3.9: Difference between revisions

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{{DISPLAYTITLE:Numerical Methods - 3.9 Acceleration of Convergence}}
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Revision as of 16:31, 25 May 2021


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))

(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]]
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
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