Mortici Accelerated Series

📐 Definition

A Mortici accelerated series is obtained by correcting the partial sums of a slowly convergent series using asymptotic information about its remainder.

Given a convergent series

$$\sum_{n=1}^{\infty} a_n$$

with partial sums

$$S_N = \sum_{n=1}^N a_n,$$

the Mortici method constructs a correction term $\Delta_N$ such that

$$S_N^{(\mathrm{acc})} = S_N + \Delta_N$$

Copied

converges significantly faster to the series limit.

The correction $\Delta_N$ is chosen to cancel the dominant asymptotic behavior of the truncation error.

Domain & Codomain

Applies to real or complex series whose remainder admits an asymptotic expansion

Values in the same codomain as the original series sum

---

⚙️ Key Properties

Asymptotic Error Cancellation

If the truncation error satisfies

$$S - S_N \sim \frac{c}{N^p},$$

then choosing

$$\Delta_N = \frac{c}{N^p}$$

Copied

removes the leading-order error and improves convergence by one asymptotic order.

---

Recursive Improvement

If higher-order asymptotics are known,

$$S - S_N \sim \frac{c_1}{N^p} + \frac{c_2}{N^{p+1}} + \cdots,$$

then Mortici acceleration can be iterated to obtain

$$\Delta_N = \frac{c_1}{N^p} + \frac{c_2}{N^{p+1}}$$

Copied

yielding polynomial or even exponential convergence improvement.

---

Relation to Difference Equations

The Mortici method can be interpreted as solving an approximate difference equation for the remainder term:

$$R_{N+1} - R_N \approx a_{N+1}$$

Copied

with asymptotic matching used to determine $R_N$ explicitly.

---

Compatibility with Special Functions

Mortici acceleration is especially effective when applied to:

• harmonic-type series,
• series defining special functions,
• expansions involving logarithmic or rational remainders.

It is frequently used to accelerate series representations of $\gamma$, $\zeta(s)$, and $\psi(z)$.

---

🎯 Canonical Example

Consider the harmonic series remainder defining the Euler–Mascheroni constant:

$$\gamma = \lim_{N\to\infty} \left( \sum_{n=1}^N \frac{1}{n} - \ln N \right)$$

Using the asymptotic expansion

$$\sum_{n=1}^N \frac{1}{n} - \ln N = \gamma + \frac{1}{2N} - \frac{1}{12N^2} + \cdots,$$

a first-order Mortici acceleration gives

$$\gamma \approx \sum_{n=1}^N \frac{1}{n} - \ln N - \frac{1}{2N}$$

Copied

which converges substantially faster than the uncorrected sequence.

---

🎯 Special Forms

Common correction terms used in practice include:

Remainder type	Correction $\Delta_N$
$\mathcal{O}(N^{-1})$	$\tfrac{c}{N}$
$\mathcal{O}(N^{-2})$	$\tfrac{c_1}{N} + \tfrac{c_2}{N^2}$
$\mathcal{O}(\ln N / N)$	$\tfrac{c \ln N}{N}$

These coefficients are typically determined via asymptotic matching or symbolic expansion.

---

🔗 Related Functions

Stirling Series Asymptotic expansion used in combination with Mortici acceleration

Digamma Function Special function with series amenable to Mortici acceleration

Riemann Zeta Function Series representation accelerated via Mortici methods

---

Usage in Oakfield

Applications in Oakfield

Oakfield uses Mortici-style “speedy” approximants inside its special-function implementations:

• Speed-first digamma/trigamma variants in core/math/hyperexponential.c implement Mortici-style log-shift rational approximants (see sim_digamma_f64_mortici / sim_trigamma_f64_mortici in core/math/hyperexponential.h)
• These approximants complement the more accurate Stirling-tail “daily driver” paths and are used when throughput matters more than last-bit accuracy

---

Historical Foundations

1. 📜 Development (21st Century)

   Cristinel Mortici introduced this acceleration framework in the early 2000s through a series of papers on asymptotic methods and convergence acceleration. His approach emphasized explicit asymptotic correction rather than purely algorithmic transformation.

2. 🌍 Modern Impact

   Mortici’s method has been widely adopted in the study of special functions, constant evaluation, and symbolic asymptotics, influencing both theoretical research and computational practice.

---

📚 References

• C. Mortici, On new sequences converging towards the Euler–Mascheroni constant, Comput. Math. Appl.
• C. Mortici, Asymptotic methods for series acceleration, J. Number Theory
• Borwein & Borwein, Pi and the AGM