By Xin-She Yang
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If necessary, rows can be exchanged so the the largest element is moved so that it becomes the leading coefficient, especially on the diagonal position. This makes all the scaled elements to be in the range [−1, 1]. Thus, exceptionally large numbers are removed, which makes the scheme more numerically stable. An important issue in both Gauss elimination and GaussJordan elimination is the non-zero requirement of leading coefficients such as a11 = 0. For a11 , it is possible to re-arrange the equations to achieve this requirement.
In this chapter, we will introduce the fundamentals of the numerical techniques concerning root-finding algorithms. 1 Simple Iterations The essence of root-finding algorithms is to use iteration procedure to obtain the approximate (well sometimes quite accurate) solutions, starting from some initial guess solution. For example, even ancient Babylonians knew how to find the square root of 2 using the iterative method. From the numerical technique we learnt at school, we know that we can numerically√compute the square root of any real number k ( so that x = k) using 25 26 Chapter 3.
From the diagonal matrix D, its largest absolute eigenvalue is 1. So ρ(D−1 ) = max(|λi |) = 1 seems to be no problem. How about the following matrix? 0 4 0 0 −2. 111. 59 > 1. The iteration scheme will diverge. 43) 50 Chapter 4. 0644i. 4739 < 1. 45) That is why the earlier iteration procedure is convergent. ). If the vector size u is large (it usually is), then we can devise other iteration procedure to save memory using the running update. So only one vector storage is needed. The Gauss-Seidel iteration procedure is such an iteration procedure to use the running update and also provides an efficient way of solving the linear matrix equation Au = b.