§3.6 Linear Difference Equations

Keywords:: difference equations
Referenced by:: §11.13(v), §14.32, §18.40, §28.34(iii), §29.20(i), Ch.3, §30.16(ii), §33.23(iv)
Permalink:: http://dlmf.nist.gov/3.6

§3.6(i) Introduction
§3.6(ii) Homogeneous Equations
§3.6(iii) Miller’s Algorithm
§3.6(iv) Inhomogeneous Equations
§3.6(v) Olver’s Algorithm
§3.6(vi) Examples
§3.6(vii) Linear Difference Equations of Other Orders

§3.6(i) Introduction

Defines:: $\Delta$ : forward difference operator
Referenced by:: §2.9(i), §26.8(v)
Permalink:: http://dlmf.nist.gov/3.6.i

Many special functions satisfy second-order recurrence relations, or difference equations, of the form

3.6.1 $a_{n}w_{{n+1}}-b_{n}w_{n}+c_{n}w_{{n-1}}=d_{n},$

Symbols:: $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient, $c_{n}$ : coefficient and $d_{n}$ : coefficient
Referenced by:: §3.6(iv), §3.6(v)
Permalink:: http://dlmf.nist.gov/3.6.E1
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or equivalently,

3.6.2 $a_{n}\Delta^{2}w_{{n-1}}+(2a_{n}-b_{n})\Delta w_{{n-1}}+(a_{n}-b_{n}+c_{n})w_{{n-1}}=d_{n},$

Symbols:: $\Delta$ : forward difference operator, $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient, $c_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E2
Encodings:: TeX, pMML, png

where $\Delta w_{{n-1}}=w_{n}-w_{{n-1}}$ , $\Delta^{2}w_{{n-1}}=\Delta w_{n}-\Delta w_{{n-1}}$ , and $n\in\Integer$ . If $d_{n}=0$ , $\forall n$ , then the difference equation is homogeneous; otherwise it is inhomogeneous.

Difference equations are simple and attractive for computation. In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome.

§3.6(ii) Homogeneous Equations

Keywords:: backward recursion, difference equations
Referenced by:: §15.19(iv), §3.6(i)
Permalink:: http://dlmf.nist.gov/3.6.ii

Given numerical values of $w_{0}$ and $w_{1}$ , the solution $w_{n}$ of the equation

3.6.3 $a_{n}w_{{n+1}}-b_{n}w_{n}+c_{n}w_{{n-1}}=0,$

Symbols:: $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient and $c_{n}$ : coefficient
Referenced by:: §3.6(ii), §3.6(ii), §3.6(v), §3.6(v)
Permalink:: http://dlmf.nist.gov/3.6.E3
Encodings:: TeX, pMML, png

with $a_{n}\neq 0$ , $\forall n$ , can be computed recursively for $n=2,3,\dots$ . Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors. These errors have the effect of perturbing the solution by unwanted small multiples of $w_{n}$ and of an independent solution $g_{n}$ , say. This is of little consequence if the wanted solution is growing in magnitude at least as fast as any other solution of (3.6.3), and the recursion process is stable.

But suppose that $w_{n}$ is a nontrivial solution such that

3.6.4 $w_{n}/g_{n}\to 0,$ $n\to\infty$ .

Symbols:: $w_{n}$ : sequence and $g_{n}$ : solution
Permalink:: http://dlmf.nist.gov/3.6.E4
Encodings:: TeX, pMML, png

Then $w_{n}$ is said to be a recessive (equivalently, minimal or distinguished) solution as $n\to\infty$ , and it is unique except for a constant factor. In this situation the unwanted multiples of $g_{n}$ grow more rapidly than the wanted solution, and the computations are unstable. Stability can be restored, however, by backward recursion, provided that $c_{n}\neq 0$ , $\forall n$ : starting from $w_{N}$ and $w_{{N+1}}$ , with $N$ large, equation (3.6.3) is applied to generate in succession $w_{{N-1}},w_{{N-2}},\dots,w_{0}$ . The unwanted multiples of $g_{n}$ now decay in comparison with $w_{n}$ , hence are of little consequence.

The values of $w_{N}$ and $w_{{N+1}}$ needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). However, there are alternative procedures that do not require $w_{N}$ and $w_{{N+1}}$ to be known in advance. These are described in §§ 3.6(iii) and 3.6(v).

