Metodo Gauss Seidel 21599
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Transcript of Metodo Gauss Seidel 21599
SISTEMAS LINEALES n ECUACIONES n INCOGNITASMETODO DE GAUSS-SEIDEL
[A]*[X]=[C]
EJEMPLO Resolver1 4 2 3 1 X1 1-5 1 2 -3 -5 X2 22 -2 1 -6 3 * X3 = 33 2 5 2 3 X4 4-1 3 -3 3 7 X5 5
14determinante 4455
PASO 1
diagonal dominante
-5 1 2 -3 -5 X1 21 4 2 3 1 X2 13 2 5 2 3 * X3 = 42 -2 1 -6 3 X4 3-1 3 -3 3 7 X5 5
4200En el sistema de ecuaciones
paso 2
despejamos
[X]=[A]-1*[C]
TIENE UNA UNICA SOLUCION SI det(A)≠0
ordenar la matriz buscando la diagonal dominante
es la de maximo producto de sus terminos en valor absoluto
nnnnnnn
nn
nn
nn
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
...
......
...
...
...
332211
33333232131
22323222121
11313212111
......
/))...((
/))...((
/))...((
33)(
3)(
434)1(
232)1(
1313)1(
3
22)(
2)(
323)1(
1212)1(
2
11)(
1)(
313)(
2121)1(
1
axaxaxaxabx
axaxaxabx
axaxaxabx
inn
iiii
inn
iii
inn
iii
nnnnnnn
nn
nn
nn
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
bxaxaxaxa
...
......
...
...
...
332211
33333232131
22323222121
11313212111
concluyendo :
Usamos siempre el ultimo valor calculado de cada variablerepetimos el calculo hasta que para todo r
-5 1 2 -3 -5 X1 2SOLUCION 1 4 2 3 1 X2 1
3 2 5 2 3 * X3 = 42 -2 1 -6 3 X4 3-1 3 -3 3 7 X5 5
X1 X2 X3 X4 X50 0 0 0 0
-0.4 0.35 0.9 -0.6 1.15-0.76 0.1525 0.745 -0.105 0.90464286
-0.91314286 -0.041625 0.86375 -0.19422619 1.05509439-1.00138367 -0.04963304 0.86531726 -0.14548348 1.02570252-1.00221214 -0.07941862 0.87586661 -0.14876881 1.04427857-1.02377262 -0.08774832 0.88012739 -0.14321569 1.04421451 2
-1.0237838 -0.08775961 0.88013169 -0.1432122 1.0442181-1.02379002 -0.08776372 0.88013352 -0.1432108 1.04421916-1.02379201 -0.08776545 0.88013421 -0.14321024 1.04421967
-1.02379201 1 1-0.08776545 2 20.88013421 3 3
-0.14321024 4 41.04421967 5 5
aij ≠ 0
......
/))...((
/))...((
/))...((
33)(
3)(
434)1(
232)1(
1313)1(
3
22)(
2)(
323)1(
1212)1(
2
11)(
1)(
313)(
2121)1(
1
axaxaxaxabx
axaxaxabx
axaxaxabx
inn
iiii
inn
iii
inn
iii
)()1( kr
kr xx
)0(Ix
)0(Ix
......
/))...((
/))...((
/))...((
33)(
3)(
434)1(
232)1(
1313)1(
3
22)(
2)(
323)1(
1212)1(
2
11)(
1)(
313)(
2121)1(
1
axaxaxaxabx
axaxaxabx
axaxaxabx
inn
iiii
inn
iii
inn
iii
TIENE UNA UNICA SOLUCION SI det(A)≠0
-0.08772346
1 4 2 3 1-5 1 2 -3 -52 -2 1 -6 3 *3 2 5 2 3-1 3 -3 3 7
determinante
-5 1 2 -3 -51 4 2 3 13 2 5 2 3 *2 -2 1 -6 3-1 3 -3 3 7
solucion
-5 1 2 -3 -51 4 2 3 13 2 5 2 3 *2 -2 1 -6 3-1 3 -3 3 7
X1 X2 X3 X4 X50 0 0 0 0
-0.4 0.35 0.9 -0.6 1.15-0.76 0.1525 0.745 -0.105 0.90464286
-0.91314286 -0.041625 0.86375 -0.19422619 1.05509439-1.00138367 -0.04963304 0.86531726 -0.14548348 1.02570252-1.00221214 -0.07941862 0.87586661 -0.14876881 1.04427857-1.02055436 -0.08228775 0.8781881 -0.14425157 1.04194684-1.02057821 -0.08624753 0.87937846 -0.14390706 1.04400299-1.02315688 -0.08697046 0.87984334 -0.14342009 1.04393497-1.02333967 -0.08750543 0.88001303 -0.14330843 1.044163-1.02367382 -0.08764749 0.88008886 -0.1432458 1.0441818-1.02372828 -0.08772346 0.88011559 -0.14322477 1.04420903-1.02377262 -0.08774832 0.88012739 -0.14321569 1.04421451
X1 1X2 2X3 = 3X4 4X5 5
144455
X1 2X2 1X3 = 4X4 3X5 5
4200
X1 2X2 1X3 = 4X4 3X5 5