Iterative methods
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The document discusses two iterative methods for solving systems of linear equations: Jacobi and Gauss-Seidel. It provides an example of applying each method to solve a 3x3 matrix. Jacobi method calculates each variable based on the previous iteration while Gauss-Seidel uses the most recent value calculated. The example shows Gauss-Seidel converges to the solution in fewer iterations, making it more efficient than Jacobi method.
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Iterative methods
This document discusses two iterative methods for solving systems of linear equations: Jacobi and Gauss-Seidel. Jacobi method works by solving each equation for the unknown variable, using initial guesses. Gauss-Seidel is similar but uses the most recently calculated value in each step to improve on Jacobi. An example applying each method to a 3x3 system is shown, with Gauss-Seidel converging faster in fewer iterations.
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Maths in daytoday life by Gilad Lerman Department of Mathematics University o...
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Department of Mathematics
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Method of simple gauss
The method of Gauss, also known as the method of simple elimination of Gauss, is a technique used to solve systems of equations. It involves two phases: elimination of the unknowns and back substitution. The method works by multiplying rows by scalars and subtracting rows to eliminate unknowns from the system one by one until a solution is reached. An example applying the Gauss method to a system of 3 equations with 3 unknowns is shown. Potential issues with the Gauss method include rounding errors, division by zero, and non-well conditioned systems where small changes in coefficients lead to large changes in solutions.
Method Of Simple Gauss
The method of Gauss, also known as the method of simple elimination of Gauss, is a technique used to solve systems of equations. It involves two phases: elimination of the unknowns and back substitution. The method works by multiplying rows by scalars and subtracting rows to eliminate unknowns from the system one by one until a solution is reached. An example applying the Gauss method to a system of 3 equations with 3 unknowns is shown. Potential issues with the Gauss method include rounding errors, division by zero, and non-well conditioned systems where small changes in coefficients lead to large changes in solutions.
Determination of exact shortest paths based on mechanical analogies doi 10.1...
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The document discusses several numerical methods for solving systems of linear equations, including Jacobi, Gauss-Seidel, and Cholesky decomposition methods. Jacobi and Gauss-Seidel methods are iterative methods that start with an initial approximation and iteratively improve it until converging on a solution. The key difference between them is that Gauss-Seidel uses the most recent updates in each iteration while Jacobi uses the previous iteration's values. Cholesky decomposition rewrites a symmetric positive-definite matrix as the product of a lower triangular matrix and its transpose.
Metodos jacobi y gauss seidel
The document discusses several numerical methods for solving systems of linear equations, including Jacobi, Gauss-Seidel, LU decomposition, and Cholesky decomposition. It provides the algorithms and formulas for each method. As an example, it applies the Jacobi and Gauss-Seidel methods to solve a system of 3 equations with 3 unknowns, showing how Gauss-Seidel converges faster by immediately using updated values at each step.
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Numerical Method Analysis: Algebraic and Transcendental Equations (Linear)
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The document discusses several methods for finding the roots or zeroes of equations, where the value of x makes the function f(x) equal to 0. It explains graphical and numerical methods, including closed methods like bisection and false position which divide intervals, and open methods like fixed point, Newton-Raphson, and secant which use calculus to find successive approximations of the root. Examples are provided to demonstrate applying each method to calculate the root of sample equations.
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The document discusses different methods for finding the roots or x-intercepts of equations, where the function f(x) equals 0. It focuses on closed methods like bisection, which works by repeatedly dividing an interval in half and determining if the root is in the upper or lower segment based on the sign change of f(x). As an example, it applies the bisection method to find the root of an equation within 7 iterations, obtaining a result with low error.
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Iterative methods
- 1. RUBEN DARIO ARISMENDI RUEDA
- 2. CHAPTER 4: ‘Iterative Methods to solve lineal ecuation systems’
- 3. INTRODUCTION In the next presentetion, there will be some examples that are already solved in excel document but the interesting part is to see how each different method behaves. To use Jacobi method, the matrix has to has a prevailing diagonal. ( the addition of the other terms are less than the term of the diagonal)
- 5. n 0 1 2 3 4 5 6 7 8 x1 0 1,9 1,1375 0,91833333 0,99848958 1,00789757 0,99926888 0,99935904 1,00014015 x2 0 3,625 3,05416667 2,91927083 3,00222222 3,0069553 2,99898315 2,99947252 3,00015295 x3 0 1,33333333 2,25416667 2,03194444 1,97293403 2,00011863 2,00247548 1,99970867 1,99980526 Tol x1 1,9 -0,7625 -0,21916667 0,08015625 0,00940799 -0,00862869 9,0162E-05 0,0007811 x2 3,625 -0,57083333 -0,13489583 0,08295139 0,00473307 -0,00797215 0,00048938 0,00068043 x3 1,33333333 0,92083333 -0,22222222 -0,05901042 0,02718461 0,00235684 -0,00276681 9,659E-05
- 6. n 9 10 11 12 13 14 x1 1,00004313 0,99998223 0,99999805 1,00000186 0,99999997 0,99999983 x2 3,00003117 2,9999824 2,99999913 3,00000172 2,99999989 2,99999985 x3 2,00004885 2,00001238 1,9999941 1,99999953 2,0000006 1,99999998 Tol x1 -9,702E-05 -6,0898E-05 1,5817E-05 3,8105E-06 -1,8866E-06 -1,369E-07 x2 -0,00012179 -4,877E-05 1,6729E-05 2,5923E-06 -1,8324E-06 -3,0965E-08 x3 0,00024359 -3,6468E-05 -1,8278E-05 5,4244E-06 1,0671E-06 -6,1983E-07
- 7. n 15 16 17 x1 1,00000002 1,00000001 1 x2 3,00000003 3,00000001 3 x3 1,99999995 2,00000001 2 Tol x1 1,8904E-07 -8,8137E-09 -1,6392E-08 x2 1,7207E-07 -1,6636E-08 -1,3945E-08 x3 -2,7978E-08 6,0186E-08 -4,2416E-09
- 8. 2. GAUSS-SEIDEL. The algoritm is the same for the Jacobi method. But it makes an exception with the next expression that improve the Jacobi’s method.
- 9. 3x3 MATRIX 10 1 3 19 1 8 2 29 -1 -1 6 8
- 10. n 0 1 2 3 4 5 x1 0 1,9 0,896875 1,01126953 0,99874858 1,00013821 x2 0 3,3875 2,95924479 3,00458632 2,99949577 3,00005588 x3 0 2,21458333 1,97601997 2,00264264 1,99970739 2,00003235 Tolerancia x1 1,9 -1,003125 0,11439453 -0,01252096 0,00138963 x2 3,3875 -0,42825521 0,04534153 -0,00509055 0,00056011 x3 2,21458333 -0,23856337 0,02662268 -0,00293525 0,00032496
- 11. n 6 7 8 9 10 x1 0,99998471 1,00000169 0,99999981 1,00000002 1 x2 2,99999382 3,00000068 2,99999992 3,00000001 3 x3 1,99999642 2,0000004 1,99999996 2 2 Tolerancia x1 -0,0001535 1,6983E-05 -1,8779E-06 2,0769E-07 -2,2969E-08 x2 -6,2052E-05 6,8584E-06 -7,5864E-07 8,3894E-08 -9,2783E-09 x3 -3,5925E-05 3,9735E-06 -4,3942E-07 4,8598E-08 -5,3745E-09
- 12. With the two last examples, we can see that the Gauss-Seidel method is more efficient than the Jacobi method. Because it gets to the answer in a less number of iterations.