Birge-Vieta method Example-3 f(x)=9x^4+12x^3+13x^2+12x+4 and p0=-0.5

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Home > Numerical methods calculators > Birge-Vieta method example

Birge-Vieta method example ( Enter your problem )

( Enter your problem )

Algorithm & Example-1 `f(x)=x^3-x^2-x+1` and `p0=0.5`
Example-2 `f(x)=x^4-3x^3+3x^2-3x+2` and `p0=0.5`
Example-3 `f(x)=9x^4+12x^3+13x^2+12x+4` and `p0=-0.5`

2. Example-2 `f(x)=x^4-3x^3+3x^2-3x+2` and `p0=0.5`
(Previous example)

3. Example-3 `f(x)=9x^4+12x^3+13x^2+12x+4` and `p0=-0.5`

Find a root of polynomial using Birge-Vieta method
`f(x)=9x^4+12x^3+13x^2+12x+4` and `p0=-0.5`

Solution:
`9x^4+12x^3+13x^2+12x+4=0`

In this problem the coefficients are `a_0=9,a_1=12,a_2=13,a_3=12,a_4=4`

Let the initial approximation to p be `p_0=-0.5`

`1^(st)` iteration :

`p_0` `-0.5`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_0*b_0` `-4.5`	`p_0*b_1` `-3.75`	`p_0*b_2` `-4.625`	`p_0*b_3` `-3.6875`
	`b_0` `9`	`b_1=a_1+p_0*b_0` `7.5`	`b_2=a_2+p_0*b_1` `9.25`	`b_3=a_3+p_0*b_2` `7.375`	`b_4=a_4+p_0*b_3` `0.3125`
		`p_0*c_0` `-4.5`	`p_0*c_1` `-1.5`	`p_0*c_2` `-3.875`
	`c_0` `9`	`c_1=b_1+p_0*c_0` `3`	`c_2=b_2+p_0*c_1` `7.75`	`c_3=b_3+p_0*c_2` `3.5`

`p_1=p_0-b_4/c_3=-0.5-(0.3125)/(3.5)=-0.5893`

`2^(nd)` iteration :

`p_1` `-0.5893`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_1*b_0` `-5.3036`	`p_1*b_1` `-3.9461`	`p_1*b_2` `-5.3353`	`p_1*b_3` `-3.9274`
	`b_0` `9`	`b_1=a_1+p_1*b_0` `6.6964`	`b_2=a_2+p_1*b_1` `9.0539`	`b_3=a_3+p_1*b_2` `6.6647`	`b_4=a_4+p_1*b_3` `0.0726`
		`p_1*c_0` `-5.3036`	`p_1*c_1` `-0.8208`	`p_1*c_2` `-4.8516`
	`c_0` `9`	`c_1=b_1+p_1*c_0` `1.3929`	`c_2=b_2+p_1*c_1` `8.2331`	`c_3=b_3+p_1*c_2` `1.813`

`p_2=p_1-b_4/c_3=-0.5893-(0.0726)/(1.813)=-0.6293`

`3^(rd)` iteration :

`p_2` `-0.6293`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_2*b_0` `-5.664`	`p_2*b_1` `-3.9875`	`p_2*b_2` `-5.6719`	`p_2*b_3` `-3.9825`
	`b_0` `9`	`b_1=a_1+p_2*b_0` `6.336`	`b_2=a_2+p_2*b_1` `9.0125`	`b_3=a_3+p_2*b_2` `6.3281`	`b_4=a_4+p_2*b_3` `0.0175`
		`p_2*c_0` `-5.664`	`p_2*c_1` `-0.4229`	`p_2*c_2` `-5.4057`
	`c_0` `9`	`c_1=b_1+p_2*c_0` `0.672`	`c_2=b_2+p_2*c_1` `8.5896`	`c_3=b_3+p_2*c_2` `0.9224`

`p_3=p_2-b_4/c_3=-0.6293-(0.0175)/(0.9224)=-0.6483`

`4^(th)` iteration :

`p_3` `-0.6483`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_3*b_0` `-5.8349`	`p_3*b_1` `-3.997`	`p_3*b_2` `-5.8368`	`p_3*b_3` `-3.9957`
	`b_0` `9`	`b_1=a_1+p_3*b_0` `6.1651`	`b_2=a_2+p_3*b_1` `9.003`	`b_3=a_3+p_3*b_2` `6.1632`	`b_4=a_4+p_3*b_3` `0.0043`
		`p_3*c_0` `-5.8349`	`p_3*c_1` `-0.2141`	`p_3*c_2` `-5.698`
	`c_0` `9`	`c_1=b_1+p_3*c_0` `0.3303`	`c_2=b_2+p_3*c_1` `8.7889`	`c_3=b_3+p_3*c_2` `0.4652`

`p_4=p_3-b_4/c_3=-0.6483-(0.0043)/(0.4652)=-0.6576`

`5^(th)` iteration :

`p_4` `-0.6576`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_4*b_0` `-5.9181`	`p_4*b_1` `-3.9993`	`p_4*b_2` `-5.9186`	`p_4*b_3` `-3.9989`
	`b_0` `9`	`b_1=a_1+p_4*b_0` `6.0819`	`b_2=a_2+p_4*b_1` `9.0007`	`b_3=a_3+p_4*b_2` `6.0814`	`b_4=a_4+p_4*b_3` `0.0011`
		`p_4*c_0` `-5.9181`	`p_4*c_1` `-0.1077`	`p_4*c_2` `-5.8478`
	`c_0` `9`	`c_1=b_1+p_4*c_0` `0.1637`	`c_2=b_2+p_4*c_1` `8.8931`	`c_3=b_3+p_4*c_2` `0.2336`

