The document describes 15 iterations of the Gauss-Seidel method to solve a system of equations. In each iteration, the method calculates updated values for variables w, x, y, and z. The errors between the current and previous values of each variable decrease with each iteration, with errors below 0.001 after 10 iterations, indicating convergence to the solution.

1. 1.- Lee el documentoMETODO GAUSS SEIDEL.pdf y determinalasoluciónde lossiguientes
sistemasde ecuaciones
Iteracion1
w = (8.70- (1^2) - 1 - 1)/5 = 1.14
x = (7.3-(1.14^ 2)+(2*1)+1)/-6 = -1.5001
y = (17.29- (1.14) + (-1.5001)-(1 ^ 2))/4 =3.4125
z = (34.7 - (2*1.14) - (-1.5001)-(3.4125^2))/11 = 2.025
errorx1= |xn - xn-1|= |1.14 - (1) |= 0.14
errorx2= |xn - xn-1|= |-1.5001 - (1) |= 2.5001
errorx3= |xn - xn-1|= |3.4125 - (1) |= 2.4125
errorx4= |xn - xn-1|= |2.025 - (1) |= 1.025
Iteración2
w = (8.70- (-1.5001^2) - 3.4125 - 2.025)/5 = 0.2024
x = (7.3-(0.2024^ 2)+(2*3.4125)+2.025)/-6 = -2.6848
y = (17.29- (0.2024) + (-2.6848)-(2.025 ^ 2))/4=2.5755
z = (34.7 - (2*0.2024) - (-2.6848)-(2.5755^2))/11 = 2.7588
errorx1= |xn - xn-1|= |0.2024 - (1.14) |= 0.9376
errorx2= |xn - xn-1|= |-2.6848 - (-1.5001) |= 1.1847
errorx3= |xn - xn-1|= |2.5755 - (3.4125) |= 0.837
errorx4= |xn - xn-1|= |2.7588 - (2.025) |= 0.7338

2. Iteración3
w = (8.70- (-2.6848^2) - 2.5755 - 2.7588)/5 = -0.7685
x = (7.3-(-0.7685^ 2)+(2*2.5755)+2.7588)/-6 = -2.4365
y = (17.29- (-0.7685) + (-2.4365)-(2.7588 ^ 2))/4=2.0028
z = (34.7 - (2*-0.7685) - (-2.4365)-(2.0028^2))/11 = 3.1511
errorx1= |xn - xn-1|= |-0.7685 - (0.2024) |= 0.9709
errorx2= |xn - xn-1|= |-2.4365 - (-2.6848) |= 0.2483
errorx3= |xn - xn-1|= |2.0028 - (2.5755) |= 0.5727
errorx4= |xn - xn-1|= |3.1511 - (2.7588) |= 0.3923
Iteración4
w = (8.70- (-2.4365^2) - 2.0028 - 3.1511)/5 = -0.4781
x = (7.3-(-0.4781^ 2)+(2*2.0028)+3.1511)/-6 = -2.3714
y = (17.29- (-0.4781) + (-2.3714)-(3.1511 ^ 2))/4=1.3668
z = (34.7 - (2*-0.4781) - (-2.3714)-(1.3668^2))/11 = 3.2872
errorx1= |xn - xn-1|= |-0.4781 - (-0.7685) |= 0.2904
errorx2= |xn - xn-1|= |-2.3714 - (-2.4365) |= 0.0651
errorx3= |xn - xn-1|= |1.3668 - (2.0028) |= 0.636
errorx4= |xn- xn-1|= |3.2872 - (3.1511) |= 0.1361
Iteración5
w = (8.70- (-2.3714^2) - 1.3668 - 3.2872)/5 = -0.3155
x = (7.3-(-0.3155^ 2)+(2*1.3668)+3.2872)/-6 = -2.2035
y = (17.29- (-0.3155) + (-2.2035)-(3.2872 ^ 2))/4=1.1491
z = (34.7 - (2*-0.3155) - (-2.2035)-(1.1491^2))/11 = 3.2922
errorx1= |xn - xn-1|= |-0.3155 - (-0.4781) |= 0.1626
errorx2= |xn - xn-1|= |-2.2035 - (-2.3714) |= 0.1679
errorx3= |xn - xn-1|= |1.1491 - (1.3668) |= 0.2177
errorx4= |xn - xn-1|= |3.2922 - (3.2872) |= 0.005

