If I wanted to generate a sequence that follows the pattern
$$x_{n}=x_{n-1}+\dfrac{1}{2}\left(\sqrt{1-y_{n-1}\ ^{2}}-x_{n-1}\right)\\ y_{n}=y_{n-1}+\dfrac{1}{2}\left(\sqrt{1-x_{n}\ ^{2}}-y_{n-1}\right)$$
rather than writing the whole thing out:
x0 = N[0 + 1/2 (Sqrt[1 - 0^2] - 0)]; y0 = N[0 + 1/2 (Sqrt[1 - x0^2] - 0)]; x1 = N[x0 + 1/2 (Sqrt[1 - y0^2] - x0)]; y1 = N[y0 + 1/2 (Sqrt[1 - x1^2] - y0)]; ... etc. What is the best way to do it?
The previous answers correspond to the recursive model of the form $$[x_n,y_n]=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1})]$$. However, the state-space model of the question is of the form \begin{align} [x_n,y_n]&=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},x_n)]\\ &=[f(x_{n-1},y_{n-1}),g(x_{n-1},y_{n-1},f(x_{n-1},y_{n-1}))] \end{align} The latter equation is the correct form that should be applied with the methods proposed by the previous answers. Another method could be to use NestList
h[{x_, y_}] := {x + 1/2 (Sqrt[1 - y^2] - x), y + 1/2 (Sqrt[1 - (x + 1/2 (Sqrt[1 - y^2] - x))^2] - y)}; x0 = N[0 + 1/2 (Sqrt[1 - 0^2] - 0)]; y0 = N[0 + 1/2 (Sqrt[1 - x0^2] - 0)]; NestList[h, {x0, y0}, 10] {{0.5, 0.433013}, {0.700694, 0.573237}, {0.760042, 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}, {0.780556, 0.624467}, {0.780804, 0.624622}, {0.780865, 0.624661}, {0.780881, 0.62467}, {0.780885, 0.624673}, {0.780886, 0.624673}}
In an earlier edit, I had misread one of the subscripts in the y equation (thanks to Stelios for pointing this out). Here is the corrected version:
x[n_] := x[n] = x[n - 1] + 1/2 (Sqrt[1 - y[n - 1]^2] - x[n - 1]); y[n_] := y[n] = y[n - 1] + 1/2 (Sqrt[1 - x[n]^2] - y[n - 1]); x[0] = 0.5; y[0] = 1/2 Sqrt[1 - x[0]^2]; x[10] To get many values:
{x[#], y[#]} & /@ Range[0,10] RecurrenceTable:
RecurrenceTable[{ x[n + 1] == x[n] + (1/2) (Sqrt[1 - y[n]^2] - x[n]), y[n + 1] == y[n] + (1/2) (Sqrt[1 - x[n + 1]^2] - y[n]), x[0] == 0, y[0] == 0}, {x, y}, {n, 1, 5}] // N
{{0.5, 0.433013}, {0.700694, 0.573237}, {0.760042, 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}}
Credit goes to Stelios for spotting a mistake in the original answer :)
Using:
f[x_, y_] := x + 0.5 (Sqrt[1 - y^2] - x) Initial condition {0,0} first 10:
NestList[Apply[Function[{x, y}, {f[x, y], f[y, f[x, y]]}], #] &, {0, 0}, 10] yields:
{{0, 0}, {0.5, 0.433013}, {0.700694, 0.573237}, {0.760042, 0.611556}, {0.775621, 0.621377}, {0.779567, 0.623848}, {0.780556, 0.624467}, {0.780804, 0.624622}, {0.780865, 0.624661}, {0.780881, 0.62467}, {0.780885, 0.624673}} Note fixed points on $y=\sqrt{1-x^2}$:
Manipulate[ Show[ListPlot[ NestList[ Apply[Function[{x, y}, {f[x, y], f[y, f[x, y]]}], #] &, {a[[1]], a[[2]]}, 10]], Plot[Sqrt[1 - x^2], {x, 0, 1}], PlotRange -> Table[{0, 1}, {2}], Frame -> True, FrameLabel -> {"x", "y"}, BaseStyle -> 16, AxesOrigin -> {0, 0}], {a, {0, 0}, {1, 1}}] 
Firstly,I simplify your formular:
$$x_{n}=\frac{1}{2}\left(x_{n-1}+\sqrt{1-y_{n-1}\ ^{2}}\right)\\ y_{n}=\frac{1}{2}\left(y_{n-1}+\sqrt{1-x_{n}\ ^{2}}\right)$$
Seeing the sequences as below:
$$x_1,y_1,x_2,y_2,x_3,y_3\ldots\\ $$ I define a function:$f(x,y)=1/2(x+\sqrt{1-y^2})$ $$ x_2=f(x_1,y_1)\\ y_2=f(y_1,x_2)\\ x_3=f(x_2,y_2)\ldots $$ So considering the underlying permutation :
$$(x_1,y_1),(y_1,x_2),(x_2,y_2),(y_2,x_3) \\ \ldots$$
$x_0=1/2,y_0=\sqrt{3}/4$
To achieve:$x_0,y_0,x_1,y_1,\dots x_{12},y_{12}$
f[x_, y_] := 1/2 (x + Sqrt[1 - y^2]); xyList= DeleteDuplicates@ Flatten@NestList[{#[[2]], f[Sequence @@ ##]} &, {1/2, Sqrt[3]/4}, 10] // N {0.5, 0.433013, 0.700694, 0.573237, 0.760042, 0.611556, 0.775621, 0.621377, 0.779567, 0.623848, 0.780556, 0.624467}
Lastly
xlst=xyList[[Range[1, 12, 2]]] {0.5, 0.700694, 0.760042, 0.775621, 0.779567, 0.780556}
ylst=xyList[[Range[2, 12, 2]]] {0.433013, 0.573237, 0.611556, 0.621377, 0.623848, 0.624467}
