## Abstract

The strict superlinear order (superlinear convergence) was usually regarded as having an intermediate speed between between linear and with order \(p>1\).

In this paper we analyze the superlinear convergence (superlinear order/rate) of sequences and we show that actually there are four distinct classes of strict superlinear order: “weak”, “medium”, “strong” and “mixed”. The speed of the sequences from the first three classes is increasingly much faster, term-by-term big Oh, i.e.,

\[|x^\ast-x_k|=\mathcal{O}(|y^\ast-y_k|^{\alpha}), \mbox{ as } k\rightarrow \infty, \forall \alpha>1 \mbox{ given},

\]whereas the speed of the “mixed” class cannot be assessed.

We prove that the speed of the sequences from the “medium” and “weak” classes is term-by-term slower than the speed of the sequences with high classical C-orders \(p>1\) (in the sense of big Oh above), while an example shows that certain sequences from the “mixed” class may be term-by-term faster than some sequences with infinite C-order.

We also show that for a given sequence with strict superlinear convergence, one can evaluate numerically to which class it belongs.

Some recent results of Rodomanov and Nesterov (Math. Program., 2022), resp. Ye et al. (Math. Program., 2023) show that certain classical quasi-Newton methods (DFP, BFGS and SR1) belong to the “weak” class.

## Authors

**Emil Cătinaş**

“Tiberiu Popoviciu” Institute of Numerical Analysis

## Keywords

convergent sequence, convergence order, convergence speed, superlinear order, Q-order, convergence rate, big Oh, quasi-Newton method, DFP, BFGS, SR1.

## Paper coordinates

E. Cătinaş, *The strict superlinear order can be faster than the infinite order*, Numer. Algor., (2023). https://doi.org/10.1007/s11075-023-01604-y

## About this paper

##### Journal

Numerical Algorithms

##### Publisher Name

Springer Nature

##### Print ISSN

1017-1398

##### Online ISSN

1572-9265

google scholar link

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Linear convergence orders, weak superlinear convergence orders, medium superlinear convergence orders, strong superlinear convergence orders, mixed superlinear convergence orders, quadratic superlinear convergence orders and infinite superlinear convergence orders

Verifying numerically the superlinear convergence order / the superlinear order / the superlinear rate.