arxiv: 2605.06698 · v2 · submitted 2026-05-04 · 🧮 math.GM

Recognition: no theorem link

A fixed point iteration method for the arctangent with any odd order of convergence based on sine and cosine

Alois Schiessl

Authors on Pith no claims yet

Pith reviewed 2026-05-13 01:54 UTC · model grok-4.3

classification 🧮 math.GM

keywords arctangentfixed point iterationconvergence ordersine cosinenumerical approximationpi computation

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The pith

A fixed-point iteration using sine and cosine converges to arctan(t) with exact order 2P+1 for any natural number P.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a family of iteration functions T that combine the current approximation x with a partial sum of the arctangent series applied to a ratio built from sine and cosine of x. For any positive t and any natural number P, the map subtracts the truncated series in powers of that ratio, which equals the tangent of the angle difference. Starting from an initial guess sufficiently near arctan(t), repeated application produces a sequence that reaches the target with convergence order precisely 2P+1. The construction therefore supplies a direct way to obtain any desired odd order of convergence while using only evaluations of sine and cosine. A numerical example computing pi/4 illustrates that the scheme runs efficiently in practice.

Core claim

We define T(x) = x - sum_{k=1 to P} (-1)^{k-1}/(2k-1) * [(sin(x) - t cos(x))/(cos(x) + t sin(x))]^{2k-1}. For every initial value x0 sufficiently close to arctan(t), the sequence x_{n+1} = T(x_n) converges to arctan(t) with order of convergence exactly 2P+1.

What carries the argument

The iteration map T(x) obtained by subtracting a P-term truncation of the arctangent power series from the tangent-difference formula expressed in sine and cosine.

If this is right

Increasing P raises the convergence order by two each time without changing the form of the iteration.
The method applies unchanged to any positive real t.
Each step requires only a fixed number of sine and cosine calls independent of the desired order.
The same truncation technique yields an explicit error expansion whose leading term determines the asymptotic constant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction may be adapted to other inverse trigonometric functions by replacing the tangent-difference identity with the appropriate addition formula.
Pairing the iteration with an initial coarse approximation obtained from a different method could enlarge the basin of attraction.
The exact odd order suggests the underlying error is governed by the next odd power in the arctangent series remainder.

Load-bearing premise

The starting value must lie sufficiently close to arctan(t) for the local convergence analysis to apply, and sine and cosine must be evaluated with accuracy high enough that rounding errors remain smaller than the iteration truncation.

What would settle it

Compute successive error ratios |e_{n+1}| divided by |e_n| raised to the power 2P+1 for a sequence generated by T; if the ratios tend to a nonzero finite limit the claimed exact order holds, while systematic deviation from that limit would show the order is not 2P+1.

read the original abstract

In this paper, we present a fixed point method for the arctangent based on sine and cosine. Let $t\in \mathbb{R}^{+}$ and $P\in \mathbb{N}$. We define: \[T\left(x\right)=x-\sum_{k=1}^{P}\,\frac{\left(-1\right)^{k-1}}{2\,k-1} \left(\frac {\sin\!\left(x\right)-t\cos\!\left(x\right)} {\cos\!\left(x\right)+t\sin\!\left(x\right)} \right)^{2\,k-1}.\] For every initial value $x_0$ sufficiently close to $\arctan\left(t\right)$, the sequence \[x_{n+1}=T\left(x_{n}\right)\;;\,n=0,1,\ldots\] is converging to $\arctan\left(t\right)$ with order of convergence exactly $\left(2\,P+1\right)$. The computational test we performed demonstrates the efficiency of the method. \selectlanguage{ngerman} \[\] \[\textbf{Zusammenfassung}\] In dieser Abhandlung stellen wir ein Fixpunktverfahren zur Berechnung des arcustangens auf Basis von sinus und cosinus vor. Es sei $t\in \mathbb{R}^{+}$ und $P\in\mathbb{N}$. Wir definieren: \[T\left(x\right)=x-\sum_{k=1}^{P}\,\frac{\left(-1\right)^{k-1}}{2\,k-1} \left(\frac {\sin\!\left(x\right)-t\cos\!\left(x\right)} {\cos\!\left(x\right)+t\sin\!\left(x\right)}\right) ^{2\,k-1}.\] F\"ur jeden Startwert $x_0$ hinreichend nahe bei $\arctan\left(t\right)$ konvergiert die Folge \[x_{n+1}=T\left(x_{n}\right)\;;\,n=0,1,\ldots\] gegen $\arctan\left(t\right)$ mit Konvergenzordnung genau $\left(2\,P+1\right)$. Anhand einer praktischen Berechnung von $\frac{\pi}{4}$ zeigen wir die Effizienz des Verfahrens. \[\text{Deutsche Version ab Seite 17}\]

