Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Jacobi and gauss-seidel by arunsmm 10500 views
- Directive-based approach to Heterog... by Ruymán Reyes 1406 views
- Jawaban uts astl ganjil analisa sis... by suparman unkhair 462 views
- Iterative methods by Ketan Nayak 45 views
- System of linear equations by Diler4 968 views
- Chapter 5 by universidad indus... 1283 views

No Downloads

Total views

1,782

On SlideShare

0

From Embeds

0

Number of Embeds

1

Shares

0

Downloads

31

Comments

0

Likes

1

No embeds

No notes for slide

- 1. ITERATIVE METHODS FOR THE SOLUTION OF SYSTEM OF LINEAR EQUATIONS<br />OSCAR EDUARDO MENDIVELSO OROZCO<br />I studyPetroleumEngineering<br /> Numerical Methods <br />
- 2. JACOBI METHOD<br />The Jacobi method is the iterative method for solving systems of linear equations more simple and applies only square systems, ie systems with as many unknowns as equations.<br />First, we determine the equation of recurrence. For this order the equations and unknowns. Of the equation i cleared the unknown i. In matrix notation is written as:<br />𝒙=𝒄+𝑩𝒙<br /> where x is the vector of unknowns.<br /> <br />
- 3. 2.It takes an approach to the solutions and this is designated by 𝒙𝒐<br />3. Is iterated in the cycle that changes the approach 𝒙𝒊+𝟏=𝒄+𝑩𝒙𝒊<br /> <br />
- 4. Example:<br />From (x = 1, y = 2) apply two iterations of the Jacobi method to solve the system: 𝟓𝒙 +𝟐𝒚 =𝟏𝒙 −𝟒𝒚 =𝟎<br />Solution:<br />We must first solve the equation for the unknown<br />x= 0.20+0.00x-0.40y<br /> y= 0.00+0.25x+0.00y<br />Written in vector notation would be:<br />𝑥𝑦=0.200.00+0.00−0.400.250.00𝑥𝑦<br /> <br />
- 5. We apply the first iteration starting from𝑥0=1.00 y 𝑦0=2.00:<br />𝑥1=0.20 + 0.00 (1.00) - 0.40 (2.00) = -0.60<br />𝑦1=0.00 + 0.25 (1.00) + 0.00 (2.00) = 0.25<br />We apply the second iteration starting from 𝑥1= -0.60 y 𝑦1=0.25 <br />𝑥2= 0.20 + 0.00 (-0.60) - 0.40 (0.25) = 0.10<br />𝑦2=0.00 + 0.25 (−0.60) +0.00 (0.25) = −0.15<br />We apply the next iteration from 𝑥2= 0.10 y 𝑦1=0.25:<br />𝑥3= 0.20 + 0.00 (0.10) - 0.40 (-0.15) = 0.26<br />𝑦3= 0.00 +0.25 (0.10) +0.00 (−0.15) = 0.025<br />We apply the next iteration from 𝑥3=0.26 y 𝑦3=0.025:<br />𝑥4=0.20 + 0.00 (0.26) − 0.40 (0.025) = 0.190<br />𝑦4=0.00 + 0.25 (0.26) + 0.00 (0.025) = 0.065<br /> <br />
- 6. We apply the next iteration from 𝑥4=0.190 y 𝑦4=0.065:<br />𝑥5=0.20 + 0.00 (0.19) − 0.40 (0.065) = 0.174<br />𝑦5=0.00 + 0.25 (0.19) + 0.00 (0.065) = 0.0475<br /> We apply the next iteration from 𝑥5=0.174 y 𝑦4=0.0475:<br />𝑥6=0.20 + 0.00 (0.174) − 0.40 (0.0475) = 0.181<br />𝑦6=0.00 + 0.25 (0.174) + 0.00 (0.0475) = 0.0435<br />Where<br /> <br />
- 7.
- 8. THE GAUSS-SEIDEL METHOD<br />The Gauss-Seidel method is very similar to the method of Jacobi. While using the Jacobi the value of the unknowns to determine a new approach in the Gauss-Seidel is using values of the unknowns recently calculated in the same iteration, and not the next.<br />For example, Jacobi method is obtained in the first calculation 𝑋𝐼+1, But the value of x is not used until the next iteration. In the Gauss-Seidel method instead of 𝑥𝑖+1is used i +1 instead of 𝑥𝑖 immediatelyto calculate the value of 𝑦𝑖+1 equally applicable to the following variables are used whenever newly calculated variables.<br /> <br />
- 9. Example: From (x = 1, y = 2, z = 0) apply two iterations of Gauss-Seidel method to solve the system:<br />10𝑥+0𝑦 +𝑧 =−14𝑥 +12𝑦 +4𝑧 = 84𝑥 +4𝑦 +10𝑧 = 4<br />Solution:<br />We must first solve the equation for the unknown.<br />x = −0.10 + 0.00 x + 0.00 y + 0.10 z<br /> y = 0.66 − 0.33 x + 0.00 y + 0.33 z<br />z = 0.40 −0.40 x − 0.40 y + 0.00 z<br /> <br />
- 10. We apply the first iteration starting from𝑥0=1.00, 𝑦0=2.00 𝑦 𝑧=0.00:<br />x1 = −0.10 + 0.00 (1.00) + 0.00 (2.00) + 0.10 (0.00) = −0.1<br /> y1 = 0.66 − 0.33 (−0.10) + 0.00 (2.00) + 0.33 (0.00) = 0.70<br />z1 = 0.40 − 0.40(−0.10) − 0.40 (0.70) + 0.00 (0.00) = 0.16<br />We apply the second iteration starting from 𝑥1= -0.10, 𝑦1=0.70 y 𝑧1=0.16:<br />x1 = −0.10 + 0.00(−0.10) + 0.00 (0.70) + 0.10 (0.16) = −0.084<br />y1 = 0.66 − 0.33(−0.084) + 0.00 (0.70) + 0.33 (0.16) = 0.748<br />z1 = 0.40 − 0.40(−0.084) − 0.40 (0.748) + 0.00 (0.16) = 0.134<br />We apply the third iteration starting from 𝑥1= -0.084, 𝑦1=0.748 y 𝑧1=0.134:<br />x1 = −0.10 +0.00(−0.084) +0.00(0.748) +0.10 (0.134) = −0.086<br /> y1 = 0.66 − 0.33(−0.086) + 0.00 (0.748) + 0.33 (0.134) = 0.740<br />z1 = 0.40 − 0.40(−0.086) − 0.40 (0.740) + 0.00 (0.134) = 0.138<br /> <br />
- 11. Computationalcost<br />It is difficult to estimate the computational cost of an iterative method, it is not known beforehand how many iterations required to obtain an answer that satisfies the user. Usually proceed to calculate computational cost per iteration. In the case of the Jacobi method used the recurrence relation is:<br />𝒙𝒊+𝟏=𝐜+𝑩𝒙𝒊<br />It is difficult to estimate the computational cost involved: the product of the matrix B, n × n by the vector 𝑥𝑖 takes n × (2n-1) FLOPs, and the sum of two vectors in ℜ𝑛 with n FLOPs making which gives a total of 2𝑛2 FLOPs at each iteration of the Jacobi method. <br /> <br />
- 12. Using this information we can conclude that if the algorithm takes m iterations then the total FLOPs shall be:<br />2𝑚𝑛2<br />That is why the Jacobi method is preferred in problems where n is large, when you can ensure convergence and when the expected number of iterations is low.<br />REFERENCES<br />http://www.mty.itesm.mx/etie/deptos/m/ma95-843/lecturas/l843-13.pdf<br /> <br />

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment