On Rate of Convergence of Sequences

Abstract

Definition 0.1 The sequence (x _n ) converges to σ faster than the sequence ( y _n ) converges to λ, defined by (x _n ) < ( y _n ) if ${ \lim \atop n\rightarrow \infty } \frac{x_{n}-\sigma } {y_{n}-\lambda } = 0$, provided y _n −λ ≠ 0 for all n ∈ N.

Definition 0.2 The regular matrix A = (a _nk ) accelerates the convergence of the sequence x = (x _k ) if Ax < x. The acceleration field of the matrix A is the class of sequences {x = (x _k ) ∈ ω: Ax < x}.

At the initial stage, works on acceleration convergence sequences were done by Smith and Ford (SIAM J Numer Anal 16(2):223–240, 1979), Keagy and Ford (Pac J Math 132(2):357–362, 1988), Salzer (J Math Phys 33:356–359, 1955), Dawson (Pac J Math 24(1):51–56, 1968), Brezinski et al. (SIAM J Numer Anal 20:1099–1105, 1983), Brezinski (Rend Math 7(6):303–316, 1974), and many others.

The notion of acceleration convergence depends on the convergence of sequences. Over the years, different types of convergent sequences such as statistically convergent, I-convergent, etc. have been introduced. Accordingly, Tripathy and Sen (Ital J Pure Appl Math 17:151–158, 2005) have introduced the notion of statistical acceleration convergence. Tripathy and Mahanta (J Frankl Inst 347:591–598, 2010) have investigated about the ideal acceleration convergence of sequences. Patterson and Savas (Hacettepe J Math Stat 41(4):487–497, 2012) have studied about the acceleration convergence with respect to four dimensional matrix maps.

The notion of fuzzy sets attracted research worker on sequence spaces to study about the sequences of fuzzy real numbers. Tripathy and Dutta (Math Modell Anal 17(4):549–557, 2012) have studied about acceleration convergence of sequences of fuzzy numbers. In this talk we shall discuss about the different developments on rate of convergence of sequences.

1 Introduction and Basic Definitions

Faster convergence of sequences particularly the acceleration of convergence of sequence of partial sums of series via linear and nonlinear transformations are widely used in finding solutions of mathematical as well as different scientific and engineering problems (one may refer to Brezinski [1] and Brezinski et.al. [2]). The problem of acceleration convergence often occurs in numerical analysis. To accelerate the convergence, the standard interpolation and extrapolation methods of numerical mathematics are quite helpful. It is useful to study about the acceleration of convergence methods (we shall focus on matrix transformations), which transform a slowly converging sequence into a new sequence, converging to the same limit faster than the original sequence. The speed of convergence of sequences is of the central importance in the theory of subsequence transformation.

There are many sequences (or series) those converge very slowly. Again the rate of convergence of two convergent sequences (series) may not be equal. In order to speed the convergence of slowly convergent sequences (series) many well-known mathematicians like Salzer [8], Smith and Ford [10] established different methods. Dawson [3] has studied matrix summability over certain classes of sequences ordered with respect to rate of convergence.

Definition 1.1

The sequence (x _n ) converges to σ at the same rate as the sequence ( y _n ) converges to λ, written as (x _n ) ≈ ( y _n ) if

$$\displaystyle{ 0 < \mathrm{lim - inf}\left \vert \frac{x_{n}-\sigma } {y_{n}-\lambda }\right \vert \leq \mathrm{lim - sup}\left \vert \frac{x_{n}-\sigma } {y_{n}-\lambda }\right \vert < \infty. }$$

Example 1.1

Consider the sequences (x _k ) and ( y _k ) defined by x _k  = 1 + k ⁻¹ and y _k  = 23 + 123k ⁻¹, for all k ∈ N.

It can be easily verified that ( y _k ) converges to 23 faster than (x _k ) converges to 1.

Definition 1.2

Let A = (a _nk ) be an infinite matrix. For a sequence x = (x _k ), then the A transform of X is defined by Ax = (A _n x), where $A_{n}x = \sum \limits _{k=1}^{\infty }a_{nk}x_{k}$, for all n ∈ N.