`p_5=p_4-b_4/c_3=-0.6576-(0.0011)/(0.2336)=-0.6621`

`6^(th)` iteration :

`p_5` `-0.6621`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_5*b_0` `-5.9592`	`p_5*b_1` `-3.9998`	`p_5*b_2` `-5.9594`	`p_5*b_3` `-3.9997`
	`b_0` `9`	`b_1=a_1+p_5*b_0` `6.0408`	`b_2=a_2+p_5*b_1` `9.0002`	`b_3=a_3+p_5*b_2` `6.0406`	`b_4=a_4+p_5*b_3` `0.0003`
		`p_5*c_0` `-5.9592`	`p_5*c_1` `-0.054`	`p_5*c_2` `-5.9236`
	`c_0` `9`	`c_1=b_1+p_5*c_0` `0.0815`	`c_2=b_2+p_5*c_1` `8.9462`	`c_3=b_3+p_5*c_2` `0.117`

`p_6=p_5-b_4/c_3=-0.6621-(0.0003)/(0.117)=-0.6644`

`7^(th)` iteration :

`p_6` `-0.6644`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_6*b_0` `-5.9797`	`p_6*b_1` `-4`	`p_6*b_2` `-5.9797`	`p_6*b_3` `-3.9999`
	`b_0` `9`	`b_1=a_1+p_6*b_0` `6.0203`	`b_2=a_2+p_6*b_1` `9`	`b_3=a_3+p_6*b_2` `6.0203`	`b_4=a_4+p_6*b_3` `0.0001`
		`p_6*c_0` `-5.9797`	`p_6*c_1` `-0.027`	`p_6*c_2` `-5.9617`
	`c_0` `9`	`c_1=b_1+p_6*c_0` `0.0407`	`c_2=b_2+p_6*c_1` `8.973`	`c_3=b_3+p_6*c_2` `0.0586`

`p_7=p_6-b_4/c_3=-0.6644-(0.0001)/(0.0586)=-0.6655`

`8^(th)` iteration :

`p_7` `-0.6655`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_7*b_0` `-5.9898`	`p_7*b_1` `-4`	`p_7*b_2` `-5.9898`	`p_7*b_3` `-4`
	`b_0` `9`	`b_1=a_1+p_7*b_0` `6.0102`	`b_2=a_2+p_7*b_1` `9`	`b_3=a_3+p_7*b_2` `6.0102`	`b_4=a_4+p_7*b_3` `0`
		`p_7*c_0` `-5.9898`	`p_7*c_1` `-0.0135`	`p_7*c_2` `-5.9808`
	`c_0` `9`	`c_1=b_1+p_7*c_0` `0.0203`	`c_2=b_2+p_7*c_1` `8.9865`	`c_3=b_3+p_7*c_2` `0.0293`

`p_8=p_7-b_4/c_3=-0.6655-(0)/(0.0293)=-0.6661`

`9^(th)` iteration :

`p_8` `-0.6661`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_8*b_0` `-5.9949`	`p_8*b_1` `-4`	`p_8*b_2` `-5.9949`	`p_8*b_3` `-4`
	`b_0` `9`	`b_1=a_1+p_8*b_0` `6.0051`	`b_2=a_2+p_8*b_1` `9`	`b_3=a_3+p_8*b_2` `6.0051`	`b_4=a_4+p_8*b_3` `0`
		`p_8*c_0` `-5.9949`	`p_8*c_1` `-0.0068`	`p_8*c_2` `-5.9904`
	`c_0` `9`	`c_1=b_1+p_8*c_0` `0.0102`	`c_2=b_2+p_8*c_1` `8.9932`	`c_3=b_3+p_8*c_2` `0.0147`

`p_9=p_8-b_4/c_3=-0.6661-(0)/(0.0147)=-0.6664`

`10^(th)` iteration :

`p_9` `-0.6664`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_9*b_0` `-5.9975`	`p_9*b_1` `-4`	`p_9*b_2` `-5.9975`	`p_9*b_3` `-4`
	`b_0` `9`	`b_1=a_1+p_9*b_0` `6.0025`	`b_2=a_2+p_9*b_1` `9`	`b_3=a_3+p_9*b_2` `6.0025`	`b_4=a_4+p_9*b_3` `0`
		`p_9*c_0` `-5.9975`	`p_9*c_1` `-0.0034`	`p_9*c_2` `-5.9952`
	`c_0` `9`	`c_1=b_1+p_9*c_0` `0.0051`	`c_2=b_2+p_9*c_1` `8.9966`	`c_3=b_3+p_9*c_2` `0.0073`

`p_10=p_9-b_4/c_3=-0.6664-(0)/(0.0073)=-0.6665`

`11^(th)` iteration :

`p_10` `-0.6665`	`a_0` `9`	`a_1` `12`	`a_2` `13`	`a_3` `12`	`a_4` `4`
		`p_10*b_0` `-5.9987`	`p_10*b_1` `-4`	`p_10*b_2` `-5.9987`	`p_10*b_3` `-4`
	`b_0` `9`	`b_1=a_1+p_10*b_0` `6.0013`	`b_2=a_2+p_10*b_1` `9`	`b_3=a_3+p_10*b_2` `6.0013`	`b_4=a_4+p_10*b_3` `0`
		`p_10*c_0` `-5.9987`	`p_10*c_1` `-0.0017`	`p_10*c_2` `-5.9976`
	`c_0` `9`	`c_1=b_1+p_10*c_0` `0.0025`	`c_2=b_2+p_10*c_1` `8.9983`	`c_3=b_3+p_10*c_2` `0.0037`

`p_11=p_10-b_4/c_3=-0.6665-(0)/(0.0037)=-0.6666`

Approximate root = -0.6666

This material is intended as a summary. Use your textbook for detail explanation.
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