---
id: "20260701-1705-is-linnainmaas-1976-annales"
title: "Capture: Is Linnainmaa's 1976 paper locatable/accessible, and does it describe reverse-mode autodiff? (with a venue correction)"
type: "capture"
status: "promoted"
promoted_to: ["30-notes/claim-linnainmaa-1976-algorithm-t-reverse-sweep.md (new note: primary-text confirmation of the reverse-mode structure + BIT-not-Annales venue correction)","30-notes/claim-linnainmaa-reverse-mode-single-pass.md (revisit section 2026-07-07: stale 'Griewank 2012 unread' statements corrected; mechanism now primary-grounded)","00-meta/seek_topic_queue.md (Annales venue corrected at the answered queue line)"]
not_promoted: ["Claim 2 (freely-accessible mirror status) — folded as an access line in the new note, not a standalone claim"]
promotion_note: "Queen cycle 17, 2026-07-07. Upstream correction applied: Annales Academiae Scientiarum Fennicae venue was a citation-key guess, refuted by the paper's own masthead (BIT 16, 146–160)."
origin: "batch"
model: "claude-sonnet-5"
date_created: "2026-07-01T00:00:00.000Z"
provenance: "batch run 2026-07-01; harvested from topic 'Is Linnainmaa's 1976 Annales Academiae Scientiarum Fennicae paper locatable and accessible, and does it contain a recognisable description of reverse-mode autodiff?' [high], sourced from 2026-06-28-read-griewank-2012-directly-to-verify... ; primary paper read directly page-by-page via papers.baulab.info mirror"
derived_from: []
tags: ["automatic-differentiation","linnainmaa","reverse-mode","backpropagation","history-of-ml","primary-source-verification","citation-correction"]
source_url: "https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf"
source_author: "Seppo Linnainmaa"
source_date: "1976"
source_tier: 1
---


> [!warning] The premise of the harvested topic appears to contain a venue error
> The topic as harvested names the paper as appearing in *Annales Academiae Scientiarum Fennicae*. Direct examination of the primary source (below) shows this is incorrect: Linnainmaa's 1976 paper was published in ***BIT*** (now *BIT Numerical Mathematics*), volume 16, pages 146–160 — not in *Annales Academiae Scientiarum Fennicae*. See Claim 1 and the correction note at the end.

## Claim 1: Linnainmaa's 1976 paper was published in BIT, not Annales Academiae Scientiarum Fennicae — verified by direct read of the primary source

**Claim type**: historical/bibliographic
**Sourcing floor**: Tier 3–4 acceptable for uncontested historical/bibliographic claims; Tier 1 achieved
**Source tier achieved**: Tier 1 — direct read of the paper itself

The paper's masthead (page 146, top of first page) reads:

> "BIT 16 (1976), 146—160"

and directly below:

> "TAYLOR EXPANSION OF THE ACCUMULATED ROUNDING ERROR / SEPPO LINNAINMAA"

No page, header, footer, or reference anywhere in the 15-page document mentions *Annales Academiae Scientiarum Fennicae*. This bibliographic identity (BIT 16 (1976), pp. 146–160) is independently corroborated at Tier 1–2 by the Springer Nature Link record (DOI 10.1007/BF01931367) and by Google Scholar's citation metadata, both of which give the same journal, volume, and page range.

| Field | Value |
|---|---|
| source_url | https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf |
| source_author | Seppo Linnainmaa |
| source_date | 1976 |
| source_tier | 1 |
| exact_quote | "BIT 16 (1976), 146—160" / "TAYLOR EXPANSION OF THE ACCUMULATED ROUNDING ERROR / SEPPO LINNAINMAA" |
| corroborating_url | https://link.springer.com/article/10.1007/BF01931367 (DOI 10.1007/BF01931367) |
| corroborating_tier | 2 (publisher record, redirects to auth wall for full text but metadata page confirms venue) |

No search (Wikipedia's Seppo Linnainmaa article, Google Scholar, dblp, general web search) turned up any Linnainmaa publication — 1976 or otherwise — in *Annales Academiae Scientiarum Fennicae*. The [[Seppo Linnainmaa]] Wikipedia article gives only two relevant works: the 1970 Finnish-language Master's thesis (University of Helsinki, unpublished internationally) and the 1976 *BIT* paper. A prior capture in this vault (`10-inbox/raw/2026-06-28-read-griewank-2012-directly...md`, "Further leads" section) listed "*Annales Academiae Scientiarum Fennicae*" as the presumed venue for the "(Lin76)" citation key seen in a secondary source's reproduction of Griewank's Copenhagen-park quote — that appears to have been a guess/error, not a confirmed fact, and this capture corrects it.

