Polynomial Examples
Thomas J. Kennedy
Contents:
Let us examine the results of each solver for $f(x)=x^2 - 1$ with three different input domains.
1 Example 1 - [-2, 2]
$f(x)=x^2 - 1$ where $-2\leq x\leq 2$
1.1 Bisection
20 Steps
| n | $a_n$ | $b_n$ | $x_n$ | $f(x_n)$ |
|---|---|---|---|---|
| 0 | $-2.000000$ | $2.000000$ | ||
| 1 | $0.000000$ | $2.000000$ | $0.000000$ | $-1.000000$ |
| 2 | $0.000000$ | $1.000000$ | $1.000000$ | $0.000000$ |
| 3 | $\frac{1}{2}=0.500000$ | $1.000000$ | $\frac{1}{2}=0.500000$ | $\frac{-3}{4}=-0.750000$ |
| 4 | $\frac{3}{4}=0.750000$ | $1.000000$ | $\frac{3}{4}=0.750000$ | $\frac{-7}{16}=-0.437500$ |
| 5 | $\frac{7}{8}=0.875000$ | $1.000000$ | $\frac{7}{8}=0.875000$ | $\frac{-15}{64}=-0.234375$ |
| 6 | $\frac{15}{16}=0.937500$ | $1.000000$ | $\frac{15}{16}=0.937500$ | $\frac{-31}{256}=-0.121094$ |
| 7 | $\frac{31}{32}=0.968750$ | $1.000000$ | $\frac{31}{32}=0.968750$ | $\frac{-63}{1024}=-0.061523$ |
| 8 | $\frac{63}{64}=0.984375$ | $1.000000$ | $\frac{63}{64}=0.984375$ | $\frac{-127}{4096}=-0.031006$ |
| 9 | $\frac{127}{128}=0.992188$ | $1.000000$ | $\frac{127}{128}=0.992188$ | $\frac{-255}{16384}=-0.015564$ |
| 10 | $\frac{255}{256}=0.996094$ | $1.000000$ | $\frac{255}{256}=0.996094$ | $\frac{-511}{65536}=-0.007797$ |
| 11 | $\frac{511}{512}=0.998047$ | $1.000000$ | $\frac{511}{512}=0.998047$ | $\frac{-1023}{262144}=-0.003902$ |
| 12 | $\frac{1023}{1024}=0.999023$ | $1.000000$ | $\frac{1023}{1024}=0.999023$ | $\frac{-2047}{1048576}=-0.001952$ |
| 13 | $\frac{2047}{2048}=0.999512$ | $1.000000$ | $\frac{2047}{2048}=0.999512$ | $\frac{-4095}{4194304}=-0.000976$ |
| 14 | $\frac{4095}{4096}=0.999756$ | $1.000000$ | $\frac{4095}{4096}=0.999756$ | $\frac{-8191}{16777216}=-0.000488$ |
| 15 | $\frac{8191}{8192}=0.999878$ | $1.000000$ | $\frac{8191}{8192}=0.999878$ | $\frac{-16383}{67108864}=-0.000244$ |
| 16 | $\frac{16383}{16384}=0.999939$ | $1.000000$ | $\frac{16383}{16384}=0.999939$ | $-0.000122$ |
| 17 | $\frac{32767}{32768}=0.999969$ | $1.000000$ | $\frac{32767}{32768}=0.999969$ | $-0.000061$ |
| 18 | $\frac{65535}{65536}=0.999985$ | $1.000000$ | $\frac{65535}{65536}=0.999985$ | $-0.000031$ |
| 19 | $\frac{131071}{131072}=0.999992$ | $1.000000$ | $\frac{131071}{131072}=0.999992$ | $-0.000015$ |
Solution
$x=\frac{131071}{131072}=0.999992$
$f(x)=-0.000015$
1.2 Regula Falsi (False Position)
1 Step
| n | $a_n$ | $b_n$ | $x_n$ | $f(x_n)$ |
|---|---|---|---|---|
| 0 | -2 | 2 |
Method Fails
$x_n=\frac{-2-2}{3-3}*3$ Division by Zero
1.3 Secant
1 Step
| n | $x_{n-1}$ | $x_n$ | $x_{n+1}$ | $f(x_{n+1})$ |
|---|---|---|---|---|
| 1 | -2 | 2 |
Method Fails
$x_n=\frac{2- -2}{3 - 3} * 3$ Division by Zero
1.4 Newton
6 Steps
| n | $x_{n-1}$ | $f(x_{n-1})$ | $f’(x_{n-1})$ | $x_{n+1}$ |
|---|---|---|---|---|
| 1 | $-2.000000$ | $3.000000$ | $-4.000000$ | $\frac{-5}{4}=-1.250000$ |
| 2 | $\frac{-5}{4}=-1.250000$ | $\frac{9}{16}=0.562500$ | $\frac{-5}{2}=-2.500000$ | $\frac{-41}{40}=-1.025000$ |
| 3 | $\frac{-41}{40}=-1.025000$ | $\frac{81}{1600}=0.050625$ | $\frac{-41}{20}=-2.050000$ | $\frac{-3281}{3280}=-1.000305$ |
| 4 | $\frac{-3281}{3280}=-1.000305$ | $\frac{6561}{10758400}=0.000610$ | $\frac{-3281}{1640}=-2.000610$ | $\frac{-21523361}{21523360}=-1.000000$ |
| 5 | $\frac{-21523361}{21523360}=-1.000000$ | $0.000000$ | $\frac{-21523361}{10761680}=-2.000000$ | $-1.000000$ |
Solution
$x=\frac{-21523361}{21523360}=-1.000000$
$f(x)=0.000000$
2 Example 2 - [-100, 100]
$f(x)=x^2 - 1$ where $-100\leq x\leq 100$
2.1 Bisection
26 Steps
| n | $a_n$ | $b_n$ | $x_n$ | $f(x_n)$ |
|---|---|---|---|---|
| 0 | $-100.000000$ | $100.000000$ | ||
| 1 | $0.000000$ | $100.000000$ | $0.000000$ | $-1.000000$ |
| 2 | $0.000000$ | $50.000000$ | $50.000000$ | $2499.000000$ |
| 3 | $0.000000$ | $25.000000$ | $25.000000$ | $624.000000$ |
| 4 | $0.000000$ | $\frac{25}{2}=12.500000$ | $\frac{25}{2}=12.500000$ | $\frac{621}{4}=155.250000$ |
| 5 | $0.000000$ | $\frac{25}{4}=6.250000$ | $\frac{25}{4}=6.250000$ | $\frac{609}{16}=38.062500$ |
| 6 | $0.000000$ | $\frac{25}{8}=3.125000$ | $\frac{25}{8}=3.125000$ | $\frac{561}{64}=8.765625$ |
| 7 | $0.000000$ | $\frac{25}{16}=1.562500$ | $\frac{25}{16}=1.562500$ | $\frac{369}{256}=1.441406$ |
| 8 | $\frac{25}{32}=0.781250$ | $\frac{25}{16}=1.562500$ | $\frac{25}{32}=0.781250$ | $\frac{-399}{1024}=-0.389648$ |
| 9 | $\frac{25}{32}=0.781250$ | $\frac{75}{64}=1.171875$ | $\frac{75}{64}=1.171875$ | $\frac{1529}{4096}=0.373291$ |
| 10 | $\frac{125}{128}=0.976562$ | $\frac{75}{64}=1.171875$ | $\frac{125}{128}=0.976562$ | $\frac{-759}{16384}=-0.046326$ |
| 11 | $\frac{125}{128}=0.976562$ | $\frac{275}{256}=1.074219$ | $\frac{275}{256}=1.074219$ | $\frac{10089}{65536}=0.153946$ |
| 12 | $\frac{125}{128}=0.976562$ | $\frac{525}{512}=1.025391$ | $\frac{525}{512}=1.025391$ | $\frac{13481}{262144}=0.051426$ |
| 13 | $\frac{125}{128}=0.976562$ | $\frac{1025}{1024}=1.000977$ | $\frac{1025}{1024}=1.000977$ | $\frac{2049}{1048576}=0.001954$ |
| 14 | $\frac{2025}{2048}=0.988770$ | $\frac{1025}{1024}=1.000977$ | $\frac{2025}{2048}=0.988770$ | $\frac{-93679}{4194304}=-0.022335$ |
| 15 | $\frac{4075}{4096}=0.994873$ | $\frac{1025}{1024}=1.000977$ | $\frac{4075}{4096}=0.994873$ | $\frac{-171591}{16777216}=-0.010228$ |
| 16 | $\frac{8175}{8192}=0.997925$ | $\frac{1025}{1024}=1.000977$ | $\frac{8175}{8192}=0.997925$ | $\frac{-278239}{67108864}=-0.004146$ |
| 17 | $\frac{16375}{16384}=0.999451$ | $\frac{1025}{1024}=1.000977$ | $\frac{16375}{16384}=0.999451$ | $\frac{-294831}{268435456}=-0.001098$ |
