Numerical Methods - 3.9 Acceleration of Convergence
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 3.9.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{t_{n}-\sigma}{s_{n}-\sigma} = 0}
\lim_{n\to\infty}\frac{t_{n}-\sigma}{s_{n}-\sigma} = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | limit((t[n]- sigma)/(s[n]- sigma), n = infinity) = 0 |
Limit[Divide[Subscript[t, n]- \[Sigma],Subscript[s, n]- \[Sigma]], n -> Infinity, GenerateConditions->None] == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle t_{n} = s_{n}-\frac{(\Delta s_{n})^{2}}{\Delta^{2}s_{n}}}
t_{n} = s_{n}-\frac{(\Delta s_{n})^{2}}{\Delta^{2}s_{n}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | t[n] = s[n]-((Delta*s[n])^(2))/((Delta)^(2)* s[n]) |
Subscript[t, n] == Subscript[s, n]-Divide[(\[CapitalDelta]*Subscript[s, n])^(2),\[CapitalDelta]^(2)* Subscript[s, n]] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma} = \rho}
\lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma} = \rho |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \abs{\rho} < 1} | limit((s[n + 1]- sigma)/(s[n]- sigma), n = infinity) = rho |
Limit[Divide[Subscript[s, n + 1]- \[Sigma],Subscript[s, n]- \[Sigma]], n -> Infinity, GenerateConditions->None] == \[Rho] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle t_{n,2k} = \frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})}}
t_{n,2k} = \frac{H_{k+1}(s_{n})}{H_{k}(\Delta^{2}s_{n})} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | t[n , 2*k] = (H[k + 1](s[n]))/(H[k]((Delta)^(2)* s[n])) |
Subscript[t, n , 2*k] == Divide[Subscript[H, k + 1][Subscript[s, n]],Subscript[H, k][\[CapitalDelta]^(2)* Subscript[s, n]]] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \varepsilon_{-1}^{(n)} = 0}
\varepsilon_{-1}^{(n)} = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (varepsilon[- 1])^(n) = 0 |
(Subscript[\[CurlyEpsilon], - 1])^(n) == 0 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \varepsilon_{0}^{(n)} = s_{n}}
\varepsilon_{0}^{(n)} = s_{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (varepsilon[0])^(n) = s[n] |
(Subscript[\[CurlyEpsilon], 0])^(n) == Subscript[s, n] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \varepsilon_{m+1}^{(n)} = \varepsilon_{m-1}^{(n+1)}+\frac{1}{\varepsilon_{m}^{(n+1)}-\varepsilon_{m}^{(n)}}}
\varepsilon_{m+1}^{(n)} = \varepsilon_{m-1}^{(n+1)}+\frac{1}{\varepsilon_{m}^{(n+1)}-\varepsilon_{m}^{(n)}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (varepsilon[m + 1])^(n) = (varepsilon[m - 1])^(n + 1)+(1)/((varepsilon[m])^(n + 1)- (varepsilon[m])^(n)) |
(Subscript[\[CurlyEpsilon], m + 1])^(n) == (Subscript[\[CurlyEpsilon], m - 1])^(n + 1)+Divide[1,(Subscript[\[CurlyEpsilon], m])^(n + 1)- (Subscript[\[CurlyEpsilon], m])^(n)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s_{n} = \sum_{j=1}^{n}\frac{(-1)^{j+1}}{j^{2}}}
s_{n} = \sum_{j=1}^{n}\frac{(-1)^{j+1}}{j^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | s[n] = sum(((- 1)^(j + 1))/((j)^(2)), j = 1..n) |
Subscript[s, n] == Sum[Divide[(- 1)^(j + 1),(j)^(2)], {j, 1, n}, GenerateConditions->None] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\cal L}_{k}^{(n)}(s) = \frac{\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}c_{j,k,n}\ifrac{s_{n+j}}{a_{n+j+1}}}{\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}c_{j,k,n}/a_{n+j+1}}}
