1 Introduction
2 Adams-Bashforth methods
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n = 1$$ y_{k}=y_{k-1}+h_{k}f_{k-1}, $$(10)
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n = 2where$$ y_{k}=y_{k-1}+h_{k}\gamma_{2k}, $$(11)$$ \gamma_{2k}=f_{k-1}+\frac{1}{2}\frac{h_{k}}{h_{k-1}}\left( f_{k-1}-f_{k-2}\right) , $$
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n = 3where$$ y_{k}=y_{k-1}+h_{k}\gamma_{3k}, $$(12)$$ \begin{array}{@{}rcl@{}} \gamma_{3k} &=&\gamma_{2k}+\frac{1}{2}\left( 1-\frac{1}{3}\frac{h_{k}}{ h_{k}+h_{k-1}}\right) \frac{h_{k}}{h_{k-1}}\frac{h_{k}+h_{k-1}}{ h_{k-1}+h_{k-2}} \\ &&{\kern2.3pc}\times\left[ f_{k-1} -f_{k-2}-\frac{h_{k-1}}{h_{k-2}}\left( f_{k-2}-f_{k-3}\right) \right] , \end{array} $$
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n = 4where$$ y_{k}=y_{k-1}+h_{k}\gamma_{4k}, $$(13)$$ \begin{array}{@{}rcl@{}} \gamma_{4k} &=&\gamma_{3k} \\ &&+\left[ \frac{1}{2}\left( 1 - \frac{1}{3}\frac{h_{k}}{h_{k}+h_{k-1}}\right) - \frac{1}{6}\left( 1-\frac{1}{2}\frac{h_{k}}{h_{k}+h_{k-1}}\right) \frac{h_{k}}{h_{k}+h_{k-1}+h_{k-2}}\right] \\ &&{\kern.6pc}\times \frac{h_{k}}{h_{k-1}}\frac{h_{k}+h_{k-1}}{h_{k-1}+h_{k-2}}\frac{ h_{k}+h_{k-1}+h_{k-2}}{h_{k-1}+h_{k-2}+h_{k-3}} \\ &&{\kern.6pc}\times \left\{ f_{k-1}-f_{k-2}-\frac{h_{k-1}}{h_{k-2}}\left( f_{k-2}-f_{k-3}\right) \right. \\ &&{\kern1.5pc}\left. -\frac{h_{k-1}}{h_{k-2}}\frac{h_{k-1}+h_{k-2}}{h_{k-2}+h_{k-3}} \left[ f_{k-2}-f_{k-3}-\frac{h_{k-2}}{h_{k-3}}\left( f_{k-3}-f_{k-4}\right) \right] \right\}. \end{array} $$
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n = 1 (Euler’s method)$$ y_{k}=y_{k-1}+hf_{k-1}, $$
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n = 2$$ y_{k}=y_{k-1}+\frac{h}{2}\left( 3f_{k-1}-f_{k-2}\right) , $$
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n = 3$$ y_{k}=y_{k-1}+\frac{h}{12}\left( 23f_{k-1}-16f_{k-2}+5f_{k-3}\right) , $$
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n = 4$$ y_{k}=y_{k-1}+\frac{h}{24}\left( 55f_{k-1}-59f_{k-2}+37f_{k-3}-9f_{k-4}\right) . $$
3 Interval versions of Adams-Bashforth methods with variable step sizes
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$$ {\Delta}_{t}=\left\{ t\in \mathbb{R}:0\leq t\leq a\right\},\quad {\Delta}_{y}=\left\{ y \in \mathbb{R}:\underline{b}\leq y\leq \overline{b},\right\}, $$
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F (T,Y)—an interval extension of f(t,y), where an interval extension of the functionwe call a function$$ f:\mathbb{R}\times \mathbb{R}\supset {\Delta}_{t}\times {\Delta}_{y}\rightarrow \mathbb{R} $$such that$$ F:\mathbb{I}\mathbb{R}\times \mathbb{I}\mathbb{R}\supset \mathbb{I}{\Delta}_{t}\times \mathbb{I}{\Delta}_{y}\rightarrow \mathbb{I}\mathbb{R} $$where \(\mathbb {I}\mathbb {R}\) denotes the space of real intervals.$$ \left( t,y\right) \in \left( T,Y\right) \Rightarrow f\left( t,y\right) \in F\left( T,Y\right), $$
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Ψ (T,Y)—an interval extension of f(n) (t,y (t)) ≡ y(n+ 1) (t),
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the function F (T,Y) is monotonic with respect to inclusion, i.e.,$$ T_{1}\subset T_{2}\wedge Y_{1}\subset Y_{2}\Rightarrow F\left( T_{1},Y_{1}\right) \subset F\left( T_{2},Y_{2}\right), $$
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for each T ⊂Δt and for each Y ⊂Δy, there exists a constant Λ > 0 such thatwhere w (A) denotes the width of the interval A,$$ w\left( F\left( T,Y\right) \right) \leq {\Lambda} \left( w\left( T\right) +w\left( Y\right) \right) , $$
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the function Ψ (T,Y) is defined for all T ⊂Δt and Y ⊂Δy,
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the function Ψ (T,Y) is monotonic with respect to inclusion.