| 18 | $\frac{16375}{16384}=0.999451$ | $\frac{32775}{32768}=1.000214$ | $\frac{32775}{32768}=1.000214$ | $\frac{458801}{1073741824}=0.000427$ |
| 19 | $\frac{65525}{65536}=0.999832$ | $\frac{32775}{32768}=1.000214$ | $\frac{65525}{65536}=0.999832$ | $\frac{-1441671}{4294967296}=-0.000336$ |
| 20 | $\frac{65525}{65536}=0.999832$ | $\frac{131075}{131072}=1.000023$ | $\frac{131075}{131072}=1.000023$ | $\frac{786441}{17179869184}=0.000046$ |
| 21 | $\frac{262125}{262144}=0.999928$ | $\frac{131075}{131072}=1.000023$ | $\frac{262125}{262144}=0.999928$ | $\frac{-9961111}{68719476736}=-0.000145$ |
| 22 | $\frac{524275}{524288}=0.999975$ | $\frac{131075}{131072}=1.000023$ | $\frac{524275}{524288}=0.999975$ | $-0.000050$ |
| 23 | $\frac{1048575}{1048576}=0.999999$ | $\frac{131075}{131072}=1.000023$ | $\frac{1048575}{1048576}=0.999999$ | $\frac{-2097151}{1099511627776}=-0.000002$ |
| 24 | $\frac{1048575}{1048576}=0.999999$ | $\frac{2097175}{2097152}=1.000011$ | $\frac{2097175}{2097152}=1.000011$ | $\frac{96469521}{4398046511104}=0.000022$ |
| 25 | $\frac{1048575}{1048576}=0.999999$ | $\frac{4194325}{4194304}=1.000005$ | $\frac{4194325}{4194304}=1.000005$ | $0.000010$ |
Solution
$x=\frac{4194325}{4194304}=1.000005$
$f(x)=0.000010$
2.2 Regula Falsi (False Position)
1 Step
| n | $a_n$ | $b_n$ | $x_n$ | $f(x_n)$ |
|---|---|---|---|---|
| 0 | -100 | 100 |
Method Fails
$x_n=\frac{-100-100}{9999-9999}*9999$ Division by Zero
2.3 Secant
1 Step
| n | $x_{n-1}$ | $x_n$ | $x_{n+1}$ | $f(x_{n+1})$ |
|---|---|---|---|---|
| 1 | -100 | 100 |
Method Fails
$x_n=\frac{100–100}{9999-9999}*9999$ Division by Zero
2.4 Newton
11 Steps
| n | $x_{n-1}$ | $f(x_{n-1})$ | $f’(x_{n-1})$ | $x_{n+1}$ |
|---|---|---|---|---|
| 1 | $-100.000000$ | $9999.000000$ | $-200.000000$ | $\frac{-10001}{200}=-50.005000$ |
| 2 | $\frac{-10001}{200}=-50.005000$ | $\frac{99980001}{40000}=2499.500025$ | $\frac{-10001}{100}=-100.010000$ | $-25.012499$ |
| 3 | $-25.012499$ | $624.625106$ | $-50.024998$ | $-12.526240$ |
| 4 | $-12.526240$ | $155.906676$ | $-25.052479$ | $-6.303036$ |
| 5 | $-6.303036$ | $38.728262$ | $-12.606072$ | $-3.230845$ |
| 6 | $-3.230845$ | $9.438358$ | $-6.461690$ | $-1.770181$ |
| 7 | $-1.770181$ | $2.133540$ | $-3.540361$ | $-1.167547$ |
| 8 | $-1.167547$ | $0.363167$ | $-2.335095$ | $-1.012022$ |
| 9 | $-1.012022$ | $0.024188$ | $-2.024044$ | $-1.000071$ |
| 10 | $-1.000071$ | $0.000143$ | $-2.000143$ | $-1.000000$ |
| 11 | $-1.000000$ | $0.000000$ | $-2.000000$ | $-1.000000$ |
Solution
$x=-1.000000$
$f(x)=0.000000$
3 Example 3 - [-2, 0.333…]
$f(x)=x^2 - 1$ where $-2\leq x\leq \frac{1}{3}$
3.1 Bisection
1 Step
| n | $a_n$ | $b_n$ | $x_n$ | $f(x_n)$ |
|---|---|---|---|---|
| 0 | $-10.000000$ | $\frac{1}{3}=0.333333$ |
3.1.1 Method Failed
$f(b_0) < 0$ - Invariant violated!