{\cal L}_{k}^{(n)}(s) = \frac{\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}c_{j,k,n}\ifrac{s_{n+j}}{a_{n+j+1}}}{\sum_{j=0}^{k}(-1)^{j}\binom{k}{j}c_{j,k,n}/a_{n+j+1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (L[k])^(n)(s) = (sum((- 1)^(j)*binomial(k,j)*(((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1)))*(s[n + j])/(a[n + j + 1]), j = 0..k))/(sum((- 1)^(j)*binomial(k,j)*(((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1)))/a[n + j + 1], j = 0..k))
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(Subscript[L, k])^(n)[s] == Divide[Sum[(- 1)^(j)*Binomial[k,j]*(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)])*Divide[Subscript[s, n + j],Subscript[a, n + j + 1]], {j, 0, k}, GenerateConditions->None],Sum[(- 1)^(j)*Binomial[k,j]*(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)])/Subscript[a, n + j + 1], {j, 0, k}, GenerateConditions->None]]
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Failure | Failure | Error | Skipped - Because timed out |
| 3.9.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma} = 1}
\lim_{n\to\infty}\frac{s_{n+1}-\sigma}{s_{n}-\sigma} = 1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | limit((s[n + 1]- sigma)/(s[n]- sigma), n = infinity) = 1 |
Limit[Divide[Subscript[s, n + 1]- \[Sigma],Subscript[s, n]- \[Sigma]], n -> Infinity, GenerateConditions->None] == 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 3.9.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{j,k,n} = \frac{\Pochhammersym{\beta+n+j}{k-1}}{\Pochhammersym{\beta+n+k}{k-1}}}
c_{j,k,n} = \frac{\Pochhammersym{\beta+n+j}{k-1}}{\Pochhammersym{\beta+n+k}{k-1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1))) = (pochhammer(beta + n + j, k - 1))/(pochhammer(beta + n + k, k - 1))
|
(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)]) == Divide[Pochhammer[\[Beta]+ n + j, k - 1],Pochhammer[\[Beta]+ n + k, k - 1]]
|
Failure | Failure | Failed [36 / 81] Result: -.277777778e-1
Test Values: {beta = 1.5, j = 1, k = 2, n = 1}
Result: -.181818182e-1
Test Values: {beta = 1.5, j = 1, k = 2, n = 2}
Result: -.128205129e-1
Test Values: {beta = 1.5, j = 1, k = 2, n = 3}
Result: -.805594406e-1
Test Values: {beta = 1.5, j = 1, k = 3, n = 1}
... skip entries to safe data |
Failed [36 / 81]
Result: -0.02777777777777768
Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 1], Rule[β, 1.5]}
Result: -0.018181818181818188
Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 2], Rule[β, 1.5]}
... skip entries to safe data |
| 3.9.E17 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c_{j,k,n} = \frac{\Pochhammersym{-\gamma-n-j}{k-1}}{\Pochhammersym{-\gamma-n-k}{k-1}}}
c_{j,k,n} = \frac{\Pochhammersym{-\gamma-n-j}{k-1}}{\Pochhammersym{-\gamma-n-k}{k-1}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (((n + j + 1)^(k - 1))/((n + k + 1)^(k - 1))) = (pochhammer(- gamma - n - j, k - 1))/(pochhammer(- gamma - n - k, k - 1))
|
(Divide[(n + j + 1)^(k - 1),(n + k + 1)^(k - 1)]) == Divide[Pochhammer[- \[Gamma]- n - j, k - 1],Pochhammer[- \[Gamma]- n - k, k - 1]]
|
Failure | Failure | Failed [120 / 270] Result: .295470259e-1
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 2, n = 1}
Result: .184734286e-1
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 2, n = 2}
Result: .126342713e-1
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 2, n = 3}
Result: .1117465202
Test Values: {gamma = 1/2*3^(1/2)+1/2*I, j = 1, k = 3, n = 1}
... skip entries to safe data |
Failed [120 / 270]
Result: Complex[0.004408174927732822, -0.03290306559789975]
Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 1], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[0.003359414702213348, -0.020895843920590226]
Test Values: {Rule[j, 1], Rule[k, 2], Rule[n, 2], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |