Lecture Notes Course à MA 190 Numerical Mathematics, First ...

4.2 Jacobi and Gauss-Seidel iterations4.2.1 Jacobi iterationWe consider again the linear system Ax = b which we write on the formx = Bx + f.The latter is solved by iterations. We discuss only two different methods, namelythe Jacobi and the Gauss-Seidel schemes. By Jacobi iterations we form thesequence:x l+1 = Bx l + f, l = 0, 1, . . . ,where the starting approximation may be arbitrary. Often one takes x 0 = 0.The elements of x l are updated according ton∑x l+1i = b i,j x l j + f i ,This process is easily parallelized.j=14.2.2 Gauss-Seidel iterationThe Gauss-Seidel iterations differs from the Jacobi ones, in that one uses thelatest available components of x to make the next update. Thus one updates thecomponents of x one at a time beginning with x 1 . When x 2 is updated, one usesthe just calculated value of x 1 together with the earlier values of the remainingcomponents. To update x 3 one uses the recent values of x 1 , x 2 together withearlier values for x 4 , . . . and so on. The two iteration schemes do not convergefor all matrices B but the following condition is sufficient for convergence ofboth methods:maxin∑|b i,j | < 1 .j=1Example 4.2.1 We solve the system Ax = b where⎛A = ⎝ 10 1 2⎞ ⎛3 11 1 ⎠ b = ⎝ 13152 3 12 17We write this system in the formx = Bx + f,by solving the first equation with respect to x 1 , the second with respect to x 2and the third with respect to x 3 . Then we find⎞⎠ .x 1 = 1 10 (13 − x 2 − 2x 3 ) (4.2)x 2 = 1 11 (15 − 3x 1 − x 3 ) (4.3)x 3 = 1 12 (17 − 2x 1 − 3x 2 ) (4.4)34