The problem Linnainmaa's 1970 thesis solved was computing the sensitivity of accumulated floating-point rounding error to every local rounding error simultaneously

Every numerical algorithm running in finite-precision (floating-point) arithmetic introduces small errors at each elementary operation — rounding errors that arise from the impossibility of representing most real numbers exactly in binary. These local rounding errors compound as the computation proceeds, producing an accumulated (cumulative) rounding error in the final output.

The question Linnainmaa posed and answered in 1970 is:

Given a numerical algorithm with N elementary operations, each introducing a small local rounding error δᵢ, what is the contribution of each δᵢ to the final accumulated rounding error E — and can all N contributions be computed at once?

Formally, this is the problem of computing the first-order Taylor expansion of E with respect to all δᵢ simultaneously: finding ∂E/∂δᵢ for all i in one computation rather than N separate re-evaluations.

The 1976 BIT paper, which is the accessible English publication of the thesis work, states its subject as:

"determining the coefficients of the Taylor expansion of an accumulated rounding error with respect to the local rounding errors, and hence determining the influence of the local errors on the accumulated error. Second and higher order coefficients are also discussed."

The Taylor coefficients — ∂E/∂δᵢ — are the sensitivities: they say how much the accumulated error at the output would change if local rounding error i were slightly larger or smaller. Computing them individually would require re-running the algorithm N times (once per δᵢ). Linnainmaa's contribution was an algorithm that avoids this; see claim-linnainmaa-reverse-mode-single-pass for the mechanism.

Why this structure matters

The problem has a specific mathematical shape: many inputs (N local rounding errors), one output (the accumulated error). This is exactly the regime in which a backward propagation of derivatives is efficient — it recovers all N sensitivities in roughly one additional pass through the computation, regardless of N. Linnainmaa was not the first to observe this efficiency property in the abstract (the optimal control literature anticipated it), but he was the first to formalize it as a general algorithm for numerical error analysis.

This problem structure is identical in its mathematics to the problem of training a neural network: many parameters (weights), one scalar loss. The algorithm Linnainmaa derived for rounding errors is, mathematically, backpropagation. He arrived at it from a completely different domain. See backpropagation-gap and claim-linnainmaa-priority-not-paternity.

Source access note

The full text of Linnainmaa 1976 is paywalled at Springer Nature and ACM Digital Library. The abstract quoted above was accessed via BibBase (https://bibbase.org/network/publication/linnainmaa-taylorexpansionoftheaccumulatedroundingerror-1976) and Semantic Scholar, where it appears identically — consistent with authentic publisher metadata. The DOI resolves to the official Springer record.

Sources

• Linnainmaa, S. "Taylor expansion of the accumulated rounding error." BIT Numerical Mathematics, vol. 16, no. 2, pp. 146–160, 1976. DOI: 10.1007/BF01931367. Tier 1. Abstract accessed via BibBase (Tier 3 bibliographic database reproducing publisher metadata). Full paper not directly read.
• Wikipedia. "Automatic differentiation." https://en.wikipedia.org/wiki/Automatic_differentiation (accessed 2026-06-26). Tier 3. Confirms that reverse accumulation is efficient when "the number of inputs is larger than the number of outputs" — the exact structure of Linnainmaa's rounding-error problem.