Chapter v
- 1. CHAPTER VIterative Methods to Solve Systems of Linear EquationsBy: Maria Fernanda Vergara Mendoza.PetroleumEngineeringUniversidad Industrial de SantanderColombia-2010
- 2. SPECIAL MATRICESA banded matrix is a square matrix that has all elements equal to zero, with the exception of a band centered on the main diagonal. Gauss elimination or LU decomposition can be used to solve these systems, but they are inefficient, because if pivoting is unnecessary none of elements outside the band would change from their original values of zero.If is known beforehand that pivoting is unnecessary, very efficient algorithms can be developed that do not involve the zero elements outside the band.
- 3. Thedimension of a bandedsystem can bequantifiedbytwoparameters:The band-width (BW)
- 5. THE JACOBI METHOD“Jacobi method is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges.”Given a square system of n linear equations: Ax=bWhere:Thesystem of linear equationsmayberewritten as:(D+R)x=bDx+Rx=bDx=b-Rx
- 6. THE JACOBI METHODThe Jacobi method is an iterative technique that solves the left hand side of this expression for x, using previous value for x on the right hand side. Analytically, this may be written as:The element-based formula is thus:
- 7. EXAMPLEA linear system of the form Ax = b with initial estimate x(0) is given byUse the equation x(k + 1) = D− 1(b − Rx(k)), to estimate x. First, rewrite the equation into the form D− 1(b − Rx(k)) = Tx(k) + C, where T = − D− 1R and C = D− 1b. Note that R = L + U where L and U are the strictly lower and upper parts of A. From the known valuesDetermine T = − D− 1(L + U) as
- 8. EXAMPLEC is found asWith T and C calculated, we estimate x as x(1) = Tx(0) + C:The next iteration yieldsThis process is repeated until convergence (i.e., until |Ax(n) – b| is small). The solution after 25 iterations is
- 9. GAUSS-SEIDEL METHODVery similar totheJacobiMethod.Though it can be applied to any matrix with non-zero elements on the diagonals, convergence is only guaranteed if the matrix is either diagonally dominant, or symmetric and positive definite.methodisthemostcommonlyusediterativemethod. Asume thatAx=b, and supposingthat A=LU (As weseenpreviously):The system of linear equations may be rewritten as:“The Gauss–Seidel method is an iterative technique that solves the left hand side of this expression for x, using previous value for x on the right hand side.”
- 10. The elements of x(k+1) can be computed sequentially using forward substitution:The procedure is continued until the changes made by an iteration are below some tolerance.Convergence:
- 11. EXAMPLEWe want to use the equationIn the formWhereandWe must decompose A into the sum of L* + UThe inverse of L* isNow we can find T and C:
- 12. We can use T and C to obtain the vectors X iteratively, supposing an x(0):...
- 13. As expected, the algorithm converges to the exact solution:
- 14. IMPROVEMENT OF CONVERGENCE USING RELAXATIONRelaxation represents a slight modification of the Gauss-Seidel method and is designed to enhance convergence. After each new value of x is computed, that value is modified by a weighted average of the result of the previous and the present iterations:Where λis a weighting factor thatisassigned a valuebetween 0 and 2.Ifλ= 1, theresultisunmodified.If 0<λ<1, iscalledunderrelaxationIf 1<λ<2, iscalledoverrelaxation
- 15. BIBLIOGRAPHYCHAPRA S., Numerical methods for engineers, Mc Graw Hill.http://www.math-linux.com/spip.php?article48http://www.netlib.org/linalg/html_templates/node14.html#figgs