Definition 1.3

The subsequence $x = (x_{n_{i}})$ of (x _k ) can be represented as a regular matrix transformation A = (a _nk ) times (x _k ) by defining $a_{i,n_{i}} = 1$, for all i ∈ N and a _pq  = 0, otherwise.

Definition 1.4

The convergence field of the matrix A = (a _nk ) is defined by {x = (x _k ): Ax ∈ c}, where c denotes the class of all convergent sequences.

Definition 1.5

The matrix A = (a _nk ) accelerates the convergence of x if Ax < x. The acceleration field of A is defined by {(x _k ) ∈ ω: Ax < x}.

Let S ₀ denote the set of all sequences in c ₀ with non-zero terms. If a ∈ S ₀, let [a] = { x ∈ S ₀: x ≈ a}. Also let E ₀ = { [x]: x ∈ S ₀}. If [a], [b] ∈ E ₀, then we say [a] is less than [b], [a] < ^∕[b], provided a < b. Then E ₀ is partially ordered with respect to < ^∕.

Open intervals in S ₀ will be denoted by (a, b), (a, −), (−, b), where (a, −) = { x ∈ S ₀: a < x} and (−, b) = { x ∈ S ₀: x < b}. On combining these two we have (a, b) = { x = (x _k ) ∈ S ₀: a < x < b}.

The necessary and sufficient conditions for a matrix A = (a _pq ) to be convergence preserving over (abbreviated c.p.o.) S ₀ are (deduced from the Silverman and Toeplitz conditions)

• (i) (a _pq ) _p = 1 ^∞ converges for each q = 1, 2, 3, … and

• (ii) there exists K such that $\sum \limits _{q=1}^{\infty }a_{pq} < K$, for each p = 1, 2, 3, ….

Dawson [3] has characterized the summability field of a matrix A by showing A is convergence preserving over the set of all sequences which converges faster than some fixed sequence. The following results are due to him.

Theorem 1.6

If A is c.p.o. [b], then there exists b ^∕ ∈ S ₀ such that b < b ^∕ and A is c.p.o. [b ^∕ ].

Theorem 1.7

If A is c.p.o. each of the sets [b ⁽¹⁾ ],[b ⁽²⁾ ],[b ⁽³⁾ ],…, then there exists d ∈ S ₀ such that b ^( p) < d, p = 1,2,3,… and A is c.p.o. [d].

It follows that A is convergence preserving over a set of the type (−,x).

Keagy and Ford [5] proved that if a subsequence transformation A accelerates x ∈ S ₀, then it accelerates each y ∈ S ₀ which converge at the same rate as x. They also proved the following two results.

Theorem 1.8

If A is a subsequence transformation and x ∈ S ₀ , then there exists y,z ∈ S ₀ , such that y < x < z and A does not accelerate y or z.

Theorem 1.9

If A is a subsequence transformation and x ∈ S ₀ , then there exists y ∈ S ₀ such that y < x and A accelerates y.

It is also proved by Keagy and Ford [5] that an analog to the above theorem does not exist for x < z, that is the acceleration field of a subsequence transformation cannot be any of the forms (x, −), [x, −), (−, x], (x, y), [x, y], (x, y] or [x, y); nor it include any of the first four of these forms. They showed that the acceleration field for each subsequence transformation A is the union of collection of sets of the form (x, y). The result is given below.

Theorem 1.10

If x ∈ S ₀ and A is a subsequence matrix that accelerates x, then there exist y and z such that y < x < z and A accelerates each r ∈ ( y,z).

They also proved that this algorithm cannot be extended to a larger class of sequences defined in terms of rate of convergence.

2 On Statistical Acceleration Convergence of Sequences

The notion of statistical convergence of sequences was introduced by Fast [4] and Schoenberg [9] independently. Later on it was further investigated from different aspects of sequence spaces and summability theory by many research workers.

A subset E of N is said to have asymptotic density δ(E) if $\delta (E) ={ \lim \atop n\rightarrow \infty } \frac{1} {n}\sum \limits _{k=1}^{\infty }\chi _{ E}(k)$ exists, where χ _E is the characteristic function of E. Clearly all finite subsets of N have zero asymptotic density and δ(E ^c) = δ(N − E) = 1 −δ(E).