§3.6(iii) Miller’s Algorithm

Keywords:: Miller’s algorithm, difference equations
Referenced by:: ¶ ‣ §3.6(vi), §3.6(ii), §6.18(ii)
Permalink:: http://dlmf.nist.gov/3.6.iii

Because the recessive solution of a homogeneous equation is the fastest growing solution in the backward direction, it occurred to J.C.P. Miller (Bickley et al. (1952, pp. xvi–xvii)) that arbitrary “trial values” can be assigned to $w_{N}$ and $w_{{N+1}}$ , for example, 1 and 0. A “trial solution” is then computed by backward recursion, in the course of which the original components of the unwanted solution $g_{n}$ die away. It therefore remains to apply a normalizing factor $\Lambda$ . The process is then repeated with a higher value of $N$ , and the normalized solutions compared. If agreement is not within a prescribed tolerance the cycle is continued.

The normalizing factor $\Lambda$ can be the true value of $w_{0}$ divided by its trial value, or $\Lambda$ can be chosen to satisfy a known property of the wanted solution of the form

3.6.5 $\sum _{{n=0}}^{{\infty}}\lambda _{n}w_{n}=1,$

Symbols:: $w_{n}$ : sequence and $\lambda _{n}$ : constants
Permalink:: http://dlmf.nist.gov/3.6.E5
Encodings:: TeX, pMML, png

where the $\lambda$ ’s are constants. The latter method is usually superior when the true value of $w_{0}$ is zero or pathologically small.

For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a). See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions.

§3.6(iv) Inhomogeneous Equations

Notes:: See Olver (1967a).
Keywords:: boundary-value methods or problems, difference equations
Referenced by:: §3.6(v)
Permalink:: http://dlmf.nist.gov/3.6.iv

Similar principles apply to equation (3.6.1) when $a_{n}c_{n}\neq 0$ , $\forall n$ , and $d_{n}\neq 0$ for some, or all, values of $n$ . If, as $n\to\infty$ , the wanted solution $w_{n}$ grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable.

A new problem arises, however, if, as $n\to\infty$ , the asymptotic behavior of $w_{n}$ is intermediate to those of two independent solutions $f_{n}$ and $g_{n}$ of the corresponding inhomogeneous equation (the complementary functions). More precisely, assume that $f_{0}\neq 0$ , $g_{n}\neq 0$ for all sufficiently large $n$ , and as $n\to\infty$

3.6.6

$f_{n}/g_{n}\to 0,$

$w_{n}/g_{n}\to 0.$

Symbols:: $w_{n}$ : sequence and $g_{n}$ : solution
Permalink:: http://dlmf.nist.gov/3.6.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

Then computation of $w_{n}$ by forward recursion is unstable. If it also happens that $f_{n}/w_{n}\to 0$ as $n\to\infty$ , then computation of $w_{n}$ by backward recursion is unstable as well. However, $w_{n}$ can be computed successfully in these circumstances by boundary-value methods, as follows.

Let us assume the normalizing condition is of the form $w_{0}=\lambda$ , where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_{1}^{{(N)}},w_{2}^{{(N)}},\dots,w_{{N-1}}^{{(N)}}$ ; see §3.2(ii). Here $N$ is an arbitrary positive integer.

3.6.7 $\begin{bmatrix}-b_{1}&a_{1}&&&0\\ c_{2}&-b_{2}&a_{2}\\ &\ddots&\ddots&\ddots\\ &&c_{{N-2}}&-b_{{N-2}}&a_{{N-2}}\\ 0&&&c_{{N-1}}&-b_{{N-1}}\end{bmatrix}\begin{bmatrix}w_{1}^{{(N)}}\\ w_{2}^{{(N)}}\\ \vdots\\ w_{{N-2}}^{{(N)}}\\ w_{{N-1}}^{{(N)}}\end{bmatrix}=\begin{bmatrix}d_{1}-c_{1}\lambda\\ d_{2}\\ \vdots\\ d_{{N-2}}\\ d_{{N-1}}\end{bmatrix}.$

Symbols:: $w_{n}$ : sequence, $N$ : arbitrary positive integer, $a_{n}$ : coefficient, $b_{n}$ : coefficient, $c_{n}$ : coefficient, $d_{n}$ : coefficient and $\lambda$ : constant
Permalink:: http://dlmf.nist.gov/3.6.E7
Encodings:: TeX, pMML, png

Then as $N\to\infty$ with $n$ fixed, $w_{n}^{{(N)}}\to w_{n}$ .