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n = 1$$ \begin{array}{@{}rcl@{}} Y_{k}&=&Y_{k-1}+h_{k}F_{k-1}\\ &&+\frac{{h_{k}^{2}}}{2}{\Psi} \left( T_{k-1}+\left[0,h_{k}\right] ,Y_{k-1}+\left[ 0,h_{k}\right] F\left( {\Delta}_{t},{\Delta}_{y}\right) \right), \end{array} $$(16)
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n = 2where$$ \begin{array}{@{}rcl@{}} &&Y_{k} = Y_{k-1}+h_{k}{\Gamma}_{2k}\\ &&{\kern1.7pc}+{h_{k}^{3}}g_{2}\left( k\right) {\Psi} \left( T_{k-1}+\left[ -h_{k-1},h_{k} \right] ,\right. \\ &&{\kern6.5pc}\left. Y_{k-1}+\left[ -h_{k-1},h_{k}\right] F\left( {\Delta}_{t},{\Delta}_{y}\right) \right), \end{array} $$(17)$$ \begin{array}{@{}rcl@{}} &{\Gamma}_{2k} = F_{k-1}+\frac{1}{2}\frac{h_{k}}{h_{k-1}}\left( F_{k-1}-F_{k-2}\right), \\ &{\kern-.1pc}g_{2}\left( k\right) = \frac{1}{2}\left( \frac{1}{3}+\frac{1}{2}\frac{ h_{k-1}}{h_{k}}\right), \end{array} $$
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n = 3where$$ \begin{array}{@{}rcl@{}} Y_{k} &=& Y_{k-1}+h_{k}{\Gamma}_{3k} \\ &&+{h_{k}^{4}}g_{3}\left( k\right) {\Psi} \left( T_{k-1}+\left[ -h_{k-2}-h_{k-1},h_{k}\right] ,\right. \\ &&{\kern4.5pc}\left. Y_{k-1}+\left[ -h_{k-2}-h_{k-1},h_{k}\right] F\left( {\Delta}_{t},{\Delta}_{y}\right) \right), \end{array} $$(18)$$ \begin{array}{@{}rcl@{}} {\Gamma}_{3k} &=& {\Gamma}_{2k} \\ &&+\frac{1}{2}\left( 1-\frac{1}{3}\frac{h_{k}}{h_{k}+h_{k-1}}\right) \frac{ h_{k}}{h_{k-1}}\frac{h_{k}+h_{k-1}}{h_{k-1}+h_{k-2}} \\ &&{\kern.3pc}\times\left[ F_{k-1} -F_{k-2}-\frac{h_{k-1}}{h_{k-2}}\left( F_{k-2}-F_{k-3}\right)\right], \end{array} $$$$ g_{3}\left( k\right) =\frac{1}{6}\left( \frac{1}{4}+\frac{1}{3}\frac{ 2h_{k-1}+h_{k-2}}{h_{k}}+\frac{1}{2}\frac{h_{k-1}}{h_{k}}\frac{ h_{k-1}+h_{k-2}}{h_{k}}\right) , $$
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n = 4where$$ \begin{array}{@{}rcl@{}} Y_{k} &=& Y_{k-1}+h_{k}{\Gamma}_{4k} \\ &&+{h_{k}^{5}}g_{4}\left( k\right) {\Psi} \left( T_{k-1}+\left[ -h_{k-3}-h_{k-2}-h_{k-1},h_{k}\right] ,\right. \\ &&{\kern4.5pc}\left. Y_{k-1}+\left[ -h_{k-3}-h_{k-2}-h_{k-1},h_{k}\right] F\left( {\Delta}_{t},{\Delta}_{y}\right) \right), \end{array} $$(19)$$ \begin{array}{@{}rcl@{}} {\Gamma}_{4k} &=& {\Gamma}_{3k}\\ &&+\left[ \frac{1}{2}\left( 1-\frac{1}{3}\frac{h_{k}}{h_{k}+h_{k-1}}\right) - \frac{1}{6}\left( 1 - \frac{1}{2}\frac{h_{k}}{h_{k}+h_{k-1}}\right) \frac{h_{k}}{h_{k}+h_{k-1}+h_{k-2}}\right] \\ &&{\kern10pt} \times \frac{h_{k}}{h_{k-1}}\frac{h_{k}+h_{k-1}}{h_{k-1}+h_{k-2}}\frac{ h_{k}+h_{k-1}+h_{k-2}}{h_{k-1}+h_{k-2}+h_{k-3}} \\ &&{\kern10pt} \times \left\{ F_{k-1}-F_{k-2}-\frac{h_{k-1}}{h_{k-2}}\left( F_{k-2}-Ff_{k-3}\right) \right. \\ &&{\kern1.7pc}\left. -\frac{h_{k-1}}{h_{k-2}}\frac{h_{k-1}+h_{k-2}}{h_{k-2}+h_{k-3}} \left[ F_{k-2}-F_{k-3}-\frac{h_{k-2}}{h_{k-3}}\left( F_{k-3}-F_{k-4}\right) \right] \right\}, \end{array} $$$$ \begin{array}{@{}rcl@{}} g_{4}\left( k\right) &=& \frac{1}{24}\left( \frac{1}{5}+\frac{1}{4}\frac{3h_{k-1}+2h_{k-2}+h_{k-3}}{h_{k}}\right.\\ &&{\kern16pt}+\frac{1}{3}\frac{h_{k-1}}{h_{k}}\frac{3h_{k-1}+4h_{k-2}+2h_{k-3}}{h_{k}}+ \frac{1}{3}\frac{h_{k-2}}{h_{k}}\frac{h_{k-2}+h_{k-3}}{h_{k}} \\ &&{\kern16pt}+\left. \frac{1}{2}\frac{h_{k-1}}{h_{k}}\frac{h_{k-1}+h_{k-2}}{h_{k}}\frac{ h_{k-1}+h_{k-2}+h_{k-3}}{h_{k}}\right). \end{array} $$
4 Using variable step sizes
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n = 1$$ p_{2}\left( h_{k}\right) =\frac{{h_{k}^{2}}}{12}w\left( {\Psi} \left( {\Delta}_{t},{\Delta}_{y}\right) \right) +h_{k}{\Lambda} w\left( Y_{k-1}\right) +w\left( Y_{k-1}\right) -eps=0, $$
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n = 2where$$ \begin{array}{@{}rcl@{}} &&p_{3}\left( h_{k}\right) =\frac{{h_{k}^{2}}}{12}\left( 2h_{k}+3h_{k-1}\right) w\left( {\Psi} \left( {\Delta}_{t},{\Delta}_{y}\right) \right)\\ &&{\kern3.5pc}+{\Lambda} q_{2}\left( h_{k}\right) \left[ 2w\left( Y_{k-1}\right) +w\left( Y_{k-2}\right) \right] +w\left( Y_{k-1}\right) - eps=0, \end{array} $$$$ q_{2}\left( h_{k}\right) =h_{k}\cdot \max \left\{ 1,\frac{h_{k}}{h_{k-1}} \right\} , $$
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n = 3where$$ \begin{array}{@{}rcl@{}} p_{4}\left( h_{4}\right) &=&\frac{{h_{k}^{2}}}{72}\left[ 3{h_{k}^{2}}+4h_{k}\left( 2h_{k-1}+h_{k-2}\right) \right. \\ &&{\kern1.2pc}\left. +6h_{k-1}\left( h_{k-1}+h_{k-2}\right) \right] w\left( {\Psi} \left( {\Delta}_{t},{\Delta}_{y}\right) \right) \\ &&+{\Lambda} q_{3}\left( h_{k}\right) \left[ 3w\left( Y_{k-1}\right) +2w\left( Y_{k-2}\right) +w\left( Y_{k-3}\right) \right] \\ &&+ w\left( Y_{k-1}\right) - eps=0, \end{array} $$$$ q_{3}\left( h_{k}\right) =h_{k}\cdot \max \left\{ 1,\frac{h_{k}}{h_{k-1}}, \frac{h_{k}\left( h_{k}+h_{k-1}\right) }{h_{k-1}h_{k-2}}\right\} , $$
