3.8

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
3.8.E2	${\displaystyle{\displaystyle z=\phi(z)}}$ z = \phi(z)		z = phi(z)	z == \[Phi][z]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8.E3	${\displaystyle{\displaystyle\left|z_{n+1}-\zeta\right|<A{\left|z_{n}-\zeta\right|^{p}}}}$ \abs{z_{n+1}-\zeta} < A\abs{z_{n}-\zeta}^{p}		abs(z[n + 1]- zeta) < A*(abs(z[n]- zeta))^(p)	Abs[Subscript[z, n + 1]- \[Zeta]] < A*(Abs[Subscript[z, n]- \[Zeta]])^(p)	Failure	Failure	Failed [30 / 300] Result: 0. < 0. Test Values: {A = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, z[n] = 1/2*3^(1/2)+1/2*I, z[n+1] = 1/2*3^(1/2)+1/2*I, n = 1} Result: 0. < 0. Test Values: {A = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, z[n] = 1/2*3^(1/2)+1/2*I, z[n+1] = 1/2*3^(1/2)+1/2*I, n = 2} Result: 0. < 0. Test Values: {A = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, z[n] = 1/2*3^(1/2)+1/2*I, z[n+1] = 1/2*3^(1/2)+1/2*I, n = 3} Result: 1.414213562 < 0. Test Values: {A = 1/2*3^(1/2)+1/2*I, p = 1/2*3^(1/2)+1/2*I, zeta = 1/2*3^(1/2)+1/2*I, z[n] = 1/2*3^(1/2)+1/2*I, z[n+1] = -1/2+1/2*I*3^(1/2), n = 1} ... skip entries to safe data	Failed [300 / 300] Result: False Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: False Test Values: {Rule[A, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[z, Plus[1, n]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
3.8#Ex1	${\displaystyle{\displaystyle x_{n+1}=\phi(x_{n})}}$ x_{n+1} = \phi(x_{n})		x[n + 1] = phi(x[n])	Subscript[x, n + 1] == \[Phi][Subscript[x, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8#Ex2	${\displaystyle{\displaystyle\phi(x)=x+x{\cot^{2}}x-\cot x}}$ \phi(x) = x+x\cot^{2}@@{x}-\cot@@{x}		phi(x) = x + x*(cot(x))^(2)- cot(x)	\[Phi][x] == x + x*(Cot[x])^(2)- Cot[x]	Failure	Failure	Failed [30 / 30] Result: -.137590423+.7500000000*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 1.5} Result: .881577740e-1+.2500000000*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = .5} Result: -1.144507621+1.*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 2} Result: -2.186628529+1.299038106*I Test Values: {phi = -1/2+1/2*I*3^(1/2), x = 1.5} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.1375904227343937, 0.7499999999999999] Test Values: {Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-2.186628528411051, 1.299038105676658] Test Values: {Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
3.8.E6	${\displaystyle{\displaystyle x_{2}=x_{1}-\frac{x_{1}-x_{0}}{f_{1}-f_{0}}f_{1}}}$ x_{2} = x_{1}-\frac{x_{1}-x_{0}}{f_{1}-f_{0}}f_{1}		x[2] = x[1]-(x[1]- x[0])/(f[1]- f[0])*f[1]	Subscript[x, 2] == Subscript[x, 1]-Divide[Subscript[x, 1]- Subscript[x, 0],Subscript[f, 1]- Subscript[f, 0]]*Subscript[f, 1]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8.E7	${\displaystyle{\displaystyle z_{n+1}=z_{n}-\frac{(\phi(z_{n})-z_{n})^{2}}{\phi(\phi(z_{n}))-2\phi(z_{n})+z_{n}}}}$ z_{n+1} = z_{n}-\frac{(\phi(z_{n})-z_{n})^{2}}{\phi(\phi(z_{n}))-2\phi(z_{n})+z_{n}}		z[n + 1] = z[n]-((phi(z[n])- z[n])^(2))/(phi(phi(z[n]))- 2*phi(z[n])+ z[n])	Subscript[z, n + 1] == Subscript[z, n]-Divide[(\[Phi][Subscript[z, n]]- Subscript[z, n])^(2),\[Phi][\[Phi][Subscript[z, n]]]- 2*\[Phi][Subscript[z, n]]+ Subscript[z, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8#Ex3	${\displaystyle{\displaystyle z_{n+1}=\phi(z_{n})}}$ z_{n+1} = \phi(z_{n})		z[n + 1] = phi(z[n])	Subscript[z, n + 1] == \[Phi][Subscript[z, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8#Ex4	${\displaystyle{\displaystyle\phi(z)=\frac{3z^{4}+1}{4z^{3}}}}$ \phi(z) = \frac{3z^{4}+1}{4z^{3}}		phi(z) = (3*(z)^(4)+ 1)/(4*(z)^(3))	\[Phi][z] == Divide[3*(z)^(4)+ 1,4*(z)^(3)]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8#Ex7	${\displaystyle{\displaystyle\Delta s=\frac{r_{3}q_{0}-r_{2}q_{1}}{r_{2}^{2}-\ell r_{3}}}}$ \Delta s = \frac{r_{3}q_{0}-r_{2}q_{1}}{r_{2}^{2}-\ell r_{3}}		Delta*s = (r[3]*q[0]- r[2]*q[1])/((r[2])^(2)- ell*r[3])	\[CapitalDelta]*s == Divide[Subscript[r, 3]*Subscript[q, 0]- Subscript[r, 2]*Subscript[q, 1],(Subscript[r, 2])^(2)- \[ScriptL]*Subscript[r, 3]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8#Ex8	${\displaystyle{\displaystyle\Delta t=\frac{\ell q_{1}-r_{2}q_{0}}{r_{2}^{2}-\ell r_{3}}}}$ \Delta t = \frac{\ell q_{1}-r_{2}q_{0}}{r_{2}^{2}-\ell r_{3}}		Delta*t = (ell*q[1]- r[2]*q[0])/((r[2])^(2)- ell*r[3])	\[CapitalDelta]*t == Divide[\[ScriptL]*Subscript[q, 1]- Subscript[r, 2]*Subscript[q, 0],(Subscript[r, 2])^(2)- \[ScriptL]*Subscript[r, 3]]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8#Ex9	${\displaystyle{\displaystyle\ell=sr_{2}+tr_{3}}}$ \ell = sr_{2}+tr_{3}		ell = s*r[2]+ t*r[3]	\[ScriptL] == s*Subscript[r, 2]+ t*Subscript[r, 3]	Skipped - no semantic math	Skipped - no semantic math	-	-
3.8.E13	${\displaystyle{\displaystyle\frac{\mathrm{d}z}{\mathrm{d}\alpha}=-\ifrac{\frac{\partial f}{\partial\alpha}}{\frac{\partial f}{\partial z}}}}$ \deriv{z}{\alpha} = -\ifrac{\pderiv{f}{\alpha}}{\pderiv{f}{z}}		diff(z, alpha) = -(diff(f, alpha))/(diff(f, z))	D[z, \[Alpha]] == -Divide[D[f, \[Alpha]],D[f, z]]	Error	Failure	-	Failed [210 / 210] Result: Indeterminate Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 1.5]} Result: Indeterminate Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[α, 0.5]} ... skip entries to safe data
3.8.E16	${\displaystyle{\displaystyle\frac{\mathrm{d}x}{\mathrm{d}a_{19}}=-\frac{20^{19}}{19!}}}$ \deriv{x}{a_{19}} = -\frac{20^{19}}{19!}		subs( temp=a[19], diff( x, temp$(1) ) ) = -((20)^(19))/(factorial(19))	(D[x, {temp, 1}]/.temp-> Subscript[a, 19]) == -Divide[(20)^(19),(19)!]	Failure	Failure	Skip - No test values generated	Failed [30 / 30] Result: 4.309980412182177*^7 Test Values: {Rule[x, 1.5], Rule[Subscript[a, 19], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: 4.309980412182177*^7 Test Values: {Rule[x, 1.5], Rule[Subscript[a, 19], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
3.8.E16	${\displaystyle{\displaystyle-\frac{20^{19}}{19!}=(-4.30\dots)\times 10^{7}}}$ -\frac{20^{19}}{19!} = (-4.30\dots)\times 10^{7}		-((20)^(19))/(factorial(19)) = (- 4.30) * (10)^(7)	-Divide[(20)^(19),(19)!] == (- 4.30) * (10)^(7)	Translation Error	Translation Error	Skip - symbolical successful subtest	Skip - symbolical successful subtest
3.8.E17	${\displaystyle{\displaystyle z_{n+1}=\phi(z_{n})}}$ z_{n+1} = \phi(z_{n})		z[n + 1] = phi(z[n])	Subscript[z, n + 1] == \[Phi][Subscript[z, n]]	Skipped - no semantic math	Skipped - no semantic math	-	-