Mathematical Methods for Physics and Engineering - Matematica.NET

NUMERICAL METHODSappreciate how this would apply in (say) a computer program for a 1000-variablecase, perhaps with unforseeable zeros or very small numbers appearing on theleading diagonal.◮Solve the simultaneous equations(a) x 1 +6x 2 −4x 3 = 8,(b) 3x 1 −20x 2 +x 3 = 12,(c) −x 1 +3x 2 +5x 3 = 3.(27.22)Firstly, we interchange rows (a) and (b) to bring the term 3x 1 onto the leading diagonal. Inthe following, we label the important equations (I), (II), (III), and the others alphabetically.A general (i.e. variable) label will be denoted by j.(I) 3x 1 −20x 2 +x 3 = 12,(d) x 1 +6x 2 −4x 3 = 8,(e) −x 1 +3x 2 +5x 3 = 3.For (j) = (d) and (e), replace row (j) byrow (j) − a j1× row (I),3where a j1 is the coefficient of x 1 in row (j), to give the two equations(II)( )6+20 x23+ ( )−4 − 1 x33= 8− 12 3(f)( )3 −20 x23+ ( )5+ 1 x33= 3+ 12 3Now |6+ 2020| > |3 − | and so no interchange is required before the next elimination. To3 3eliminate x 2 ,replacerow(f)by( )−113row (f) − × row (II).This gives[(III) 16] (−13) x3 =7+ 11 3 38 338Collecting together and tidying up the final equations, we have(I) 3x 1 −20x 2 +x 3 = 12,(II) 38x 2 −13x 3 = 12,(III) x 3 = 2.Starting with (III) and working backwards, it is now a simple matter to obtainx 1 =10, x 2 =1, x 3 =2. ◭38327.3.2 Gauss–Seidel iterationIn the example considered in the previous subsection an explicit way of solvinga set of simultaneous equations was given, the accuracy obtainable being limitedonly by the rounding errors in the calculating facilities available, and the calculationwas planned to minimise these. However, in some situations it may be thatonly an approximate solution is needed. If, for a large number of variables, this is996