expm_sym

std.linalg.expm_sym · Level L3

The matrix exponential of a symmetric matrix, V·e^Λ·Vᵀ. Through the eigen-decomposition, so f(A) is exact in exact arithmetic. The reference is mpmath's expm at 100 digits, which does not diagonalize.

e^A = V·diag(e^λ)·Vᵀ

Signature

expm_sym(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.expm(A); exactly: mpmath at 100 digits to 80 digits (100-digit arithmetic), on all 40 test cases.
All 850 float64 results inside the running error bound; the closest uses 10% 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 4% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 4%
bit-equal to the NumPy formula in float64: 1%
largest error, in units in the last place: 4.1e+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

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Called by

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sha256:0d6c138ce42c06ef293006f21e98c0d6888ff04a519204883c6c8b65fdf787ab

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.expm_sym · Ω:expm_sym
Pins: sha256:a8793d4e5fdc348f9b501fe7a8a6553e60d23deb8ff687c2a2e8385dd373d88athis graph alone
Evidence: sha256:d3835e0c6af6d31112dd1faff494e5b97cd6ed0d715270bbfe650046fec0c6d2the 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.