SolveNt does not find all roots of equation

[MomoDeve]

Entity e = "arcsin(x) - x*pi/3";
Console.WriteLine(e.SolveEquation("x"));

Yes, this equation does not have analytical solution, but SolveNt() can (and should!) find all roots, as their number is finite
Output: {0}
Desired output: {-1/2, 0, 1/2}

[Rafael-SOWNet]

Still open — your case is not fixed. arcsin(x) - x*pi/3 still answers { 0 } and not { -1/2, 0, 1/2 }. Leaving this open deliberately, but recording what the investigation found, since two other defects were behind the same symptom and both are now fixed.

Fixed, #695. The Newton grid collapsed to a single point. The shares that lay out the starting grid were computed with EDecimal's bare /, which carries no precision context and returns NaN for any quotient that does not terminate in base ten. Only step counts of the form 2^a·5^b divide exactly, so for 3, 6, 7, 9, 12, 21 and most other values every share but the first came out NaN, and with it every starting point — the grid quietly shrank to the corner where x and y are both zero. The default of 10 divides exactly, which is why it went unseen, and why asking for a finer search made the answer worse. x^3 - 2x gave all three roots at 10 steps and one at 21.

Fixed, #696. Compiled arcsin returned the conjugate of the arcsine off the real axis — a five-year-old defect. Newton's method felt it as divergence away from a root it had been started on, which matters here because your equation is an arcsine.

Not fixed, and the reason yours still fails. The default grid is 10 steps over a region that does not tighten around the interval where the roots live. All three of your roots lie in (-1, 1), which the grid steps over; raising StepCount finds them, and now genuinely does since #695:

MathS.Settings.NewtonSolver.As(
    new MathS.Settings.NewtonSetting { From = (-1, -1), To = (1, 1), StepCount = (20, 20) },
    () => "arcsin(x) - x*pi/3".ToEntity().SolveEquation("x"));
// { -1/2, 0, 1/2 }

Making that the default is a cost decision — every solve pays for it — so it is the maintainers' call rather than mine. Three options as I see them: raise the default step count; choose the region from the expression rather than fixing it; or leave it and document the setting. Happy to implement whichever.

[Rafael-SOWNet]

Now fixed, in #729, merged to master as 552e825. This supersedes the comment above.

	before	after
"arcsin(x) - x*pi/3".SolveEquation("x")	{ 0 }	{ -1/2, 0, 1/2 }
the same via SolveNt	1 root	3 roots

Which is the desired output in the report, exactly.

The comment above had the diagnosis right — "the default grid is 10 steps over a region that does not tighten around the interval where the roots live" — but read it as something only a finer grid could fix. It is worth saying why that was too pessimistic.

The grid is two-dimensional: a step count of N costs N² Newton runs but lays real starting points only (To - From) / N apart. At the default that is 2, and the three roots span one unit in total, so they shared a single starting point. Buying real-axis resolution by raising N costs N², which is what made this look expensive.

But a sign change is a far cheaper witness of a root than a Newton run is — one evaluation against the sixty an iteration to precision 30 costs. So the real axis is now scanned at StepCount.Re * StepCount.Im points, as many as the grid has starting points, and Newton runs only from the brackets it finds. The spacing that matters for a real root becomes (To - From) / N² rather than (To - From) / N — 0.2 at the default rather than 2 — for about 2% of what the grid already spends. No setting has to change.

One thing the comment above got wrong and worth correcting: it treated close roots as generally unreachable. They are not. x*(x - 1/2)*(x + 1/2) has roots the same 0.5 apart and gave all three even before this, because a polynomial's basins interleave across the whole plane. arcsin is the hard case because outside [-1, 1] every starting point hands it a complex value and the iteration wanders off — so the failure was specific to an expression that is real only on a small interval, not to close roots as such.

The fix is additive and the grid is untouched. A sign change witnesses a root of odd multiplicity on an interval where the expression is real, so repeated roots (x^2 + 2x + 1) and roots off the real axis (x^2 + 1) stay the grid's to find; tests pin both. Regression tests in SolverRegressionTest.cs and NewtonGridTest.cs. Full suite green, 117-problem corpus unchanged.