Lay out the whole Newton grid, not one corner of it (#115)

[Rafael-SOWNet]

The defect

SolveNt searches from a grid of starting points. Each point's position is computed as

var xShare = ((EDecimal)x) / settings.StepCount.Re;

EDecimal's division operator carries no context, and PeterO's EDecimal answers NaN when the quotient does not terminate in base ten. Only step counts of the form 2^a * 5^b divide exactly. 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 — so the search covered the single corner where both indices are zero, and said nothing about it.

The visible effect is that asking for a finer search made the answer worse:

StepCount	real roots of x^3 - 2x found
8, 10, 16, 20, 40	-1.414214, 0, 1.414214
9, 12, 21, 41	1.414214

sin(x) - x/2 loses both outlying roots the same way. The default of (10, 10) divides exactly, which is why nothing caught this.

The fix

Divide with CtxDivide, which is what the rest of the library divides decimals with (Entity.Number.CtxDivide, Core/Entity/Continuous/Number/Operators.cs) and which rounds to the working precision. One line each for the real and imaginary share.

Nothing changes at any step count that already divided exactly, the default included. The full suite is unchanged: Failed: 0, Passed: 4018, Skipped: 14, Total: 4032.

Tests

Sources/Tests/UnitTests/Algebra/SolveTest/NewtonGridTest.cs, 9 cases. 5 of them fail without the source change (verified by reverting it in place and re-running):

• EveryStepCountSearchesTheWholeRegion at 9, 21, 41
• AFinerSearchKeepsWhatACoarserOneFound — states the broken property without naming a step count
• AStepCountThatDoesNotDivideExactlyStillReachesTheOutlyingRoots

The four passing cases (8, 10, 12, 20) are there to pin that the working step counts stay working. 12 passes here because x^3 - 2x happens to be reachable from the corner; it is a NaN grid all the same.

What this does not do

This is not the whole of #115. arcsin(x) - x*pi/3 still returns {0} at the default step count: its other two roots are at ±1/2, inside (-1, 1), and the default grid has spacing 2 over [-10, 10], so it steps over them. At 40 steps it now finds 1/2 but not -1/2 — an asymmetry I have not diagnosed and am not guessing at here.

What this fixes is that raising the step count, the documented way to look harder, was not reliably possible. That is a prerequisite for the rest of #115 rather than the answer to it.

I also noticed but did not change that the grid samples To and never From — x runs 0 .. StepCount-1, so at the default the real starts are 10, 8, ..., -8 and -10 is never tried. Making the grid endpoint-inclusive changes the default sample points, which has a blast radius this fix does not, so it belongs in its own change if it is wanted.

🤖 Generated with Claude Code

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 81.59%. Comparing base (90c00a8) to head (bc447f2).
⚠️ Report is 73 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #695      +/-   ##
==========================================
+ Coverage   80.99%   81.59%   +0.60%     
==========================================
  Files         155      157       +2     
  Lines       13687    13347     -340     
  Branches     1957     2198     +241     
==========================================
- Hits        11086    10891     -195     
+ Misses       1990     1849     -141     
+ Partials      611      607       -4     

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[Rafael-SOWNet]

CI, run on my fork across the full matrix, since the run here is held at action_required:

https://github.com/Rafael-SOWNet/AngouriMath/actions/runs/30919850130

job	result	tests
Test (ubuntu-latest)	success	Failed: 0, Passed: 4018, Skipped: 14, Total: 4032
Test (macos-latest)	success	Failed: 0, Passed: 4018, Skipped: 14, Total: 4032
Test (windows-latest)	success	Failed: 0, Passed: 4018, Skipped: 14, Total: 4032

The branch it ran on, ci-check/newton-grid, is this PR's head plus one commit that only touches .github/workflows/CSharpTest.yml (adding workflow_dispatch: and ci-check/** to the push branches, so the run can be triggered at all). Everything under Sources/ is byte-identical to this PR:

$ git diff --stat fix/newton-grid-step-count ci-check/newton-grid
 .github/workflows/CSharpTest.yml | 2 ++
 1 file changed, 2 insertions(+)