A sequence (x _k ) is said to be statistically convergent to L if for every ɛ > 0, δ({k ∈ N:  | x _k − L | ≥ ɛ}) = 0. We write stat−lim x _k  = L.

Throughout ω, ℓ _∞ , c, c ₀, $\bar{c}$, $\bar{c_{0}}$, m ₀, represent the spaces of all, bounded, convergent, null, statistically convergent, statistically null and bounded statistically null sequences, respectively. Further S ₀, $\bar{S_{0}}$ denote the subsets of the spaces c ₀ and m ₀, respectively, with non-zero terms.

Example 2.1

The sequence (x _k ) defined by x _k  = i, for k = i ², i ∈ N and x _k  = k ⁻², otherwise, is statistically convergent to 0 and is unbounded.

Tripathy and Sen [13] introduced the notion of statistical acceleration convergence of sequences as follows.

Definition 2.1

Let the sequence (x _k ) be statistically convergent to σ and the sequence ( y _k ) statistically convergent to λ with $(x_{k}-\sigma )\notin \bar{S_{0}}$ and $(\,y_{k}-\sigma )\notin \bar{S_{0}}$, then the sequence (x _k ) statistically converges to σ statistically faster than ( y _k ) statistically convergent to λ, written as (x _k ) < ^stat( y _k ) if

$\mathrm{stat - lim}\frac{x_{k}-\sigma } {y_{k}-\sigma } = 0$, provided ( y _k −σ) ≠ 0 for all k ∈ N.

Definition 2.2

The sequence (x _k ) statistically converges to σ statistically at the same rate as the sequence ( y _k ) statistically converges to λ, written as (x _k ) ≈ ( y _k ) if

$$\displaystyle{ 0 < \mathrm{stat - lim - inf}\left \vert \frac{x_{k}-\sigma } {y_{k}-\lambda }\right \vert \leq \mathrm{stat - lim - sup}\left \vert \frac{x_{k}-\sigma } {y_{k}-\lambda }\right \vert < \infty. }$$

Tripathy and Sen [13] proved the statistical analogue of most of the above results as well as they established the following decomposition theorem for acceleration convergence.

Theorem 2.3

Let (x _k ), $(\,y_{k}) \in \bar{ S_{0}}$ , then the following are equivalent.

• (i) (x _k ) < ^stat ( y _k ).

• (ii) there exist (x _k ^∕ ) and ( y _k ^∕ ) in S ₀ such that x _k = x _k ^∕ for a.a.k, y _k = y _k ^∕ for a.a.k and (x _k ^∕ ) < ( y _k ^∕ ).

• (iii) there exists a subset K ={ k _i : I ∈ N} of N such that δ(K) = 1 and $(x_{k_{i}}) < (\,y_{k_{i}})$.

Remark 2.4

Keagy and Ford [5] conjectured “If A is any subsequence transformation and (x _k ) ∈ S ₀, then either Ax < x or Ax ≈ x”. Tripathy and Sen [13] provided the following example, which shows that this conjecture fails.

Example 2.2

Let the sequence (x _k ) be defined by the subsequence $(x_{k_{i}}) = (x_{1},x_{3},x_{5},\ldots )$.

3 On I-Acceleration Convergence of Sequences

The notion of I-convergence was introduced by Kostyrko et al. [6]. Later on it was further investigated from sequence space point of view and linked with summability theory by many others.

The notion depends on the notion of ideals. Let X be a non-empty set, then a family of sets I ⊂ 2^X is an ideal if and only if for each A, B ∈ I, we have A ∪ B ∈ I and for A ∈ I and for each B ⊂ A, we have B ∈ I. A non-empty family of sets F ⊂ 2^X is a filter on X if and only if ∅ ∉ F, for each A, B ∈ F, we have A ∩ B ∈ F and for each A ∈ F and for each B ⊃ A, we have B ∈ F. An ideal I is called non-trivial if I ≠ ∅ and X ∉ I. Hence I ⊂ 2^X is a non-trivial ideal if and only if F = F(I) = { X − A: A ∈ I} is a filter on X.

