROR RELATIF
1.73
ERROR RELATIF
2.26
ERROR RELATIF
1.46
ERROR RELATIF
1.16
ERROR RELATIF
1.39
ERROR RELATIF
0.79
ERROR RELATIF
0.83
ERROR RELATIF
0.82
ERROR RELATIF
0.45
ERROR RELATIF
0.57
ERROR RELATIF
0.45
ERROR RELATIF
0.30
ERROR RELATIF
0.37
ERROR RELATIF
0.24
ERROR RELATIF
0.21
ERROR RELATIF
0.23
ERROR RELATIF
0.13
ERROR RELATIF
0.15
ERROR RELATIF
0.13
ERROR RELATIF
0.08
ERROR RELATIF
0.10
2.67
1.73
2.26
1.46
Penyelesaian persamaan linier simultan Matriks 3x3 dengan menggunakan Metode Gauss Said Persamaan : 4x1 - 2x2 + 2x3 = 2 4x1 - 5x2 + 2x3 = -4 -2x1 + 4x2 + 4 = 10 Solusi : A=
x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 = x1 = x2 = x3 =
4 4 -2
-2 -5 4
2 2 4
2 -4 10
4 4 -2
x1 x1 x1
2 -4 10 2 -4 10 2 -4 10 2 -4 10 2 -4 10
4 4 -2
x1 x1 x1
4 4 -2
x1 x1 x1
4 4 -2
x1 x1 x1
4 4 -2
x1 x1 x1
4 4 -2
x1 x1 x1
-2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4 -2 -5 4
Verifikasi menggunakan excel : 0.47 -0.27 -0.10 A-1 = 0.33 -0.33 0.00 -0.10 0.20 0.20 x = A-1
1 2 1
B=
2 -4 10
x2 x2 x2
2 2 4
x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2 x2
2 2 4 2 2 4 2 2 4 2 2 4 2 2 4
x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3 x3
= = = = = = = = = = = = = = = = = =
0.000 0.000 0.000 NILAI ERROR 0.500 1.200 1.550 NILAI ERROR 0.325 1.680 0.983 NILAI ERROR 0.849 1.872 1.052 NILAI ERROR 0.910 1.949 1.006 NILAI ERROR 0.971 1.980 1.006
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nggunakan Metode Gauss Saidel
ILAI ERROR RELATIF
2.02
ILAI ERROR RELATIF
0.76
ILAI ERROR RELATIF
0.56
ILAI ERROR RELATIF
0.11
ILAI ERROR RELATIF
0.07
Penyelesaian persamaan linier simultan Matriks 3x3 dengan menggunakan metode eliminasi gauss Persamaan : 4x1 - 2x2 + 2x3 4x1 - 5x2 + 2x3 -2x1 + 4x2 + 4x3
=2 = -4 = 10
A=
4 4 -2
-2 -5 4
2 2 4
A=
a11 a21 a31
a12 a22 a32
a13 a23 a33
b1 b2 b3
4 4 -2
-2 -5 4
2 2 4
2 -4 10
B=
Faktor Pengali 1 -0.5
Eliminasi 4 0 0
-2 3 -3
2 0 -5
2 -4 10
2 6 -11
Faktor Pengali -1.000
OKE
4 0 0
-2 3 0
2 0 5
Subtitusi
X3 X2 X1
Verifikasi menggunakan excel : A-1 = 0.47 -0.27 0.33 -0.33 -0.10 0.20 x = A-1.b
1 2 1
2 6 5 1 2 1
-0.10 0.00 0.20
Penyelesaian persamaan linier simultan Matriks 3x3 dengan menggunakan metode gauss-jordan
Persamaan : 4x1 - 2x2 + 2x3 4x1 - 5x2 + 2x3 -2x1 + 4x2 + 4x3
=2 = -4 = 10
A=
4 4 -2
-2 -5 4
2 2 4
A=
a11 a21 a31
a12 a22 a32
a13 a23 a33
4 4 -2
-2 -5 4
B=
b1 b2 b3 2 2 4
2 -4 10 Faktor Pengali 1 -0.5
Eliminasi 4 0 0
-2 3 -3
2 -4 10
2 0 -5
2 6 -11
Faktor Pengali -1.000 4 0 0
-2 3 0
Eliminasi segitiga atas -4 0 0 OKE
2 0 5
2 -3 0
2 6 5
0 0 5
0 -6 5
4 0 0
0 -3 0
0 0 5
4 -6 5
1 0 0
0 1 0
0 0 1
1 2 1
X1 X2 X3
1 2 1
Faktor Pengali 0.400 0
Faktor Pengali -0.667
Verifikasi menggunakan excel : 0.47 -0.27 A-1 = 0.33 -0.33 -0.10 0.20 x = A-1.b
1 2 1
-0.10 0.00 0.20
auss-jordan
