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7

Just for fun: f[a_, t_] := a {t - Sin[t], 1 - Cos[t]} Manipulate[ ParametricPlot[{f[1, 4 t], f[2, 2 t], f[4, t]}, {t, 0, 4 Pi}, PlotStyle -> {Red, Green, Blue}, Epilog -> {{Orange, Circle[{4 p, 1}, 1], Black, PointSize[0.015], Point[f[1, 4 p]]}, {Orange, Circle[{4 p, 2}, 2], Black, PointSize[0.015], Point[f[2, 2 p]]}, {Orange, ...


5

You can use Fold instead: f[n_Integer] := Fold[#2/(1 + #) &, n, Reverse@Range[n - 1]] f[3] $\frac{1}{1+\frac{2}{1+3}}$ It not very useful analytically, but it allows you to invoke the CPU gods: f /@ Range[50] // ListLinePlot[#, PlotRange -> All] &


5

No, it is not doing a Gram-Schmidt procedure. One way to note that this is distinct from Gram-Schmidt is that Gram-Schmidt produces an orthonormal basis, whereas the example outputs in the Documentation Center page for LatticeReduce are neither normalized nor orthogonal. Instead, LatticeReduce returns a basis $B$ comprised of linear combinations of integer ...


3

LyapunovSolve and DiscreteLyapunovSolve solve several equations Lyapunov, Sylvester, Stein, generalized versions, etc., and as such there is no one standard form. Since they are linear solvers their design was based on the precedent set by LinearSolve. For $\dot{x}=A.x$ to be stable, $P=\text{LyapunovSolve}\left[A^{\mathsf{T}},-Q\right]$ has to be positive ...


1

Seeing as you're trying to evaluate hydrogen wavefunctions, note that the necessary special functions are already built-in, so you can skip the step of defining the special functions entirely, and just do this: ψ[n_, l_, m_, ρ_, θ_, ϕ_] := Sqrt[(2/(n a0))^3 (n - l - 1)!/(2 n (n + l)!)] Exp[-ρ/2] ρ^ l LaguerreL[n - l - 1, 2 l + 1, ρ] ...


1

You can define partial derivatives in specified slots without using #1 or #2: p[x_, y_] := x^5 y^7; d02p = Derivative[0, 2][p]; d02p[a, b] which returns 42 a^5 b^5 Likewise, to obtain the pure function fDer for the mixed partial derivative of a function f of vector argument, try this: f[list_] := (Times @@ list)^8; ind = RandomInteger[{0, 8}, 10]; ...


1

Conjugate by default assumes that all symbolic quantities are potentially complex. This may seem annoying at first, but there is a very good reason for it, and one way to see why is to define your own version of Conjugate, and see it fail. For educational purposes, I do that below. Define $Conjugate as follows: $Conjugate[x_] := x /. Complex[a_, b_] :> ...


1

I'm afraid I don't know what realized covariance means. Perhaps the easiest solution is to use RLink and directly use the R implementation. Here are some links to the documentation to get you started. http://reference.wolfram.com/language/RLink/guide/RLink.html http://reference.wolfram.com/language/RLink/tutorial/UsingRLink.html


1

Here is what I believe you are seeking: ParametricPlot[{{1 (Theta - Sin[Theta]), 1 (1 - Cos[Theta])}, {2 (Theta - Sin[Theta]), 2 (1 - Cos[Theta])}, {4 (Theta - Sin[Theta]), 4 (1 - Cos[Theta])}}, {Theta, -10 Pi, 10 Pi}, AspectRatio -> .5, PlotRange -> {{-8 Pi, 8 Pi}, Automatic}] The resulting Plot:



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