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n = 4where$$ \begin{array}{@{}rcl@{}} p_{5}\left( h_{4}\right) &=&\frac{{h_{k}^{2}}}{1440}\left\{ 12{h_{k}^{3}}+15{h_{k}^{2}}\left( 3h_{k-1}+2h_{k-2}+h_{k-3}\right) \right.\\ &&{\kern2pc}+20h_{k}\left[ h_{k-1}\left( h_{k-1}+h_{k-2}\right) \right. \\ &&{\kern4.7pc}\left. +\left( 2h_{k-1}+h_{k-2}\right) \left( h_{k-1}+h_{k-2}+h_{k-3}\right) \right] \\ &&{\kern2pc}\left. +30h_{k-1}\left( h_{k-1}+h_{k-2}\right) \left( h_{k-1}+h_{k-2}+h_{k-3}\right) \right\} \\ &&{\kern2pc}\times w\left( {\Psi} \left( {\Delta}_{t},{\Delta}_{y}\right) \right) \\ &&+{\Lambda} q_{4}\left( h_{4}\right) \left[ 4w\left( Y_{k-1}\right) +3w\left( Y_{k-2}\right) +2w\left( Y_{k-3}\right) +w\left( Y_{k-4}\right) \right] \\ &&+w\left( Y_{k-1}\right) -eps=0, \end{array} $$$$ \begin{array}{@{}rcl@{}} q_{4}\left( h_{k}\right) &=& h_{k}\cdot \max \left\{ 1, \frac{h_{k}}{h_{k-1}}, \frac{h_{k}\left( h_{k}+h_{k-1}\right) }{h_{k-1}h_{k-2}}, \right. \\ &&{\kern3pc}\frac{h_{k}\left( h_{k}+h_{k-1}\right) \left( h_{k}+h_{k-1}+h_{k-2}\right) }{ h_{k-1}h_{k-2}\left( h_{k-2}+h_{k-3}\right) }, \\ &&{\kern2.7pc}\left. \frac{h_{k}\left( h_{k}+h_{k-1}\right) \left( h_{k}+h_{k-1}+h_{k-2}\right) }{\left( h_{k-1}+h_{k-2}\right) h_{k-2}h_{k-3}} \right\}. \end{array} $$
5 A note on a system of differential equations
6 Numerical examples
Method | \(Y_{k}\) | Width | CPU time |
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(s) | |||
(16) | [ 2.7169134040046282E + 0000, 2.7235393714205552E + 0000] | \(\approx 6.63 \cdot 10^{-3}\) | 0.042 |
(17) | [ 2.7179091592957537E + 0000, 2.7187125466868537E + 0000] | \(\approx 8.03 \cdot 10^{-4}\) | 0.054 |
(18) | [ 2.7182298899088899E + 0000, 2.7183323624455116E + 0000] | \(\approx 1.02 \cdot 10^{-4}\) | 0.055 |
(19) | [ 2.7182739085121117E + 0000, 2.7182894852166692E + 0000] | \(\approx 1.56 \cdot 10^{-5}\) | 0.076 |
Method | k | \(t_{k}\) | \(Y_{k}\) | Width |
---|---|---|---|---|
(16) | 1 | \(\approx \) 0.0002 | [ 1.0001078385871569E + 0000, 1.0001078385888622E + 0000] | \(\approx 1.71 \cdot 10^{-12}\) |
15852 | \(\approx \) 1.5766 | [ 2.1996396812768524E + 0000, 2.1996396912768528E + 0000] | \(< 1.00 \cdot 10^{-8}\) | |
(17) | 2 | \(\approx \) 0.0815 | [ 1.0415997191016650E + 0000, 1.0415997197462058E + 0000] | \(\approx 6.45 \cdot 10^{-10}\) |
278 | \(\approx \) 0.8333 | [ 1.5168547254474315E + 0000, 1.5168547354474316E + 0000] | \(< 1.00 \cdot 10^{-8}\) | |
