radius_estimates

std.seq.radius_estimates · Level L2

Two estimates of the radius of convergence R of Σ aₙ(x − c)ⁿ from its coefficients a₀ … a_N: the ratio |a_{N−1}/a_N| and the root |a_N|^(−1/N). Each converges to R when the corresponding limit exists, and for aₙ = qⁿ both equal 1/|q|. A finite prefix of coefficients does not determine R, so these are estimates, not the radius.

R ≈ |a_{N−1}/a_N|, R ≈ |a_N|^(−1/N)

Signature

radius_estimates(a: f64[n]) → f64[], f64[]

Requires: n ≥ 2

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 with n ≥ 2.
Agrees with the reference |a[N−1]/a[N]| and |a[N]|^(−1/N) to 80 digits (100-digit arithmetic), on all 40 test cases.
All 80 float64 results inside the running error bound; the closest uses 70% 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 89% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 88%
bit-equal to the NumPy formula in float64: 88%
largest error, in units in the last place: 3.02

Identity

Calls

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Called by

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sha256:8c512c2d3dca4b9e307ff2132af619d653be1c84eb1eb87c7e167f5f5b9e7df2

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.radius_estimates · Ω:radius_estimates
Pins: sha256:3296ace761609cc030756e3338c6babab4a789455c3772cee3decf43fea36853this graph alone
Evidence: sha256:11e3b231ec6777713d6bfb064fa2bb4a789d2d4eecc3af674e5c118ad1b5f78dthe 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.