A subset E of N is said to have logarithmic density d(E) if $d(E) ={ \lim \atop n\rightarrow \infty } \frac{1} {s_{n}}\sum \limits _{k=1}^{n}\frac{\chi _{E}(k)} {k}$ exists, where $s_{n} = \sum \limits _{k=1}^{n}\frac{1} {k}$, for all n ∈ N. Clearly all finite subsets of N have zero logarithmic density and d(E ^c) = d(N − E) = 1 − d(E).

Let T = (t _nk ) be a regular non-negative matrix. Then for E ⊂ N, if $d_{T}(E) ={ \lim \atop n\rightarrow \infty } \sum \limits _{k=1}^{\infty }t_{nk}\chi _{E}(k)$ exists, it is called the T-density of E. From the regularity of T it follows that ${ \lim \atop n\rightarrow \infty } \sum \limits _{k=1}^{\infty }t_{ nk} = 1$ and from this and non-negativeness of T it follows that d _T (E) ∈ [0, 1].

Clearly the asymptotic density and logarithmic density can be obtained as the particular cases of T-density. If one considers $t_{nk} = \frac{1} {n}$, for k ≤ n and t _nk  = 0, otherwise, then d _T (E) = δ(E). If one considers $t_{nk} = \frac{k^{-1}} {s_{n}}$, for k ≤ n and t _nk  = 0, otherwise, then one will get d _T (E) = d(E).

The uniform density of a subset E of N is defined as follows: For integers t ≥ 0 and s ≥ 1, let E(t + 1, s + 1) = Card{n ∈ E: t + 1 ≤ n ≤ t + s}. Let $\beta _{s} ={ \lim \inf \atop t\rightarrow \infty } E(t + 1,t + s)$ and $\beta ^{s} ={ \lim \sup \atop t\rightarrow \infty } E(t + 1,t + s)$. Then $\underline{u}(E) ={ \lim \atop s\rightarrow \infty } \frac{\beta _{s}} {s}$ and $\bar{u}(E) ={ \lim \atop s\rightarrow \infty } \frac{\beta ^{s}} {s}$ exist. If $\underline{u}(E) =\bar{ u}(E)$, then we say that the uniform density of E exists and $u(E) =\underline{ u}(E) = \overline{u}(E)$.

Remark 3.1

Throughout we consider I to be a non-trivial ideal of subsets of N, the set of natural numbers

Definition 3.2

A sequence (x _k ) is said to be I − convergent to L if for each ɛ > 0, {k ∈ N: x _k − L ≥ ɛ} ∈ I. We write I −limx _k  = L.

The following are the examples of ideals:

Example 3.1

The class I _f of all finite subsets of N is an ideal of 2^N.

Example 3.2

The class I _C  = { E ⊂ N: } is an ideal of 2^N.

Example 3.3

The class I = { E ⊂ N: (E) = 0} is an ideal of 2^N.

Example 3.4

The class I _d  = { EN: d(E) = 0} is an ideal of 2^N.

Example 3.5

The class = { E ⊂ N: T _d (E) = 0} is an ideal of 2^N.

Example 3.6

The class I _u  = { E ⊂ N: u(E) = 0} is an ideal of 2^N.

Remark 3.3

All the ideals considered in the above examples are non-trivial ideals.

Tripathy and Mahanta [12] introduce the following definition on acceleration convergence related to I-convergence of sequences:

Definition 3.4

Let I −limx _k  = σ and I −limy _k  = λ with (x _k −σ), ( y _k −λ) ∈ S ₀ ^I. Then we say (x _k ) I-converges to σ, I-faster than ( y _k ) I-converges to λ, written as (x _k ) < ^I( y _k ) if $I -{ \mathrm{lim} \atop k\rightarrow \infty } \frac{x_{k}-\sigma } {y_{k}-\lambda } = 0$, provided y _k −λ ≠ 0, for all k ∈ N.

Peterson and Savas [7] have studied the acceleration convergence for double sequences, Tripathy and Dutta [11] have studied I-acceleration convergence for sequences of fuzzy real numbers.