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper shows how to build fixed-point arctan iterations of any odd order 2P+1 from the tangent subtraction identity and the arctan power series.

read the letter

The paper gives a clean construction for fixed-point iterations that compute arctan to any odd order of convergence you choose. You pick a natural number P, and the iteration subtracts the first P terms of the arctan series applied to (sin x - t cos x)/(cos x + t sin x). By the tangent subtraction formula, that argument is tan(x - arctan(t)), so the whole thing is like applying a truncated inverse tan to the error. The remainder term in the series then ensures the error multiplies by a factor that vanishes to order 2P, giving overall convergence order exactly 2P+1. What the paper does well is spell this out without unnecessary complication. The abstract and the construction make the order claim transparent once you recall the series expansion of arctan. The computational test they include for pi/4 shows the method in action and confirms faster convergence for larger P. A soft spot is that everything is local: the analysis assumes the starting point is close enough that |tan(error)| stays small. In practice, for computing arctan of a given t, you might still need a separate way to get a rough estimate first. The paper does not explore global convergence or how to combine this with other starters. Also, while the order is high, the per-step work involves evaluating sine and cosine, so the efficiency gain depends on how those are implemented. This kind of paper suits readers who care about high-order methods for elementary functions or fixed-point iterations in general. Someone looking for new ways to accelerate convergence in numerical libraries might find the tunable order interesting. The math is self-contained and the claims line up with the derivation. I think it deserves peer review. The result is reproducible from the given formulas, and the exact order is a nice feature even if arctan computation is a solved problem in many ways.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a fixed-point iteration T(x) for arctan(t) with t > 0. For each natural number P, T(x) subtracts the first P terms of the arctan Taylor series from x, where the series argument is the tangent-subtraction expression (sin(x) - t cos(x)) / (cos(x) + t sin(x)). The central claim is that, for any initial x0 sufficiently close to arctan(t), the iterates converge to arctan(t) with exact order of convergence 2P + 1. A numerical example computing pi/4 is provided to illustrate efficiency.

Significance. If the local convergence analysis and exact-order claim are fully established, the construction supplies a simple, tunable family of high-order fixed-point methods for arctan that require only sine and cosine evaluations. The fact that the order is exactly 2P + 1 (rather than at least 2P + 1) follows directly from the nonzero leading coefficient in the arctan remainder and is a clear technical strength.

minor comments (3)

The abstract states the exact-order claim but does not display the leading error term; the manuscript should explicitly note that the coefficient of the (2P+1) term in the arctan remainder is nonzero (as follows from the standard series expansion) to confirm the order is not higher.
The numerical demonstration for pi/4 should report the observed error ratios or convergence orders for at least two values of P to allow direct verification of the theoretical rate.
The English and German versions of the abstract and main text should be cross-checked for identical mathematical statements and consistent notation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The manuscript provides a complete local convergence analysis establishing the exact order 2P+1; we address the referee's points on the claim and its verification below.

read point-by-point responses

Referee: The central claim is that, for any initial x0 sufficiently close to arctan(t), the iterates converge to arctan(t) with exact order of convergence 2P + 1. If the local convergence analysis and exact-order claim are fully established, the construction supplies a simple, tunable family of high-order fixed-point methods for arctan that require only sine and cosine evaluations. The fact that the order is exactly 2P + 1 (rather than at least 2P + 1) follows directly from the nonzero leading coefficient in the arctan remainder.

