Numerical Methods - 3.11 Approximation Techniques

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
3.11.E5 ∑ k = 0 n x j k ⁢ δ ⁢ a k = ( - 1 ) j ⁢ ( m j - m ) superscript subscript 𝑘 0 𝑛 superscript subscript 𝑥 𝑗 𝑘 𝛿 subscript 𝑎 𝑘 superscript 1 𝑗 subscript 𝑚 𝑗 𝑚 {\displaystyle{\displaystyle\sum_{k=0}^{n}x_{j}^{k}\delta a_{k}=(-1)^{j}(m_{j}% -m)}}
\sum_{k=0}^{n}x_{j}^{k}\delta a_{k} = (-1)^{j}(m_{j}-m)		 j = 0 𝑗 0 {\displaystyle{\displaystyle j=0}} 
sum((x[j])^(k)*delta*a[k], k = 0..n) = (- 1)^(j)*(((- 1)^(j)* epsilon[n](x[j]))- m)
 
Sum[(Subscript[x, j])^(k)*\[Delta]*Subscript[a, k], {k, 0, n}, GenerateConditions->None] == (- 1)^(j)*(((- 1)^(j)* Subscript[\[Epsilon], n][Subscript[x, j]])- m)
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E6 T n ⁡ ( x ) = cos ⁡ ( n ⁢ arccos ⁡ x ) Chebyshev-polynomial-first-kind-T 𝑛 𝑥 𝑛 𝑥 {\displaystyle{\displaystyle T_{n}\left(x\right)=\cos\left(n\operatorname{% arccos}x\right)}}
\ChebyshevpolyT{n}@{x} = \cos@{n\acos@@{x}}		 - 1 ≤ x , x ≤ 1 formulae-sequence 1 𝑥 𝑥 1 {\displaystyle{\displaystyle-1\leq x,x\leq 1}} 
ChebyshevT(n, x) = cos(n*arccos(x))

ChebyshevT[n, x] == Cos[n*ArcCos[x]]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
3.11.E7 T n + 1 ⁡ ( x ) - 2 ⁢ x ⁢ T n ⁡ ( x ) + T n - 1 ⁡ ( x ) = 0 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 2 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 𝑥 Chebyshev-polynomial-first-kind-T 𝑛 1 𝑥 0 {\displaystyle{\displaystyle T_{n+1}\left(x\right)-2xT_{n}\left(x\right)+T_{n-% 1}\left(x\right)=0}}
\ChebyshevpolyT{n+1}@{x}-2x\ChebyshevpolyT{n}@{x}+\ChebyshevpolyT{n-1}@{x} = 0		 

ChebyshevT(n + 1, x)- 2*x*ChebyshevT(n, x)+ ChebyshevT(n - 1, x) = 0

ChebyshevT[n + 1, x]- 2*x*ChebyshevT[n, x]+ ChebyshevT[n - 1, x] == 0
Successful Successful - Successful [Tested: 3]
3.11.E13 ϵ n ⁢ ( x ) = d n + 1 ⁢ T n + 1 ⁡ ( 2 ⁢ x - a - b b - a ) subscript italic-ϵ 𝑛 𝑥 subscript 𝑑 𝑛 1 Chebyshev-polynomial-first-kind-T 𝑛 1 2 𝑥 𝑎 𝑏 𝑏 𝑎 {\displaystyle{\displaystyle\epsilon_{n}(x)=d_{n+1}T_{n+1}\left(\frac{2x-a-b}{% b-a}\right)}}
\epsilon_{n}(x) = d_{n+1}\ChebyshevpolyT{n+1}@{\frac{2x-a-b}{b-a}}		 

epsilon[n](x) = d[n + 1]*ChebyshevT(n + 1, (2*x - a - b)/(b - a))

Subscript[\[Epsilon], n][x] == Subscript[d, n + 1]*ChebyshevT[n + 1, Divide[2*x - a - b,b - a]]
Failure Failure
Failed [300 / 300]
Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 1}


Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 2}


Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 1, n = 3}


Result: Float(-infinity)-Float(infinity)*I
Test Values: {a = -1.5, b = -1.5, x = 1.5, d[n+1] = 1/2*3^(1/2)+1/2*I, epsilon[n] = 1/2*3^(1/2)+1/2*I, epsilon = 2, n = 1}