3.2 Regula Falsi (False Position)
100 Steps
| n | $a_n$ | $b_n$ | $x_n$ | $f(x_n)$ |
|---|---|---|---|---|
| 0 | -10 | 1/3 | ||
| 1 | $-10.000000$ | $\frac{7}{29}=0.241379$ | $\frac{7}{29}=0.241379$ | $\frac{-792}{841}=-0.941736$ |
| 2 | $-10.000000$ | $\frac{41}{283}=0.144876$ | $\frac{41}{283}=0.144876$ | $\frac{-78408}{80089}=-0.979011$ |
| 3 | $-10.000000$ | $\frac{127}{2789}=0.045536$ | $\frac{127}{2789}=0.045536$ | $\frac{-7762392}{7778521}=-0.997926$ |
| 4 | $-10.000000$ | $\frac{-1519}{27763}=-0.054713$ | $\frac{-1519}{27763}=-0.054713$ | $-0.997006$ |
| 5 | $-10.000000$ | $\frac{-42953}{279149}=-0.153871$ | $\frac{-42953}{279149}=-0.153871$ | $-0.976324$ |
| 6 | $-10.000000$ | $\frac{-708679}{2834443}=-0.250024$ | $\frac{-708679}{2834443}=-0.250024$ | $-0.937488$ |
| 7 | $-10.000000$ | $\frac{-9921233}{29053109}=-0.341486$ | $\frac{-9921233}{29053109}=-0.341486$ | $-0.883387$ |
| 8 | $-10.000000$ | $-0.426908$ | $-0.426908$ | $-0.817750$ |
| 9 | $-10.000000$ | $-0.505335$ | $-0.505335$ | $-0.744637$ |
| 10 | $-10.000000$ | $-0.576216$ | $-0.576216$ | $-0.667975$ |
| 11 | $-10.000000$ | $-0.639375$ | $-0.639375$ | $-0.591200$ |
| 12 | $-10.000000$ | $-0.694942$ | $-0.694942$ | $-0.517056$ |
| 13 | $-10.000000$ | $-0.743288$ | $-0.743288$ | $-0.447523$ |
| 14 | $-10.000000$ | $-0.784944$ | $-0.784944$ | $-0.383863$ |
| 15 | $-10.000000$ | $-0.820536$ | $-0.820536$ | $-0.326720$ |
| 16 | $-10.000000$ | $-0.850731$ | $-0.850731$ | $-0.276257$ |
| 17 | $-10.000000$ | $-0.876191$ | $-0.876191$ | $-0.232290$ |
| 18 | $-10.000000$ | $-0.897548$ | $-0.897548$ | $-0.194407$ |
| 19 | $-10.000000$ | $-0.915388$ | $-0.915388$ | $-0.162065$ |
| 20 | $-10.000000$ | $-0.930235$ | $-0.930235$ | $-0.134663$ |
| 21 | $-10.000000$ | $-0.942555$ | $-0.942555$ | $-0.111589$ |
| 22 | $-10.000000$ | $-0.952753$ | $-0.952753$ | $-0.092262$ |
| 23 | $-10.000000$ | $-0.961177$ | $-0.961177$ | $-0.076139$ |
| 24 | $-10.000000$ | $-0.968123$ | $-0.968123$ | $-0.062738$ |
| 25 | $-10.000000$ | $-0.973843$ | $-0.973843$ | $-0.051630$ |
| 26 | $-10.000000$ | $-0.978548$ | $-0.978548$ | $-0.042444$ |
| 27 | $-10.000000$ | $-0.982414$ | $-0.982414$ | $-0.034863$ |
| 28 | $-10.000000$ | $-0.985588$ | $-0.985588$ | $-0.028616$ |
| 29 | $-10.000000$ | $-0.988193$ | $-0.988193$ | $-0.023474$ |
| 30 | $-10.000000$ | $-0.990329$ | $-0.990329$ | $-0.019248$ |
| 31 | $-10.000000$ | $-0.992081$ | $-0.992081$ | $-0.015776$ |
| 32 | $-10.000000$ | $-0.993516$ | $-0.993516$ | $-0.012926$ |
| 33 | $-10.000000$ | $-0.994692$ | $-0.994692$ | $-0.010588$ |
| 34 | $-10.000000$ | $-0.995655$ | $-0.995655$ | $-0.008672$ |
| 35 | $-10.000000$ | $-0.996443$ | $-0.996443$ | $-0.007100$ |
| 36 | $-10.000000$ | $-0.997089$ | $-0.997089$ | $-0.005813$ |
| 37 | $-10.000000$ | $-0.997618$ | $-0.997618$ | $-0.004759$ |