Authors: The local convergence analysis is fully developed in Section 3 of the manuscript. We expand the error e_{n+1} = T(x_n) - arctan(t) using the Taylor series of the arctangent remainder after P terms, composed with the tangent-subtraction formula for the argument. The leading term is shown to be a nonzero multiple of e_n^{2P+1} whose coefficient depends only on t and P (explicitly nonzero for t > 0), while all lower-order terms cancel by construction of the partial sum. This establishes both convergence for x0 sufficiently close to arctan(t) and the exact order 2P+1. The nonzero leading coefficient is verified by direct differentiation or by the known remainder formula for the arctangent series. revision: no
Referee: A numerical example computing pi/4 is provided to illustrate efficiency.

Authors: The numerical test in Section 4 compares the iteration for several values of P against the standard arctangent series and Newton iteration, confirming the predicted orders and the practical advantage of using only sine and cosine evaluations per step. We can add a short table of observed convergence orders if the referee finds it helpful for clarity. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses external identities and series remainder

full rationale

The iteration T(x) is obtained by composing the tangent subtraction formula (sin x - t cos x)/(cos x + t sin x) = tan(x - arctan(t)) with the partial sum of the known arctan Taylor series up to P terms. The error recurrence then reduces to e_{n+1} = remainder of arctan series after P terms evaluated at tan(e_n), whose leading term is asymptotically nonzero and of exact degree 2P+1. This follows from standard external analysis of the series remainder and the fixed-point theorem; no parameters are fitted to data, no self-citations are load-bearing for the central claim, and the order is not assumed or renamed but derived from the asymptotic expansion. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard mathematical facts and one user-chosen integer; no new entities are postulated.

free parameters (1)

P
Positive integer chosen by the user to set the number of terms and thus the convergence order; not fitted to data.

axioms (2)

standard math The Taylor series arctan(u) = sum_{k=1}^infty (-1)^{k-1} u^{2k-1}/(2k-1) for |u|<1
Truncated at P terms to produce the correction inside T(x).
standard math The identity (sin x - t cos x)/(cos x + t sin x) = tan(x - arctan(t))
Used to interpret the argument of the arctan series as the tangent of the current error.

pith-pipeline@v0.9.0 · 5745 in / 1572 out tokens · 51241 ms · 2026-05-13T01:54:34.908794+00:00 · methodology

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Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

Handbook of Mathematical Functio ns, 10th ed.; Applied Mathematics Series; National Bureau of Standards: Washington, DC, USA, 1972

Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functio ns, 10th ed.; Applied Mathematics Series; National Bureau of Standards: Washington, DC, USA, 1972

work page 1972
[2]

JUNGHENN: A COURSE IN REAL ANALYSIS CRC Press Taylor and Francis Group 2015

HUGO D. JUNGHENN: A COURSE IN REAL ANALYSIS CRC Press Taylor and Francis Group 2015

work page 2015
[3]

Walter Gautschi: Numerical Analysis Second Edition Birkh¨ auser 2012

work page 2012
[4]

Schaums’s Outline Series: Mathematical Handbook of Formulas an d Tables McGraw-Hill Book Company 1968

work page 1968
[5]

Carl St¨ ormer: Solution compl` ete en nombres entiers de l’´ eq uation m ·arctan (x) + n · arctan (y) = k · π 4 ; Bulletin de la S. M. F., tome 27 (1899), p. 160-170

work page
[6]

J¨ org Arndt: Arctan relations for Pi https://www.jjj.de/arctan/arctanpage.html

work page
[7]

Le Phuong Quan and Th´ ai Anh Nhan: Applying Computer Algebra S ystems in Approxi- mating the Trigonometric Functions MDPI; Mathematical and Computational Applications https://www.mdpi.com/2297-8747/23/3/37

work page
[8]

BRENT: Fast Algorithms for High-Precision Computat ion of Elementary Functions, Australian National University Canberra, ACT 0200, Austra lia 2006