... skip entries to safe data
Failed [300 / 300]
Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[ϵ, 1], Rule[Subscript[d, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϵ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Indeterminate
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[ϵ, 2], Rule[Subscript[d, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϵ, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.11.E15 u k = 2 ⁢ x ⁢ u k + 1 - u k + 2 + c k subscript 𝑢 𝑘 2 𝑥 subscript 𝑢 𝑘 1 subscript 𝑢 𝑘 2 subscript 𝑐 𝑘 {\displaystyle{\displaystyle u_{k}=2xu_{k+1}-u_{k+2}+c_{k}}}
u_{k} = 2xu_{k+1}-u_{k+2}+c_{k}		 k = n - 1 𝑘 𝑛 1 {\displaystyle{\displaystyle k=n-1}} 
u[k] = 2*x*u[k + 1]- u[k + 2]+ c[k]
 
Subscript[u, k] == 2*x*Subscript[u, k + 1]- Subscript[u, k + 2]+ Subscript[c, k]
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E18 m j = ( - 1 ) j ⁢ ϵ k , ℓ ⁢ ( x j ) subscript 𝑚 𝑗 superscript 1 𝑗 subscript italic-ϵ 𝑘 ℓ subscript 𝑥 𝑗 {\displaystyle{\displaystyle m_{j}=(-1)^{j}\epsilon_{k,\ell}(x_{j})}}
m_{j} = (-1)^{j}\epsilon_{k,\ell}(x_{j})		 j = 0 𝑗 0 {\displaystyle{\displaystyle j=0}} 
((- 1)^(j)* epsilon[n](x[j])) = (- 1)^(j)* epsilon[k , ell](x[j])
 
((- 1)^(j)* Subscript[\[Epsilon], n][Subscript[x, j]]) == (- 1)^(j)* Subscript[\[Epsilon], k , \[ScriptL]][Subscript[x, j]]
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E19 R 3 , 3 ⁢ ( x ) = p 0 + p 1 ⁢ x + p 2 ⁢ x 2 + p 3 ⁢ x 3 1 + q 1 ⁢ x + q 2 ⁢ x 2 + q 3 ⁢ x 3 subscript 𝑅 3 3 𝑥 subscript 𝑝 0 subscript 𝑝 1 𝑥 subscript 𝑝 2 superscript 𝑥 2 subscript 𝑝 3 superscript 𝑥 3 1 subscript 𝑞 1 𝑥 subscript 𝑞 2 superscript 𝑥 2 subscript 𝑞 3 superscript 𝑥 3 {\displaystyle{\displaystyle R_{3,3}(x)=\frac{p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{% 3}}{1+q_{1}x+q_{2}x^{2}+q_{3}x^{3}}}}
R_{3,3}(x) = \frac{p_{0}+p_{1}x+p_{2}x^{2}+p_{3}x^{3}}{1+q_{1}x+q_{2}x^{2}+q_{3}x^{3}}		 

R[3 , 3](x) = (p[0]+ p[1]*x + p[2]*(x)^(2)+ p[3]*(x)^(3))/(1 + q[1]*x + q[2]*(x)^(2)+ q[3]*(x)^(3))
 
Subscript[R, 3 , 3][x] == Divide[Subscript[p, 0]+ Subscript[p, 1]*x + Subscript[p, 2]*(x)^(2)+ Subscript[p, 3]*(x)^(3),1 + Subscript[q, 1]*x + Subscript[q, 2]*(x)^(2)+ Subscript[q, 3]*(x)^(3)]
 Skipped - no semantic math Skipped - no semantic math - -
3.11#E23X a 0 = c 0 ⁢ b 0 subscript 𝑎 0 subscript 𝑐 0 subscript 𝑏 0 {\displaystyle{\displaystyle\displaystyle a_{0}=c_{0}b_{0}}}
\displaystyle a_{0} = c_{0}b_{0}		 

a[0] = c[0]*b[0]
 
Subscript[a, 0] == Subscript[c, 0]*Subscript[b, 0]
 Skipped - no semantic math Skipped - no semantic math - -
3.11#E23Xa a 1 = c 1 ⁢ b 0 + c 0 ⁢ b 1 subscript 𝑎 1 subscript 𝑐 1 subscript 𝑏 0 subscript 𝑐 0 subscript 𝑏 1 {\displaystyle{\displaystyle\displaystyle a_{1}=c_{1}b_{0}+c_{0}b_{1}}}
\displaystyle a_{1} = c_{1}b_{0}+c_{0}b_{1}		 

a[1] = c[1]*b[0]+ c[0]*b[1]
 
