sqrtm_spd

std.linalg.sqrtm_spd · Level L3

The principal square root of a symmetric positive-definite matrix, V·√Λ·Vᵀ: the symmetric positive-definite S with S·S = A. Through the eigen-decomposition, so f(A) is exact in exact arithmetic. The reference is mpmath's sqrtm at 100 digits, which does not diagonalize.

√A = V·diag(√λ)·Vᵀ

Signature

sqrtm_spd(A: f64[n, n]) → f64[n, n]

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.
Agrees with the reference scipy.linalg.sqrtm(A); exactly: mpmath at 100 digits to 80 digits (100-digit arithmetic), on all 40 test cases.
All 783 float64 results inside the running error bound; the closest uses 4% 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 2% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 8%
bit-equal to the NumPy formula in float64: 5%
largest error, in units in the last place: 1.2e+4

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.

Note

Exact for every symmetric A in exact arithmetic, since A = VΛVᵀ. The eigenvectors' signs cancel in VΛVᵀ, but their error bound grows as eigenvalues approach each other, so the cases have eigenvalues 1, 2, …, n.

Identity

Calls

—

Called by

—

sha256:3db581fd73d00ed3e0a43ac0f550d974d2d052f9453bbcce37bf5d2121ef81fd

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

Control handle

Symbol: Ω:std.linalg.sqrtm_spd · Ω:sqrtm_spd
Pins: sha256:395623224418bcdaf5724dfd13d3fbfeb743b90b14c40683a3894833ce1471aethis graph alone
Evidence: sha256:b247c5e1b2b89bf3df9837065d1024ac3d2423660177640b4a73689218d53bb6the 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.