(18) | 3 | \(\approx \) 0.1597 | [ 1.0831027071336879E + 0000, 1.0831027083961128E + 0000] | \(\approx 1.26 \cdot 10^{-9}\) |
76 | \(\approx \) 0.6178 | [ 1.3619093684910934E + 0000, 1.3619093784910935E + 0000] | \(< 1.00 \cdot 10^{-8}\) | |
(19) | 4 | \(\approx \) 0.2529 | [ 1.1348046916372604E + 0000, 1.1348046936371020E + 0000] | \(\approx 2.00 \cdot 10^{-9}\) |
71 | \(\approx \) 0.6259 | [ 1.3674527147839997E + 0000, 1.3674527247839998E + 0000] | \(< 1.00 \cdot 10^{-8}\) |
Method | k | \(t_{k}\) | \(Y_{k}\) | Width |
---|---|---|---|---|
(16) | 3190 | \(\approx \) 0.60000 | [ 1.3498581914855010E + 0000, 1.3498581958390336E + 0000] | \(\approx 4.35 \cdot 10^{-9}\) |
3191 | 0.60000 | [ 1.3498588069670051E + 0000, 1.3498588113205398E + 0000] | \(\approx 4.35 \cdot 10^{-9}\) | |
(17) | 135 | \(\approx \) 0.59891 | [ 1.3491258950165315E + 0000, 1.3491259025297920E + 0000] | \(\approx 7.51 \cdot 10^{-9}\) |
136 | 0.60000 | [ 1.3498588036394939E + 0000, 1.3498588111590695E + 0000] | \(\approx 7.52 \cdot 10^{-9}\) | |
(18) | 32 | \(\approx \) 0.59894 | [ 1.3491410173457222E + 0000, 1.3491410270971693E + 0000] | \(\approx 9.75 \cdot 10^{-9}\) |
33 | 0.60000 | [ 1.3498588016695426E + 0000, 1.3498588114278859E + 0000] | \(\approx 9.76 \cdot 10^{-9}\) | |
(19) | 15 | \(\approx \) 0.59880 | [ 1.3490512149587291E + 0000, 1.3490512245357438E + 0000] | \(\approx 9.58 \cdot 10^{-9}\) |
16 | 0.60000 | [ 1.3498588016932220E + 0000, 1.3498588112774857E + 0000] | \(\approx 9.58 \cdot 10^{-9}\) |
k | \(t_{k}\) | \(Y_{k}\) |
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0 | 0 | [\(\underline {4.0000000000000000\mathrm {E}+0000}\), \(\overline {4.0000000000000000\mathrm {E}+0000}\)] |
1 | 0.081746227283888863 | [\(\underline {4.0801194662264153\mathrm {E}+0000}\), \(\overline {4.0801194662264155\mathrm {E}+0000}\)] |
2 | 0.163492454567777736 | [\(\underline {4.1571485011046702\mathrm {E}+0000}\), \(\overline {4.1571485011046705\mathrm {E}+0000}\)] |
3 | 0.245238681851666599 | [\(\underline {4.2313071906453238\mathrm {E}+0000}\), \(\overline {4.2313071906453243\mathrm {E}+0000}\)] |
k | \(t_{k}\) | \(Y_{k}\) | Width |
---|---|---|---|
50 | \(\approx \) 0.69774 | [ 4.5975465221505685E + 0000, 4.5975465243497029E + 0000] | \(\approx 2.20 \cdot 10^{-9}\) |
100 | \(\approx \) 1.14965 | [ 4.9030146239176995E + 0000, 4.9030146292963836E + 0000] | \(\approx 5.38 \cdot 10^{-9}\) |