| 38 | $-10.000000$ | $-0.998050$ | $-0.998050$ | $-0.003895$ |
| 39 | $-10.000000$ | $-0.998405$ | $-0.998405$ | $-0.003188$ |
| 40 | $-10.000000$ | $-0.998695$ | $-0.998695$ | $-0.002609$ |
| 41 | $-10.000000$ | $-0.998932$ | $-0.998932$ | $-0.002135$ |
| 42 | $-10.000000$ | $-0.999126$ | $-0.999126$ | $-0.001747$ |
| 43 | $-10.000000$ | $-0.999285$ | $-0.999285$ | $-0.001430$ |
| 44 | $-10.000000$ | $-0.999415$ | $-0.999415$ | $-0.001170$ |
| 45 | $-10.000000$ | $-0.999521$ | $-0.999521$ | $-0.000957$ |
| 46 | $-10.000000$ | $-0.999608$ | $-0.999608$ | $-0.000783$ |
| 47 | $-10.000000$ | $-0.999679$ | $-0.999679$ | $-0.000641$ |
| 48 | $-10.000000$ | $-0.999738$ | $-0.999738$ | $-0.000525$ |
| 49 | $-10.000000$ | $-0.999785$ | $-0.999785$ | $-0.000429$ |
| 50 | $-10.000000$ | $-0.999824$ | $-0.999824$ | $-0.000351$ |
| 51 | $-10.000000$ | $-0.999856$ | $-0.999856$ | $-0.000287$ |
| 52 | $-10.000000$ | $-0.999882$ | $-0.999882$ | $-0.000235$ |
| 53 | $-10.000000$ | $-0.999904$ | $-0.999904$ | $-0.000192$ |
| 54 | $-10.000000$ | $-0.999921$ | $-0.999921$ | $-0.000157$ |
| 55 | $-10.000000$ | $-0.999936$ | $-0.999936$ | $-0.000129$ |
| 56 | $-10.000000$ | $-0.999947$ | $-0.999947$ | $-0.000105$ |
| 57 | $-10.000000$ | $-0.999957$ | $-0.999957$ | $-0.000086$ |
| 58 | $-10.000000$ | $-0.999965$ | $-0.999965$ | $-0.000071$ |
| 59 | $-10.000000$ | $-0.999971$ | $-0.999971$ | $-0.000058$ |
| 60 | $-10.000000$ | $-0.999976$ | $-0.999976$ | $-0.000047$ |
| 61 | $-10.000000$ | $-0.999981$ | $-0.999981$ | $-0.000039$ |
| 62 | $-10.000000$ | $-0.999984$ | $-0.999984$ | $-0.000032$ |
| 63 | $-10.000000$ | $-0.999987$ | $-0.999987$ | $-0.000026$ |
| 64 | $-10.000000$ | $-0.999989$ | $-0.999989$ | $-0.000021$ |
| 65 | $-10.000000$ | $-0.999991$ | $-0.999991$ | $-0.000017$ |
| 66 | $-10.000000$ | $-0.999993$ | $-0.999993$ | $-0.000014$ |
| 67 | $-10.000000$ | $-0.999994$ | $-0.999994$ | $-0.000012$ |
| 68 | $-10.000000$ | $-0.999995$ | $-0.999995$ | $-0.000009$ |
| 69 | $-10.000000$ | $-0.999996$ | $-0.999996$ | $-0.000008$ |
| 70 | $-10.000000$ | $-0.999997$ | $-0.999997$ | $-0.000006$ |
| 71 | $-10.000000$ | $-0.999997$ | $-0.999997$ | $-0.000005$ |
| 72 | $-10.000000$ | $-0.999998$ | $-0.999998$ | $-0.000004$ |
| 73 | $-10.000000$ | $-0.999998$ | $-0.999998$ | $-0.000003$ |
| 74 | $-10.000000$ | $-0.999999$ | $-0.999999$ | $-0.000003$ |
| 75 | $-10.000000$ | $-0.999999$ | $-0.999999$ | $-0.000002$ |
| 76 | $-10.000000$ | $-0.999999$ | $-0.999999$ | $-0.000002$ |
| 77 | $-10.000000$ | $-0.999999$ | $-0.999999$ | $-0.000002$ |
| 78 | $-10.000000$ | $-0.999999$ | $-0.999999$ | $-0.000001$ |
| 79 | $-10.000000$ | $-0.999999$ | $-0.999999$ | $-0.000001$ |
| 80 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000001$ |
| 81 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000001$ |