RICHARD P. BRENT: Fast Algorithms for High-Precision Computat ion of Elementary Functions, Australian National University Canberra, ACT 0200, Austra lia 2006

work page 2006
[9]

Georg Hager and Gerhard Wellein: High performance computing fo r Scientists and En- gineers, CRC Press Taylor Francis Group, 6000 Broken Sound Parkway NW , Suite 300: (2010)

work page 2010
[10]

Barrio: Performance of the Taylor series method Appl

R. Barrio: Performance of the Taylor series method Appl. Math. Comput., vol. 163 (2005)

work page 2005
[11]

Nikolaos P. Bakas: Taylor Polynomials in High Arithmetic Precision https://www.arxiv.org/abs/1909.13565 16 Ein iteratives Verfahren zur Berechnung von arctan(x) mit beliebiger ungerader Konvergenzordnung Alois Schiessl ∗

work page arXiv 1909
[12]

Es sei t ∈ R+ und P ∈ N

Mai 2026 Zusammenfassung In dieser Abhandlung stellen wir ein Fixpunktverfahren zur Berechnung des Arcustan- gens mit beliebiger ungerader Konvergenzordnung vor. Es sei t ∈ R+ und P ∈ N. Wir deﬁnieren die Fixpunktiteration T (x) = x − P∑ k=1 (− 1)k− 1 2 k − 1 ( sin(x) − t cos(x) cos(x) + t sin(x) ) 2 k− 1 . F¨ur jeden Startwert x0 hinreichend nahe bei arc...

work page 2026
[13]

entnehmen und auf unsere Bed ¨urfnisse zuschneiden: ( ( x − arctan (t) ) 2 P ) (k) = (2 P ) ! (2 P − k) ! ( x − arctan (t) ) 2 P − k , k = 0, 1, . . . , 2 P . Insbesondere stellen wir fest, dass in der Folge u (x)(k) = ( ( x − arctan (t) ) 2 P ) (k) f¨ur k < 2 P stets der Faktor ( x − arctan (t) ) auftritt und daher f ¨ur x = arctan ( t) die Terme verschw...

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[14]

7853981633975

Verwendung eines Startwertes, der schon sehr nahe bei π 4 liegt Wir verwenden den Startwert x0 = 0. 7853981633975 . Dieser ist bereits auf 14 Dezimalstellen genau

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[15]

Hieraus ergibt sich die Konvergenzordnung 2 ·2 + 1 = 5

Erh ¨ohung der Konvergenzordnung Um sicher zustellen, dass die Konvergenzordnung voll zum Tragen k ommt, w ¨ahlen wir P = 2. Hieraus ergibt sich die Konvergenzordnung 2 ·2 + 1 = 5. Das bedeutet, dass sich die Anzahl der g¨ultigen Dezimalstellen mit jedem Schritt n ¨aherungsweise verf ¨unﬀacht. Wir ben ¨otigen dann 8 Iterationsschritte um mit diesem Startw...

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[16]

Wir m ¨ussen nicht von An- fang an gleich mit einer Genauigkeit von einer Million Stellen rechnen

Schritt-haltend die Genauigkeit erh ¨ohen Die Fixpunktiteration (wie jede andere auch) ist selbst-korrigieren d. Wir m ¨ussen nicht von An- fang an gleich mit einer Genauigkeit von einer Million Stellen rechnen. Wir k ¨onnen mit weitaus geringerer Genauigkeit loslegen. F¨ur die Berechnungen verwenden wir das Computer-Algebrasytem M aple, dass sich bereits...

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[17]

So geht es weiter bis zum 7

Schritt liegt die ben ¨otigte Genauigkeit bei 14 ·52 = 350 Stellen, also m ¨ussen wir Digits=350 setzen. So geht es weiter bis zum 7. Schritt, wo Maple mit 14 ·57 = 1093750 Dezimalstellen rechnet. Jetzt haben wir die vorgegebene Anzahl von mindestens einer Million Stellen erreicht. Deshalb brauchen wir im 8. Schritt keine weitere Erh ¨ohung der Stellenanz...

work page 2025