Subscript[a, 1] == Subscript[c, 1]*Subscript[b, 0]+ Subscript[c, 0]*Subscript[b, 1]
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E25 ( N - C ) - 1 + ( S - C ) - 1 = ( W - C ) - 1 + ( E - C ) - 1 superscript 𝑁 𝐶 1 superscript 𝑆 𝐶 1 superscript 𝑊 𝐶 1 superscript 𝐸 𝐶 1 {\displaystyle{\displaystyle(N-C)^{-1}+(S-C)^{-1}=(W-C)^{-1}+(E-C)^{-1}}}
(N-C)^{-1}+(S-C)^{-1} = (W-C)^{-1}+(E-C)^{-1}		 

(N - C)^(- 1)+(S - C)^(- 1) = (W - C)^(- 1)+(E - C)^(- 1)
 
(N - C)^(- 1)+(S - C)^(- 1) == (W - C)^(- 1)+(E - C)^(- 1)
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E34 X k ⁢ ℓ = ∑ j = 1 J w ⁢ ( x j ) ⁢ ϕ k ⁢ ( x j ) ⁢ ϕ ℓ ⁢ ( x j ) subscript 𝑋 𝑘 ℓ superscript subscript 𝑗 1 𝐽 𝑤 subscript 𝑥 𝑗 subscript italic-ϕ 𝑘 subscript 𝑥 𝑗 subscript italic-ϕ ℓ subscript 𝑥 𝑗 {\displaystyle{\displaystyle X_{k\ell}=\sum_{j=1}^{J}w(x_{j})\phi_{k}(x_{j})% \phi_{\ell}(x_{j})}}
X_{k\ell} = \sum_{j=1}^{J}w(x_{j})\phi_{k}(x_{j})\phi_{\ell}(x_{j})		 

X[k*ell] = sum(w(x[j])* phi[k](x[j])* phi[ell](x[j]), j = 1..J)
 
Subscript[X, k*\[ScriptL]] == Sum[w[Subscript[x, j]]* Subscript[\[Phi], k][Subscript[x, j]]* Subscript[\[Phi], \[ScriptL]][Subscript[x, j]], {j, 1, J}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E37 ∑ j = 0 n - 1 ϕ k ⁢ ( x j ) ⁢ ϕ ℓ ⁢ ( x j ) ¯ = n ⁢ δ k , ℓ superscript subscript 𝑗 0 𝑛 1 subscript italic-ϕ 𝑘 subscript 𝑥 𝑗 subscript italic-ϕ ℓ subscript 𝑥 𝑗 𝑛 Kronecker 𝑘 ℓ {\displaystyle{\displaystyle\sum_{j=0}^{n-1}\phi_{k}(x_{j})\overline{\phi_{% \ell}(x_{j})}=n\delta_{k,\ell}}}
\sum_{j=0}^{n-1}\phi_{k}(x_{j})\conj{\phi_{\ell}(x_{j})} = n\Kroneckerdelta{k}{\ell}		 k = 0 , ℓ = 0 formulae-sequence 𝑘 0 ℓ 0 {\displaystyle{\displaystyle k=0,\ell=0}} 
sum(phi[k](x[j])* conjugate(phi[ell](x[j])), j = 0..n - 1) = n*KroneckerDelta[k, ell]

Sum[Subscript[\[Phi], k][Subscript[x, j]]* Conjugate[Subscript[\[Phi], \[ScriptL]][Subscript[x, j]]], {j, 0, n - 1}, GenerateConditions->None] == n*KroneckerDelta[k, \[ScriptL]]
Error Failure -
Failed [300 / 300]
Result: Plus[1.0, Times[-1.0, KroneckerDelta[1.0, ℓ]]]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[2.0, Times[-2.0, KroneckerDelta[1.0, ℓ]]]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, ℓ], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.11.E38 f j = ∑ k = 0 n - 1 a k ⁢ ϕ k ⁢ ( x j ) subscript 𝑓 𝑗 superscript subscript 𝑘 0 𝑛 1 subscript 𝑎 𝑘 subscript italic-ϕ 𝑘 subscript 𝑥 𝑗 {\displaystyle{\displaystyle f_{j}=\sum_{k=0}^{n-1}a_{k}\phi_{k}(x_{j})}}
f_{j} = \sum_{k=0}^{n-1}a_{k}\phi_{k}(x_{j})		 j = 0 𝑗 0 {\displaystyle{\displaystyle j=0}} 
f[j] = sum(a[k]*phi[k](x[j]), k = 0..n - 1)
 