§3.6(v) Olver’s Algorithm

Keywords:: Olver’s algorithm, difference equations
Referenced by:: §10.74(iv), §13.29(iv), ¶ ‣ §3.6(vi), ¶ ‣ §3.6(vi), §3.6(ii), §3.7(iii)
Permalink:: http://dlmf.nist.gov/3.6.v

To apply the method just described a succession of values can be prescribed for the arbitrary integer $N$ and the results compared. However, a more powerful procedure combines the solution of the algebraic equations with the determination of the optimum value of $N$ . It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution $w_{n}$ of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied.

Suppose again that $f_{0}\neq 0$ , $w_{0}$ is given, and we wish to calculate $w_{1},w_{2},\dots,w_{M}$ to a prescribed relative accuracy $\epsilon$ for a given value of $M$ . We first compute, by forward recurrence, the solution $p_{n}$ of the homogeneous equation (3.6.3) with initial values $p_{0}=0$ , $p_{1}=1$ . At the same time we construct a sequence $e_{n}$ , $n=0,1,\dots$ , defined by

3.6.8 $a_{n}e_{n}=c_{n}e_{{n-1}}-d_{n}p_{n},$

Symbols:: $p_{n}$ : solutions, $e_{n}$ : sequence, $a_{n}$ : coefficient, $c_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E8
Encodings:: TeX, pMML, png

beginning with $e_{0}=w_{0}$ . (This part of the process is equivalent to forward elimination.) The computation is continued until a value $N$ ( $\geq M$ ) is reached for which

3.6.9 $\left|\frac{e_{N}}{p_{N}p_{{N+1}}}\right|\leq\epsilon\min _{{1\leq n\leq M}}\left|\frac{e_{n}}{p_{n}p_{{n+1}}}\right|.$

Symbols:: $N$ : arbitrary positive integer, $\epsilon$ : relative accuracy, $M$ : order, $p_{n}$ : solutions and $e_{n}$ : sequence
Referenced by:: ¶ ‣ §3.6(vi)
Permalink:: http://dlmf.nist.gov/3.6.E9
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Then $w_{n}$ is generated by backward recursion from

3.6.10 $p_{{n+1}}w_{n}=p_{n}w_{{n+1}}+e_{n},$

Symbols:: $w_{n}$ : sequence, $p_{n}$ : solutions and $e_{n}$ : sequence
Permalink:: http://dlmf.nist.gov/3.6.E10
Encodings:: TeX, pMML, png

starting with $w_{N}=0$ . (This part of the process is back substitution.)

An example is included in the next subsection. For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6).

§3.6(vi) Examples

Permalink:: http://dlmf.nist.gov/3.6.vi

¶ Example 1. Bessel Functions

Keywords:: Bessel functions, Miller’s algorithm, difference equations

The difference equation

3.6.11 $w_{{n+1}}-2nw_{n}+w_{{n-1}}=0,$ $n=1,2,\dots$ ,

Symbols:: $w_{n}$ : sequence
Permalink:: http://dlmf.nist.gov/3.6.E11
Encodings:: TeX, pMML, png

is satisfied by $\mathop{J_{{n}}\/}\nolimits\!\left(1\right)$ and $\mathop{Y_{{n}}\/}\nolimits\!\left(1\right)$ , where $\mathop{J_{{n}}\/}\nolimits\!\left(x\right)$ and $\mathop{Y_{{n}}\/}\nolimits\!\left(x\right)$ are the Bessel functions of the first kind. For large $n$ ,

3.6.12 $\mathop{J_{{n}}\/}\nolimits\!\left(1\right)\sim\frac{1}{(2\pi n)^{{1/2}}}\left(\frac{e}{2n}\right)^{n},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $e$ : base of exponential function and $\sim$ : asymptotic equality
Permalink:: http://dlmf.nist.gov/3.6.E12
Encodings:: TeX, pMML, png

3.6.13 $\mathop{Y_{{n}}\/}\nolimits\!\left(1\right)\sim\left(\frac{2}{\pi n}\right)^{{1/2}}\left(\frac{2n}{e}\right)^{n},$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $e$ : base of exponential function and $\sim$ : asymptotic equality
Permalink:: http://dlmf.nist.gov/3.6.E13
Encodings:: TeX, pMML, png

(§10.19(i)). Thus $\mathop{Y_{{n}}\/}\nolimits\!\left(1\right)$ is dominant and can be computed by forward recursion, whereas $\mathop{J_{{n}}\/}\nolimits\!\left(1\right)$ is recessive and has to be computed by backward recursion. The backward recursion can be carried out using independently computed values of $\mathop{J_{{N}}\/}\nolimits\!\left(1\right)$ and $\mathop{J_{{N+1}}\/}\nolimits\!\left(1\right)$ or by use of Miller’s algorithm (§3.6(iii)) or Olver’s algorithm (§3.6(v)).

