Question #143378
Solve by iteration method:
(i) x^3 - 2 x^2 − 5=0
1
Expert's answer
2020-11-11T19:07:03-0500
x32x25=0x^3-2x^2-5=0

x=g(x)=2x2+53x=g(x)=\sqrt[3]{2x^2+5}

x0=2.6x_0=2.6

xi+1=g(xi)=2xi2+53,i=0,1,2,...x_{i+1}=g(x_i)=\sqrt[3]{2x_i^2+5}, i=0, 1,2,...

ixig(xi)02.62.6457389712.645738972.6683955322.668395532.6796211232.679621122.6851835342.685183532.6879398652.687939862.6893057362.689305732.6899825772.689982572.6903179782.690317972.6904841892.690484182.69056654102.690566542.69060735112.690607352.69062758122.690627582.69063760132.690637602.69064257142.690642572.69064503152.690645032.69064625162.690646252.69064685172.690646852.69064715182.690647152.69064730192.690647302.69064737202.690647372.69064741212.690647412.69064743222.690647432.69064744232.690647442.69064744\def\arraystretch{1.5} \begin{array}{c:c:c} i & x_i & g(x_i) \\ \hline 0 & 2.6 & 2.64573897 \\ 1 & 2.64573897& 2.66839553 \\ 2 &2.66839553 & 2.67962112 \\ 3 & 2.67962112 & 2.68518353\\ 4 & 2.68518353 & 2.68793986 \\ 5 & 2.68793986 & 2.68930573\\ 6 & 2.68930573 & 2.68998257\\ 7 & 2.68998257 & 2.69031797\\ 8 & 2.69031797 & 2.69048418\\ 9 & 2.69048418 & 2.69056654\\ 10 & 2.69056654 & 2.69060735\\ 11 & 2.69060735 & 2.69062758\\ 12 & 2.69062758 & 2.69063760\\ 13 & 2.69063760 & 2.69064257\\ 14 & 2.69064257 & 2.69064503\\ 15 & 2.69064503 & 2.69064625\\ 16 & 2.69064625 & 2.69064685\\ 17 & 2.69064685 & 2.69064715\\ 18 & 2.69064715 & 2.69064730\\ 19 & 2.69064730 & 2.69064737\\ 20 & 2.69064737 & 2.69064741\\ 21 & 2.69064741 & 2.69064743\\ 22 & 2.69064743 & 2.69064744\\ \hdashline 23 &2.69064744 & 2.69064744\\ \end{array}

x2.69064744x\approx2.69064744



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