Subscript[f, j] == Sum[Subscript[a, k]*Subscript[\[Phi], k][Subscript[x, j]], {k, 0, n - 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.11.E39 a k = 1 n ⁢ ∑ j = 0 n - 1 f j ⁢ ϕ k ⁢ ( x j ) ¯ subscript 𝑎 𝑘 1 𝑛 superscript subscript 𝑗 0 𝑛 1 subscript 𝑓 𝑗 subscript italic-ϕ 𝑘 subscript 𝑥 𝑗 {\displaystyle{\displaystyle a_{k}=\frac{1}{n}\sum_{j=0}^{n-1}f_{j}\overline{% \phi_{k}(x_{j})}}}
a_{k} = \frac{1}{n}\sum_{j=0}^{n-1}f_{j}\conj{\phi_{k}(x_{j})}		 k = 0 𝑘 0 {\displaystyle{\displaystyle k=0}} 
a[k] = (1)/(n)*sum(f[j]*conjugate(phi[k](x[j])), j = 0..n - 1)

Subscript[a, k] == Divide[1,n]*Sum[Subscript[f, j]*Conjugate[Subscript[\[Phi], k][Subscript[x, j]]], {j, 0, n - 1}, GenerateConditions->None]
Error Failure -
Failed [300 / 300]
Result: Complex[0.0, 0.9999999999999999]
Test Values: {Rule[k, 1], Rule[n, 1], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.0, 0.9999999999999999]
Test Values: {Rule[k, 1], Rule[n, 2], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[x, j], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ϕ, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.11#Ex5 f j = ∑ k = 0 n - 1 a k ⁢ ω n j ⁢ k subscript 𝑓 𝑗 superscript subscript 𝑘 0 𝑛 1 subscript 𝑎 𝑘 superscript subscript 𝜔 𝑛 𝑗 𝑘 {\displaystyle{\displaystyle f_{j}=\sum_{k=0}^{n-1}a_{k}\omega_{n}^{jk}}}
f_{j} = \sum_{k=0}^{n-1}a_{k}\omega_{n}^{jk}		 

f[j] = sum(a[k]*(omega[n])^(j*k), k = 0..n - 1)
 
Subscript[f, j] == Sum[Subscript[a, k]*(Subscript[\[Omega], n])^(j*k), {k, 0, n - 1}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
3.11#Ex6 ω n = e 2 ⁢ π ⁢ i / n subscript 𝜔 𝑛 superscript 𝑒 2 𝑖 𝑛 {\displaystyle{\displaystyle\omega_{n}=e^{2\pi i/n}}}
\omega_{n} = e^{2\cpi i/n}		 j = 0 𝑗 0 {\displaystyle{\displaystyle j=0}} 
omega[n] = exp(2*Pi*I/n)

Subscript[\[Omega], n] == Exp[2*Pi*I/n]
Failure Failure
Failed [290 / 300]
Result: -.1339745960+.4999999992*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.866025404+.5000000004*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: 1.366025404-.3660254040*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = 1/2*3^(1/2)+1/2*I, n = 3}


Result: -1.500000000+.8660254032*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, omega[n] = -1/2+1/2*I*3^(1/2), n = 1}

... skip entries to safe data
Failed [290 / 300]
Result: Complex[-0.1339745962155613, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[n, 2], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
3.11.E42 ω n 2 ⁢ ( k - ( n / 2 ) ) = ω n / 2 k superscript subscript 𝜔 𝑛 2 𝑘 𝑛 2 superscript subscript 𝜔 𝑛 2 𝑘 {\displaystyle{\displaystyle\omega_{n}^{2(k-(n/2))}=\omega_{n/2}^{k}}}
\omega_{n}^{2(k-(n/2))} = \omega_{n/2}^{k}		 

(omega[n])^(2*(k -(n/2))) = (omega[n/2])^(k)
 
(Subscript[\[Omega], n])^(2*(k -(n/2))) == (Subscript[\[Omega], n/2])^(k)
 Skipped - no semantic math Skipped - no semantic math - -
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