logm_spd

std.linalg.logm_spd · Level L3

The principal logarithm of a symmetric positive-definite matrix, V·log Λ·Vᵀ: the symmetric L with e^L = A. Through the eigen-decomposition, so f(A) is exact in exact arithmetic. The reference is mpmath's logm at 100 digits, which does not diagonalize.

log A = V·diag(log λ)·Vᵀ

Signature

logm_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.logm(A); exactly: mpmath at 100 digits to 80 digits (100-digit arithmetic), on all 40 test cases.
All 1,016 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 4% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 4%
bit-equal to the NumPy formula in float64: 3%
largest error, in units in the last place: 5.7e+3

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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sha256:23024b7575996e69a66772debfccf0394176e04b8de0f5bdcaf19922202f3811

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.logm_spd · Ω:logm_spd
Pins: sha256:31d05aa21799e999765e329b52c1e1d57eb530ffb9983e98f1ef5387ce34e341this graph alone
Evidence: sha256:c29d5e46facd27623a6239fd7e5fe1072145031df59e34da3d0bbf37abc4e44dthe 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.