150 | \(\approx \) 1.47461 | [ 5.0930266442612574E + 0000, 5.0930266533191654E + 0000] | \(\approx 9.06 \cdot 10^{-9}\) |
200 | \(\approx \) 1.54340 | [ 5.1304617719687623E + 0000, 5.1304617819135613E + 0000] | \(\approx 9.94 \cdot 10^{-9}\) |
250 | \(\approx \) 1.54735 | [ 5.1325793100365724E + 0000, 5.1325793200333748E + 0000] | \(< 1.00 \cdot 10^{-8}\) |
464 | \(\approx \) 1.54759 | [ 5.1327090107108448E + 0000, 5.1327090207108449E + 0000] | \(< 1.00 \cdot 10^{-8}\) |
k | \(t_{k}\) | \(Y_{k}\) | Width |
---|---|---|---|
200 | \(\approx \) 0.55405 | [ 4.4886273748015889E + 0000, 4.4886273748017744E + 0000] | \(\approx 1.85 \cdot 10^{-13}\) |
400 | \(\approx \) 0.85821 | [ 4.7120930249268474E + 0000, 4.7120930249272235E + 0000] | \(\approx 3.76 \cdot 10^{-13}\) |
600 | \(\approx \) 1.13809 | [ 4.8958332122088699E + 0000, 4.8958332122095143E + 0000] | \(\approx 6.44 \cdot 10^{-13}\) |
800 | \(\approx \) 1.36239 | [ 5.0299271956108516E + 0000, 5.0299271956117965E + 0000] | \(\approx 9.45 \cdot 10^{-13}\) |
900 | \(\approx \) 1.39889 | [ 5.0507330026032329E + 0000, 5.0507330026042326E + 0000] | \(< 1.00 \cdot 10^{-12}\) |
955 | \(\approx \) 1.39915 | [ 5.0508805112730600E + 0000, 5.0508805112740601E + 0000] | \(< 1.00 \cdot 10^{-12}\) |
k | \(t_{k}\) | \(Y_{k}\) | Width |
---|---|---|---|
10 | \(\approx \) 0.67182 | [ 4.5783575612188736E + 0000, 4.5783702917087973E + 0000] | \(\approx 1.27 \cdot 10^{-5}\) |
20 | \(\approx \) 1.22301 | [ 4.9478886689807839E + 0000, 4.9479228693528902E + 0000] | \(\approx 3.42 \cdot 10^{-5}\) |
30 | \(\approx \) 1.62289 | [ 5.1725533292643105E + 0000, 5.1726121849592280E + 0000] | \(\approx 5.89 \cdot 10^{-5}\) |
100 | \(\approx \) 2.07277 | [ 5.3895608440740060E + 0000, 5.3896595445228829E + 0000] | \(\approx 9.87 \cdot 10^{-5}\) |
150 | \(\approx \) 2.08316 | [ 5.3941735254856961E + 0000, 5.3942734009330229E + 0000] | \(\approx 9.99 \cdot 10^{-5}\) |
632 | \(\approx \) 2.08426 | [ 5.3946582071369410E + 0000, 5.3947582071369411E + 0000] | \(< 1.00 \cdot 10^{-4}\) |
k | \(t_{k}\) | \(Y_{s} = Y_{sk}\) |
---|---|---|
0 | 0 | \(Y_{1} = \) [ \(\underline {0.0000000000000000\mathrm {E}+0000}\), \(\hspace {0.2cm}\overline {0.0000000000000000\mathrm {E}+0000}\)] |
\(Y_{2} =\) [ \(\underline {5.2359877559829887\mathrm {E}-0001}\), \(\hspace {0.2cm}\overline {5.2359877559829888\mathrm {E}-0001}\)] | ||