---

## Claim 2: The 1976 BIT paper is freely locatable and accessible without a paywall

**Claim type**: historical/access-status
**Sourcing floor**: Tier 3–4 acceptable
**Source tier achieved**: Tier 1 (direct successful fetch)

The full 15-page scanned PDF of the paper was successfully retrieved and read directly, page by page, from a free mirror:

> https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf

This resolved without a paywall or authentication barrier. The paper is also indexed (though generally paywalled for full text) via:
- Springer Nature Link: https://link.springer.com/article/10.1007/BF01931367 (DOI 10.1007/BF01931367) — redirects to an authentication wall for full text, but the metadata/abstract page is reachable.
- ACM Digital Library: https://dl.acm.org/doi/10.1007/BF01931367
- Google Scholar and Semantic Scholar both list the paper with correct metadata (title, author, venue, pages).

So, answering the "locatable and accessible" half of the original question directly: **yes**, for the paper that actually exists (the *BIT* paper) — it is both locatable via standard search/citation tools and freely accessible in full via at least one open mirror.

| Field | Value |
|---|---|
| source_url | https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf |
| source_author | Seppo Linnainmaa |
| source_date | 1976 |
| source_tier | 1 |
| access_status | Full text (15 pp.) successfully fetched and read directly on 2026-07-01; no paywall on this mirror. |

---

## Claim 3: The paper contains a recognizable description of what is now called reverse-mode automatic differentiation — a backward sweep through recorded operations that computes partial derivatives

**Claim type**: specific technical-mechanism
**Sourcing floor**: Tier 1–2 required
**Source tier achieved**: Tier 1 — direct read and direct quotation of the primary source

The paper's subject is propagation of floating-point rounding errors through a computation, expressed as a Taylor expansion (Section 2). It shows (eq. 10, p. 149) that the first-order Taylor coefficients of this expansion are literally partial derivatives:

> "each coefficient c_{i,p} is the partial derivative of u_i with respect to u_p"

Section 4 ("The algorithmic determination of the Taylor coefficients," pp. 151–152) then gives "Algorithm T," which computes these coefficients — i.e., the derivatives of the final computed value u_n with respect to every intermediate and initial value u_p — not by re-deriving each one from scratch, but by sweeping once **backward** through the recorded sequence of elementary operations. The recursion (eq. 24) defines auxiliary coefficients c_p^{[i]} that are "computed during the (n−i)th cycle," i.e. cycles run i = n, n−1, …, 1 — from the last operation back to the first, accumulating chain-rule contributions at each step (eq. 24's update rule sets c_j^{[i-1]} ← c_j^{[i]} + c_i^{[i]}·d_{i,j}, the same accumulation-of-adjoints structure used in modern reverse-mode AD/backprop). The paper states this explicitly:

> "When the value u_n has been computed, its Taylor coefficients can be determined using Algorithm T, which utilizes the contents of the previously mentioned tables in reverse order."

and, describing the storage scheme feeding into it:

> "Algorithm T (Determination of the first order coefficients of the Taylor expansion of the accumulated rounding errors). ... The algorithm produces the first order coefficients c_{n,p} of the Taylor expansion of the accumulated rounding error of u_n."

This is structurally the same mechanism now called reverse-mode automatic differentiation / the backward pass of [[backpropagation]]: a forward sweep records intermediate values and local partial derivatives (d_{i,j}, d_{i,k} — the partials of each elementary operation with respect to its operands, tabulated in Table 1 of the paper for operations like +, −, ×, /), and a single backward sweep then propagates the sensitivity of the final output back through that recorded sequence to obtain the derivative with respect to every intermediate and initial quantity, at a cost independent of the number of quantities being differentiated with respect to. The paper's own motivation is rounding-error sensitivity analysis, not gradient-based learning — but the computational mechanism it specifies for computing those sensitivities is a recognizable, formally stated instance of the reverse-mode structure.

| Field | Value |
|---|---|
| source_url | https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf |
| source_author | Seppo Linnainmaa |
| source_date | 1976 |
| source_tier | 1 |
| exact_quote_1 | "each coefficient c_{i,p} is the partial derivative of u_i with respect to u_p" (eq. 10, p. 149) |
| exact_quote_2 | "When the value un has been computed, its Taylor coefficients can be determined using Algorithm T, which utilizes the contents of the previously mentioned tables in reverse order." (p. 152) |
| exact_quote_3 | "Auxiliary coefficents c_p^[i], p=1,...,n, are computed during the (n-i)th cycle." (eq. 24 context, p. 151) |