3. Iteración6
w = (8.70- (-2.2035^2) - 1.1491 - 3.2922)/5 = -0.1193
x = (7.3-(-0.1193^ 2)+(2*1.1491)+3.2922)/-6 = -2.146
y = (17.29- (-0.1193) + (-2.146)-(3.2922 ^ 2))/4 =1.1062
z = (34.7 - (2*-0.1193) - (-2.146)-(1.1062^2))/11 = 3.2601
errorx1= |xn - xn-1|= |-0.1193 - (-0.3155) |= 0.1962
errorx2= |xn - xn-1|= |-2.146 - (-2.2035) |= 0.0575
errorx3= |xn - xn-1|= |1.1062 - (1.1491) |= 0.0429
errorx4= |xn - xn-1|= |3.2601 - (3.2922) |= 0.0321
Iteración7
w = (8.70- (-2.146^2) - 1.1062 - 3.2601)/5 = -0.0543
x = (7.3-(-0.0543^ 2)+(2*1.1062)+3.2601)/-6 = -2.1283
y = (17.29- (-0.0543) + (-2.1283)-(3.2601 ^ 2))/4=1.1469
z = (34.7 - (2*-0.0543) - (-2.1283)-(1.1469^2))/11 = 3.2383
errorx1= |xn - xn-1|= |-0.0543 - (-0.1193) |= 0.065
errorx2= |xn - xn-1|= |-2.1283 - (-2.146) |= 0.0177
errorx3= |xn - xn-1|= |1.1469 - (1.1062) |= 0.0407
errorx4= |xn - xn-1|= |3.2383 - (3.2601) |= 0.0218
Iteración8
w = (8.70- (-2.1283^2) - 1.1469 - 3.2383)/5 = -0.043
x = (7.3-(-0.043^ 2)+(2*1.1469)+3.2383)/-6 = -2.1384
y = (17.29- (-0.043) + (-2.1384)-(3.2383 ^ 2))/4 =1.177
z = (34.7 - (2*-0.043) - (-2.1384)-(1.177^2))/11 = 3.2308
errorx1= |xn - xn-1|= |-0.043 - (-0.0543) |= 0.0113
errorx2= |xn - xn-1|= |-2.1384 - (-2.1283) |= 0.0101

4. errorx3= |xn - xn-1|= |1.177 - (1.1469) |= 0.0301
errorx4= |xn- xn-1|= |3.2308 - (3.2383) |= 0.0075
Iteración9
w = (8.70- (-2.1384^2) - 1.177 - 3.2308)/5 = -0.0561
x = (7.3-(-0.0561^ 2)+(2*1.177)+3.2308)/-6 = -2.1469
y = (17.29- (-0.0561) + (-2.1469)-(3.2308 ^ 2))/4=1.1903
z = (34.7 - (2*-0.0561) - (-2.1469)-(1.1903^2))/11 = 3.2311
errorx1= |xn - xn-1|= |-0.0561 - (-0.043) |= 0.0131
errorx2= |xn - xn-1|= |-2.1469 - (-2.1384) |= 0.0085
errorx3= |xn - xn-1|= |1.1903 - (1.177) |= 0.0133
errorx4= |xn - xn-1|= |3.2311 - (3.2308) |= 0.0003
Iteración10
w = (8.70- (-2.1469^2) - 1.1903 - 3.2311)/5 = -0.0661
x = (7.3-(-0.0661^ 2)+(2*1.1903)+3.2311)/-6 = -2.1512
y = (17.29- (-0.0661) + (-2.1512)-(3.2311 ^ 2))/4=1.1912
z = (34.7 - (2*-0.0661) - (-2.1512)-(1.1912^2))/11 = 3.2331
errorx1= |xn - xn-1|= |-0.0661 - (-0.0561) |= 0.01
errorx2= |xn - xn-1|= |-2.1512 - (-2.1469) |= 0.0043
errorx3= |xn - xn-1|= |1.1912 - (1.1903) |= 0.0009
errorx4= |xn - xn-1|= |3.2331 - (3.2311) |= 0.002
Iteración11
w = (8.70- (-2.1512^2) - 1.1912 - 3.2331)/5 = -0.0704
x = (7.3-(-0.0704^ 2)+(2*1.1912)+3.2331)/-6 = -2.1518
y = (17.29- (-0.0704) + (-2.1518)-(3.2331 ^ 2))/4=1.1889
z = (34.7 - (2*-0.0704) - (-2.1518)-(1.1889^2))/11 = 3.2345