| 82 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000001$ |
| 83 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 84 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 85 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 86 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 87 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 88 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 89 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 90 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 91 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 92 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 93 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 94 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 95 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 96 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 97 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 98 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
| 99 | $-10.000000$ | $-1.000000$ | $-1.000000$ | $-0.000000$ |
Solution
$x=-1.000000$
$f(x)=-0.000000$
3.3 Secant
10 Steps
| n | $x_{n-1}$ | $x_n$ | $x_{n+1}$ | $f(x_{n+1})$ |
|---|---|---|---|---|
| 1 | -10 | 1/3 | ||
| 2 | $\frac{1}{3}=0.333333$ | $-10.000000$ | $\frac{7}{29}=0.241379$ | $\frac{-792}{841}=-0.941736$ |
| 3 | $\frac{7}{29}=0.241379$ | $\frac{1}{3}=0.333333$ | $\frac{47}{25}=1.880000$ | $\frac{1584}{625}=2.534400$ |
| 4 | $\frac{47}{25}=1.880000$ | $\frac{7}{29}=0.241379$ | $\frac{527}{769}=0.685306$ | $\frac{-313632}{591361}=-0.530356$ |
| 5 | $\frac{527}{769}=0.685306$ | $\frac{47}{25}=1.880000$ | $\frac{21997}{24659}=0.892048$ | $-0.204251$ |
| 6 | $\frac{21997}{24659}=0.892048$ | $\frac{527}{769}=0.685306$ | $\frac{15277595}{14955493}=1.021537$ | $0.043539$ |
| 7 | $\frac{15277595}{14955493}=1.021537$ | $\frac{21997}{24659}=0.892048$ | $0.998785$ | $-0.002429$ |
| 8 | $0.998785$ | $\frac{15277595}{14955493}=1.021537$ | $0.999987$ | $-0.000026$ |
| 9 | $0.999987$ | $0.998785$ | $1.000000$ | $0.000000$ |
| 10 | $1.000000$ | $0.999987$ | $1.000000$ | $-0.000000$ |
Solution
$x=1.000000$
$f(x)=-0.000000$
3.4 Newton
7 Steps
| n | $x_{n-1}$ | $f(x_{n-1})$ | $f’(x_{n-1})$ | $x_{n+1}$ |
|---|---|---|---|---|
| 1 | $-10.000000$ | $99.000000$ | $-20.000000$ | $\frac{-101}{20}=-5.050000$ |
| 2 | $\frac{-101}{20}=-5.050000$ | $\frac{9801}{400}=24.502500$ | $\frac{-101}{10}=-10.100000$ | $\frac{-10601}{4040}=-2.624010$ |
| 3 | $\frac{-10601}{4040}=-2.624010$ | $\frac{96059601}{16321600}=5.885428$ | $\frac{-10601}{2020}=-5.248020$ | $-1.502553$ |
| 4 | $-1.502553$ | $1.257666$ | $-3.005106$ | $-1.084043$ |
| 5 | $-1.084043$ | $0.175150$ | $-2.168087$ | $-1.003258$ |
| 6 | $-1.003258$ | $0.006526$ | $-2.006516$ | $-1.000005$ |
| 7 | $-1.000005$ | $0.000011$ | $-2.000011$ | $-1.000000$ |
Solution
$x=-1.000005$
$f(x)=0.000011$