18.3: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/18.3.E1 18.3.E1] || [[Item:Q5507|<math>\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0</syntaxhighlight> || <math>0 \leq j, j \leq N, 0 \leq k, k \leq N, j \neq k</math> || <syntaxhighlight lang=mathematica>sum(ChebyshevT(j, x[N + 1 , n])*ChebyshevT(k, x[N + 1 , n]), n = 1..N + 1) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[ChebyshevT[j, Subscript[x, N + 1 , n]]*ChebyshevT[k, Subscript[x, N + 1 , n]], {n, 1, N + 1}, GenerateConditions->None] == 0</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
| [https://dlmf.nist.gov/18.3.E1 18.3.E1] || <math qid="Q5507">\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0</syntaxhighlight> || <math>0 \leq j, j \leq N, 0 \leq k, k \leq N, j \neq k</math> || <syntaxhighlight lang=mathematica>sum(ChebyshevT(j, x[N + 1 , n])*ChebyshevT(k, x[N + 1 , n]), n = 1..N + 1) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[ChebyshevT[j, Subscript[x, N + 1 , n]]*ChebyshevT[k, Subscript[x, N + 1 , n]], {n, 1, N + 1}, GenerateConditions->None] == 0</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| [https://dlmf.nist.gov/18.3.E2 18.3.E2] || [[Item:Q5508|<math>x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x[N + 1 , n] = cos((n -(1)/(2))*Pi/(N + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[x, N + 1 , n] == Cos[(n -Divide[1,2])*Pi/(N + 1)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1432026267+.3500908026*I
| [https://dlmf.nist.gov/18.3.E2 18.3.E2] || <math qid="Q5508">x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>x[N + 1 , n] = cos((n -(1)/(2))*Pi/(N + 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[x, N + 1 , n] == Cos[(n -Divide[1,2])*Pi/(N + 1)]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .1432026267+.3500908026*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.718798807+.233214116e-1*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.718798807+.233214116e-1*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.14320262643759762, 0.350090802645732]
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [298 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[0.14320262643759762, 0.350090802645732]

Latest revision as of 11:44, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
18.3.E1 ∑ n = 1 N + 1 T j ⁡ ( x N + 1 , n ) ⁢ T k ⁡ ( x N + 1 , n ) = 0 superscript subscript 𝑛 1 𝑁 1 Chebyshev-polynomial-first-kind-T 𝑗 subscript 𝑥 𝑁 1 𝑛 Chebyshev-polynomial-first-kind-T 𝑘 subscript 𝑥 𝑁 1 𝑛 0 {\displaystyle{\displaystyle\sum_{n=1}^{N+1}T_{j}\left(x_{N+1,n}\right)T_{k}% \left(x_{N+1,n}\right)=0}}
\sum_{n=1}^{N+1}\ChebyshevpolyT{j}@{x_{N+1,n}}\ChebyshevpolyT{k}@{x_{N+1,n}} = 0		 0 ≤ j , j ≤ N , 0 ≤ k , k ≤ N , j ≠ k formulae-sequence 0 𝑗 formulae-sequence 𝑗 𝑁 formulae-sequence 0 𝑘 formulae-sequence 𝑘 𝑁 𝑗 𝑘 {\displaystyle{\displaystyle 0\leq j,j\leq N,0\leq k,k\leq N,j\neq k}} 
sum(ChebyshevT(j, x[N + 1 , n])*ChebyshevT(k, x[N + 1 , n]), n = 1..N + 1) = 0

Sum[ChebyshevT[j, Subscript[x, N + 1 , n]]*ChebyshevT[k, Subscript[x, N + 1 , n]], {n, 1, N + 1}, GenerateConditions->None] == 0
Skipped - Unable to analyze test case: Null Skipped - Unable to analyze test case: Null - -
18.3.E2 x N + 1 , n = cos ⁡ ( ( n - 1 2 ) ⁢ π / ( N + 1 ) ) subscript 𝑥 𝑁 1 𝑛 𝑛 1 2 𝜋 𝑁 1 {\displaystyle{\displaystyle x_{N+1,n}=\cos\left((n-\tfrac{1}{2})\pi/(N+1)% \right)}}
x_{N+1,n} = \cos@{(n-\tfrac{1}{2})\pi/(N+1)}		 

x[N + 1 , n] = cos((n -(1)/(2))*Pi/(N + 1))

Subscript[x, N + 1 , n] == Cos[(n -Divide[1,2])*Pi/(N + 1)]
Failure Failure
Failed [298 / 300]
Result: .1432026267+.3500908026*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.718798807+.233214116e-1*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, x[N+1,n] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [298 / 300]
Result: Complex[0.14320262643759762, 0.350090802645732]
Test Values: {Rule[n, 1], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, N], n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.7187988066024098, 0.023321411689447014]
Test Values: {Rule[n, 2], Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, Plus[1, N], n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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