euler_linear

std.seq.euler_linear · Level L2

Integrate x′ = A·x from x₀ by forward Euler over the time steps dts; returns the state after every step. A Scan of euler_step.

xₜ₊₁ = xₜ + dtₜ·A·xₜ

Signature

euler_linear(x0: f64[k], A: f64[k, k], dts: f64[n]) → f64[n, k]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

x0f64[k]Af64[k, k]dtsf64[n]Scan·euler_stepXXf64[n, k]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Equal to the reference x = x0; for each dt: x = x + dt * (A @ x) in exact rational arithmetic, on all 40 test cases.
  • All 533 float64 results inside the running error bound; the closest uses 58% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
70%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
86

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls
euler_step
Called by
—
sha256:5a9a9b87a3fb9d0929353f47bf7f95cda74549bac1962aa7b92a9e18227459e9

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol
Ω:std.seq.euler_linear · Ω:euler_linear
Pins
sha256:eaed1952e797949c6a07da6bb5a9fd911f13c0c672a10e67975e2d04af5061cbthis graph and the 1 it reaches through calls
Evidence
sha256:972f80db7ea1f5d9c59f66a6453495d789b525602a8455402fd03d8f8bccfed3the hash of its verification record
Needs
no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.