5. errorx1= |xn - xn-1|= |-0.0704 - (-0.0661) |= 0.0043
errorx2= |xn - xn-1|= |-2.1518 - (-2.1512) |= 0.0006
errorx3= |xn - xn-1|= |1.1889 - (1.1912) |= 0.0023
errorx4= |xn - xn-1|= |3.2345 - (3.2331) |= 0.0014
Iteración12
w = (8.70- (-2.1518^2) - 1.1889 - 3.2345)/5 = -0.0707
x = (7.3-(-0.0707^ 2)+(2*1.1889)+3.2345)/-6 = -2.1512
y = (17.29- (-0.0707) + (-2.1512)-(3.2345 ^ 2))/4=1.1869
z = (34.7 - (2*-0.0707) - (-2.1512)-(1.1869^2))/11 = 3.2349
errorx1= |xn - xn-1|= |-0.0707 - (-0.0704) |= 0.0003
errorx2= |xn - xn-1|= |-2.1512 - (-2.1518) |=0.0006
errorx3= |xn - xn-1|= |1.1869 - (1.1889) |= 0.002
errorx4= |xn - xn-1|= |3.2349 - (3.2345) |=0.0004
Iteración13
w = (8.70- (-2.1512^2) - 1.1869 - 3.2349)/5 = -0.0699
x = (7.3-(-0.0699^ 2)+(2*1.1869)+3.2349)/-6 = -2.1506
y = (17.29- (-0.0699) + (-2.1506)-(3.2349 ^ 2))/4=1.1862
z = (34.7 - (2*-0.0699) - (-2.1506)-(1.1862^2))/11 = 3.2348
errorx1= |xn - xn-1|= |-0.0699 - (-0.0707) |= 0.0008
errorx2= |xn - xn-1|= |-2.1506 - (-2.1512) |= 0.0006
errorx3= |xn - xn-1|= |1.1862 - (1.1869) |= 0.0007
errorx4= |xn - xn-1|= |3.2348 - (3.2349) |= 0.0001
Iteración14
w = (8.70- (-2.1506^2) - 1.1862 - 3.2348)/5 = -0.0692
x = (7.3-(-0.0692^ 2)+(2*1.1862)+3.2348)/-6 = -2.1504

6. y = (17.29- (-0.0692) + (-2.1504)-(3.2348 ^ 2))/4=1.1862
z = (34.7 - (2*-0.0692) - (-2.1504)-(1.1862^2))/11 = 3.2347
errorx1= |xn - xn-1|= |-0.0692 - (-0.0699) |= 0.0007
errorx2= |xn - xn-1|= |-2.1504 - (-2.1506) |= 0.0002
errorx3= |xn - xn-1|= |1.1862 - (1.1862) |= 0
errorx4= |xn - xn-1|= |3.2347 - (3.2348) |= 0.0001
Iteración15
w = (8.70- (-2.1504^2) - 1.1862 - 3.2347)/5 = -0.069
x = (7.3-(-0.069^ 2)+(2*1.1862)+3.2347)/-6 = -2.1504
y = (17.29- (-0.069) + (-2.1504)-(3.2347 ^ 2))/4 =1.1863
z = (34.7 - (2*-0.069) - (-2.1504)-(1.1863^2))/11 = 3.2346
errorx1= |xn - xn-1|= |-0.069 - (-0.0692) |= 0.0002
errorx2= |xn- xn-1|= |-2.1504 - (-2.1504) |= 0
errorx3= |xn - xn-1|= |1.1863 - (1.1862) |= 0.0001
errorx4= |xn - xn-1|= |3.2346 - (3.2347) |= 0.0001
Iteración16
w = (8.70- (-2.1504^2) - 1.1863 - 3.2346)/5 = -0.069
x = (7.3-(-0.069^ 2)+(2*1.1863)+3.2346)/-6 = -2.1504
y = (17.29- (-0.069) + (-2.1504)-(3.2346 ^ 2))/4 =1.1865
z = (34.7 - (2*-0.069) - (-2.1504)-(1.1865^2))/11 = 3.2346
errorx1= |xn - xn-1|= |-0.069 - (-0.069) |= 0
errorx2= |xn - xn-1|= |-2.1504 - (-2.1504) |= 0
errorx3= |xn - xn-1|= |1.1865 - (1.1863) |= 0.0002
errorx4= |xn - xn-1|= |3.2346 - (3.2346) |= 0

