recurrence_step

std.seq.recurrence_step · Level L0

One step of a linear recurrence in companion form: the window of the last r terms becomes C·window + b, where C shifts the window up and puts Σ coefficients·window in the last place. The body linear_recurrence scans.

w′ = C·w + b

Signature

recurrence_step(w: f64[r], t: f64[], C: f64[r, r], bv: f64[r]) → f64[r]

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.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference C @ w + bv in exact rational arithmetic, on all 40 test cases.
All 148 float64 results inside the running error bound; the closest uses 57% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 45% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 71%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 27

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

—

Called by

linear_recurrence

sha256:cf836ff674bc28f59be552c30fd4cc6e052a7533ff6cd0ab71989e0edec90e46

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.recurrence_step · Ω:recurrence_step
Pins: sha256:1219c1b105a9097c22c23077fdd85742bc5996779a7a4c1645a697e74ca89203this graph alone
Evidence: sha256:2234eea617f20b9c133b0fe63eca9d8cf1758bd9e017288619fbf7704b2a8902the 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.