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

Technical Summary: A Fixed Point Iteration Method for the Arctangent with Any Odd Order of Convergence

Problem Statement
The paper addresses the computational challenge of calculating the arctangent function, $\arctan(t)$, for $t \in \mathbb{R}^+$. While the standard power series expansion for $\arctan(u)$ converges for $|u| < 1$, achieving high precision often requires a large number of terms or specific transformations. The author seeks a fixed-point iteration method that can achieve an arbitrary odd order of convergence, thereby accelerating the rate at which the correct digits of the result are obtained.

Methodology
The core of the proposed method is a fixed-point function $T(x)$ derived from the truncated power series of the arctangent. The author defines a substitution variable $u$ based on trigonometric identities:
$$u = \frac{\sin(x) - t \cos(x)}{\cos(x) + t \sin(x)}$$
By substituting this $u$ into the first $P$ terms of the $\arctan(u)$ power series, the author constructs the iteration function:
$$T(x) = x - \sum_{k=1}^{P} \frac{(-1)^{k-1}}{2k-1} \left( \frac{\sin(x) - t \cos(x)}{\cos(x) + t \sin(x)} \right)^{2k-1}$$
The iteration proceeds via $x_{n+1} = T(x_n)$.

The paper provides a rigorous mathematical proof for the properties of this function:

1. Fixed Point Property: It is proven that $T(\arctan(t)) = \arctan(t)$. This relies on the fact that the substitution term becomes zero when $x = \arctan(t)$.
2. Derivative Analysis: The author demonstrates that at the fixed point, the first $2P$ derivatives of $T(x)$ vanish ($T^{(k)}(\arctan(t)) = 0$ for $0 \le k \le 2P$). The $(2P+1)$-th derivative is non-zero and equals $(-1)^P (2P)!$.
3. Convergence Proof: Using the Banach Fixed Point Theorem, the paper establishes that for an initial value $x_0$ sufficiently close to $\arctan(t)$, the sequence converges.
4. Order of Convergence: Based on the vanishing derivatives, the Taylor expansion of the error term shows that the method converges with an order of exactly $2P + 1$.

Key Contributions and Results

• Arbitrary Odd Order: The primary contribution is a unified framework allowing the user to select any natural number $P$ to achieve a specific odd order of convergence ($3, 5, 7, \dots$).
• Application to $\pi$: The method is applied to Machin-like formulas for $\pi/4$, specifically $\pi/4 = \arctan(1)$. The paper details how the iteration formula simplifies for rational arguments $t = 1/q$.
• High-Precision Computation: The author presents a practical computational experiment to calculate one million digits of $\pi/4$.
  • Configuration: The experiment used $P=2$, yielding a convergence order of 5 (quintic convergence).
  • Strategy: To optimize performance, the author employed a "step-holding" precision strategy. Instead of computing all iterations at full precision, the number of significant digits was increased geometrically ($14 \times 5^n$) at each step, matching the expected growth of correct digits.
  • Outcome: Starting with a 14-digit approximation, the method reached an accuracy exceeding one million digits in just 8 iterations. The error reduction factors observed in the experiment confirmed the theoretical order of convergence.

Significance and Claims
The paper claims that this method offers an efficient alternative for high-precision calculation of the arctangent and, by extension, $\pi$. The significance lies in the ability to tune the convergence order via the parameter $P$ without changing the fundamental structure of the algorithm. The author notes that the method is self-correcting and, when combined with fast trigonometric function evaluation (e.g., via Taylor series or other algorithms), allows for the calculation of $\arctan(t)$ with "unlimited precision" given sufficient computational resources. The work is presented as a demonstration of efficiency, validated by the successful computation of $\pi/4$ to over one million decimal places using standard hardware and computer algebra software (Maple).