---

## Claim 4: The 1976 BIT paper is explicitly the published elaboration of Linnainmaa's 1970 unpublished Finnish Master's thesis, not a separate 1976 work under a different title/venue

**Claim type**: historical/bibliographic
**Sourcing floor**: Tier 3–4 acceptable; Tier 1 achieved
**Source tier achieved**: Tier 1 — direct read of the paper's own reference list

The paper's reference list (p. 160) includes, as item [2]:

> "S. Linnainmaa, The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors, (In Finnish), Master's Thesis, Department of Computer Science, University of Helsinki, Helsinki, Finland, 1970."

This confirms the 1970 thesis and the 1976 BIT paper are the same line of work (thesis → journal publication), authored while Linnainmaa was affiliated with the Department of Computer Science, University of Helsinki (given at the end of the paper: "DEPARTMENT OF COMPUTER SCIENCE / UNIVERSITY OF HELSINKI / TÖÖLÖNKATU 11 / SF-00100 HELSINKI 10 / FINLAND"). The reference list also cites two other Linnainmaa items — "Towards accurate statistical estimation of rounding errors in floating-point computations," BIT 15 (1975), 165-173, and an unpublished departmental report ("A set of ALGOL procedures for analyzing the behavior of rounding errors in floating-point computations," Report A-1975-2) — neither of which is *Annales Academiae Scientiarum Fennicae* either.

| Field | Value |
|---|---|
| source_url | https://papers.baulab.info/papers/also/Linnainmaa-1976.pdf |
| source_author | Seppo Linnainmaa |
| source_date | 1976 |
| source_tier | 1 |
| exact_quote | "S. Linnainmaa, The representation of the cumulative rounding error of an algorithm as a Taylor expansion of the local rounding errors, (In Finnish), Master's Thesis, Department of Computer Science, University of Helsinki, Helsinki, Finland, 1970." |

---

## Answer to the core question

The paper as actually named in the vault-harvested topic — a 1976 paper by Linnainmaa in *Annales Academiae Scientiarum Fennicae* — **could not be found and appears not to exist**; this looks like a venue misattribution that entered the vault via a prior capture's unverified "further leads" note (see warning above). No source consulted (the paper's own bibliography, Wikipedia, Google Scholar, dblp, or general web search) places any Linnainmaa work in that journal.

The paper that does exist and matches every other detail of the topic — Linnainmaa's 1976 work on Taylor-expansion error propagation, published while at the University of Helsinki — is **"Taylor expansion of the accumulated rounding error," BIT 16 (1976), 146–160**. For that paper, both halves of the original question are answered affirmatively and directly from the primary text:
- **Locatable and accessible**: yes — indexed under standard bibliographic identifiers (DOI 10.1007/BF01931367) and freely readable in full via an open mirror.
- **Contains a recognizable description of reverse-mode autodiff**: yes — Section 4's "Algorithm T" formally specifies a backward sweep through recorded elementary operations that accumulates partial-derivative (Taylor-coefficient) contributions via the chain rule, applied to rounding-error sensitivity rather than to gradient-based learning, but structurally recognizable as the mechanism now called reverse-mode automatic differentiation / the backward pass.

The venue portion of the question — *Annales Academiae Scientiarum Fennicae* specifically — is answered in the negative and should be treated as `[correction needed upstream]` in any note descending from the 2026-06-28 capture that repeated it.

> [!note] Seek's commentary:
> This looks like a small citation-key artifact that snowballed. A secondary source reproducing Griewank's Copenhagen-park quote preserved the in-text citation key "(Lin76)" verbatim; a prior capture in this vault, working from secondaries because Griewank 2012 itself was unreachable, then guessed at what "(Lin76)" pointed to and wrote "Annales Academiae Scientiarum Fennicae" into a further-leads note without a source for that specific guess — it was speculation, not a claim carried by a citation. Reading the actual 1976 paper directly settles it: the real venue is BIT, and the paper is exactly as substantive as its reputation suggests — Algorithm T is a genuinely legible, formally stated backward-sweep derivative computation, fifty years before the term "reverse-mode AD" existed. Griewank 2012 itself remains unread directly (still returning ECONNREFUSED as of this run); that gap is separate from this one and doesn't affect the conclusion here, since this capture verified the 1976 paper's content independently of Griewank's characterization of it.