1 | 0.0001 | \(Y_{1} =\) [\(\underline {-5.1347498487965657\mathrm {E}-0004},\)\(\overline {-5.1347498487965656\mathrm {E}-0004}\)] |
\(Y_{2} = \) [ \(\underline {5.2359874992454941\mathrm {E}-0001}\), \(\hspace {0.2cm}\overline {5.2359874992454942\mathrm {E-}0001}\)] | ||
2 | 0.0002 | \(Y_{1} = \)\([\underline {-1.0269499194046190\mathrm {E}-0003}\), \(\overline {-1.0269499194046189\mathrm {E}-0003}\)] |
\(Y_{2} = \) [ \(\underline {5.2359867290330357\mathrm {E}-0001}\), \(\hspace {0.2cm}\overline {5.2359867290330358\mathrm {E}-0001}\)] |
k | \(t_{k}\) | \(Y_{k}\) | Width |
---|---|---|---|
20 | ≈ 0.052266 | \(Y_{1} = [\)− 2.6717739871166114E − 0001,− 2.6717739518323742E − 0001] | \(\approx 3.53 \cdot 10^{-9}\) |
\(Y_{2} = [\) 5.1660096723952543E − 0001, 5.1660096833861906E − 0001] | \(\approx 1.10 \cdot 10^{-9}\) | ||
40 | ≈ 0.098329 | \(Y_{1} = [\)− 4.9695523554187293E − 0001,− 4.9695522814330236E − 0001] | \(\approx 7.40 \cdot 10^{-9}\) |
\(Y_{2} = [\) 4.9897124759486593E − 0001, 4.9897124992870939E − 0001] | \(\approx 2.33 \cdot 10^{-9}\) | ||
60 | ≈ 0.123971 | \(Y_{1} = [\)− 6.2069000044409622E − 0001,− 6.2068999060901059E − 0001] | \(\approx 9.84 \cdot 10^{-9}\) |
\(Y_{2} = [\) 4.8463438814236487E − 0001, 4.8463439125965848E − 0001] | \(\approx 3.12 \cdot 10^{-9}\) | ||
80 | ≈ 0.125890 | \(Y_{1} = [\)− 6.2979775595258544E − 0001,− 6.2979774595297822E − 0001] | \(< 1.00 \cdot 10^{-8}\) |
\(Y_{2} = [\) 4.8343471268438440E − 0001, 4.8343471585460307E − 0001] | \(\approx 3.17 \cdot 10^{-9}\) | ||
122 | ≈ 0.125895 | \(Y_{1} = [\)− 6.2982173628611325E − 0001,− 6.2982172628611324E − 0001] | \(< 1.00 \cdot 10^{-8}\) |
\(Y_{2} = [\) 4.8343152696498447E − 0001, 4.8343153013532947E − 0001] | \(\approx 3.17 \cdot 10^{-9}\) |
eps | k | \(t_{\max }\) | CPU time |
---|---|---|---|
(s) | |||
10− 2 | 842 | \(\approx \) 2.612638 | 4.411 |
10− 3 | 731 | \(\approx \) 2.341260 | 3.751 |
10− 4 | 632 | \(\approx \) 2.084259 | 3.333 |
10− 5 | 557 | \(\approx \) 1.880208 | 2.816 |
10− 6 | 503 | \(\approx \) 1.731451 | 2.502 |
10− 7 | 472 | \(\approx \) 1.624865 | 2.352 |
10− 8 | 464 | \(\approx \) 1.547588 | 2.295 |
10− 9 | 489 | \(\approx \) 1.490991 | 2.433 |
10− 10 | 559 | \(\approx \) 1.450758 | 2.704 |
10− 11 | 705 | \(\approx \) 1.423210 | 3.496 |
10− 12 | 955 | \(\approx \) 1.399150 | 4.758 |
10− 13 | 1312 | \(\approx \) 1.321706 | 6.575 |