7. w=-0.069
x=-2.1504
y=1.1865
z=3.2346
Evaluandoenlaprimerafuncióndel sistema
(5*(-0.069))+(-2.1504)ˆ2 +(1.1865)+(3.2346) =8.70032016
b) Obtenerlasprimerassiete iteraciones,usandoel métodode Gauss- Seidel,aplicadasal
siguiente grupode ecuaciones,aceptandounerrorrelativode ꜫr=1x10-3 .
Iteración1
x1 = (30- (2 * 1.0) - (3 *1))/8 = 3.125
x2 = (1-(3.125)-(2*1))/-9= 0.4583
x3 = (31- (2 * 3.125) - (3 * 0.4583))/6= 3.8958
error x1= |xn - xn-1|= |3.125 - (1) |= 2.125
error x2= |xn - xn-1|= |0.4583 - (1) |= 0.5417
error x3= |xn- xn-1|= |3.8958 - (1) |= 2.8958

8. Iteración2
x1 = (30- (2 * 0.4583) - (3 *3.8958))/8 = 2.1745
x2 = (1-(2.1745)-(2*3.8958))/-9 = 0.9962
x3 = (31- (2 * 2.1745) - (3 * 0.9962))/6= 3.9437
error x1= |xn - xn-1|= |2.1745 - (3.125) |= 0.9505
error x2= |xn - xn-1|= |0.9962 - (0.4583) |= 0.5379
error x3= |xn - xn-1|= |3.9437 - (3.8958) |= 0.0479
Iteración3
x1 = (30- (2 * 0.9962) - (3 *3.9437))/8 = 2.0221
x2 = (1-(2.0221)-(2*3.9437))/-9 = 0.9899
x3 = (31- (2 * 2.0221) - (3 * 0.9899))/6= 3.9977
error x1= |xn - xn-1|= |2.0221 - (2.1745) |= 0.1524
error x2= |xn - xn-1|= |0.9899 - (0.9962) |= 0.0063
error x3= |xn- xn-1|= |3.9977 - (3.9437) |= 0.054
Iteración4
x1 = (30- (2 * 0.9899) - (3 *3.9977))/8 = 2.0034
x2 = (1-(2.0034)-(2*3.9977))/-9 = 0.9999
x3 = (31- (2 * 2.0034) - (3 * 0.9999))/6= 3.9989
error x1= |xn - xn-1|= |2.0034 - (2.0221) |= 0.0187
error x2= |xn- xn-1|= |0.9999 - (0.9899) |= 0.01
error x3= |xn - xn-1|= |3.9989 - (3.9977) |= 0.0012

9. Iteración5
x1 = (30- (2 * 0.9999) - (3 *3.9989))/8 = 2.0004
x2 = (1-(2.0004)-(2*3.9989))/-9 = 0.9998
x3 = (31- (2 * 2.0004) - (3 * 0.9998))/6= 4
error x1= |xn- xn-1|= |2.0004 - (2.0034) |= 0.003
error x2= |xn - xn-1|= |0.9998 - (0.9999) |= 0.0001
error x3= |xn - xn-1|= |4- (3.9989) |= 0.0011
Iteración6
x1 = (30- (2 * 0.9998) - (3 *4))/8 = 2
x2 = (1-(2)-(2*4))/-9=1
x3 = (31- (2 * 2) - (3 * 1))/6= 4
error x1= |xn - xn-1|= |2 - (2.0004) |= 0.0004
error x2= |xn - xn-1|= |1 - (0.9998) |= 0.0002
error x3= |xn - xn-1|= |4 - (4) |= 0
Iteración7
x1 = (30- (2 * 1) - (3 *4))/8 = 2
x2 = (1-(2)-(2*4))/-9=1
x3 = (31- (2 * 2) - (3 * 1))/6= 4
error x1= |xn- xn-1|= |2 - (2) |= 0
error x2= |xn - xn-1|= |1 - (1) |= 0
error x3= |xn - xn-1|= |4 - (4) |= 0

10. x1= 2
x2= 1
x3= 4
Comprobandoenlaprimerecuacióndel sistema.
8x+2x+3x=30
8(2)+2(1)+3(4)=16+2+12=30