¶ Example 2. Weber Function

Keywords:: Anger–Weber functions

The Weber function $\mathop{\mathbf{E}_{{n}}\/}\nolimits\!\left(1\right)$ satisfies

3.6.14 $w_{{n+1}}-2nw_{n}+w_{{n-1}}=-(2/\pi)(1-(-1)^{n}),$

Symbols:: $w_{n}$ : sequence
Permalink:: http://dlmf.nist.gov/3.6.E14
Encodings:: TeX, pMML, png

for $n=1,2,\dots$ , and as $n\to\infty$

3.6.15 $\mathop{\mathbf{E}_{{2n}}\/}\nolimits\!\left(1\right)\sim\frac{2}{(4n^{2}-1)\pi},$

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function and $\sim$ : asymptotic equality
Permalink:: http://dlmf.nist.gov/3.6.E15
Encodings:: TeX, pMML, png

3.6.16 $\mathop{\mathbf{E}_{{2n+1}}\/}\nolimits\!\left(1\right)\sim\frac{2}{(2n+1)\pi};$

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function and $\sim$ : asymptotic equality
Permalink:: http://dlmf.nist.gov/3.6.E16
Encodings:: TeX, pMML, png

see §11.11(ii). Thus the asymptotic behavior of the particular solution $\mathop{\mathbf{E}_{{n}}\/}\nolimits\!\left(1\right)$ is intermediate to those of the complementary functions $\mathop{J_{{n}}\/}\nolimits\!\left(1\right)$ and $\mathop{Y_{{n}}\/}\nolimits\!\left(1\right)$ ; moreover, the conditions for Olver’s algorithm are satisfied. We apply the algorithm to compute $\mathop{\mathbf{E}_{{n}}\/}\nolimits\!\left(1\right)$ to 8S for the range $n=1,2,\dots,10$ , beginning with the value $\mathop{\mathbf{E}_{{0}}\/}\nolimits\!\left(1\right)=-0.56865\;\, 663$ obtained from the Maclaurin series expansion (§11.10(iii)).

In the notation of §3.6(v) we have $M=10$ and $\epsilon=\tfrac{1}{2}\times 10^{{-8}}$ . The least value of $N$ that satisfies (3.6.9) is found to be 16. The results of the computations are displayed in Table 3.6.1. The values of $w_{n}$ for $n=1,2,\dots,10$ are the wanted values of $\mathop{\mathbf{E}_{{n}}\/}\nolimits\!\left(1\right)$ . (It should be observed that for $n>10$ , however, the $w_{n}$ are progressively poorer approximations to $\mathop{\mathbf{E}_{{n}}\/}\nolimits\!\left(1\right)$ : the underlined digits are in error.)

Table 3.6.1: Weber function $w_{n}=\mathop{\mathbf{E}_{{n}}\/}\nolimits\!\left(1\right)$ computed by Olver’s algorithm.

$n$	$p_{n}$		$e_{n}$		$e_{n}/(p_{n}p_{{n+1}})$		$w_{n}$
0	0.00000 000		−0.56865 663				−0.56865 663
1	0.10000 000	×10¹	0.70458 291		0.35229 146		0.43816 243
2	0.20000 000	×10¹	0.70458 291		0.50327 351	×10⁻¹	0.17174 195
3	0.70000 000	×10¹	0.96172 597	×10¹	0.34347 356	×10⁻¹	0.24880 538
4	0.40000 000	×10²	0.96172 597	×10¹	0.76815 174	×10⁻³	0.47850 795	×10⁻¹
5	0.31300 000	×10³	0.40814 124	×10³	0.42199 534	×10⁻³	0.13400 098
6	0.30900 000	×10⁴	0.40814 124	×10³	0.35924 754	×10⁻⁵	0.18919 443	×10⁻¹
7	0.36767 000	×10⁵	0.47221 340	×10⁵	0.25102 029	×10⁻⁵	0.93032 343	×10⁻¹
8	0.51164 800	×10⁶	0.47221 340	×10⁵	0.11324 804	×10⁻⁷	0.10293 811	×10⁻¹
9	0.81496 010	×10⁷	0.10423 616	×10⁸	0.87496 485	×10⁻⁸	0.71668 638	×10⁻¹
10	0.14618 117	×10⁹	0.10423 616	×10⁸	0.24457 824	×10⁻¹⁰	0.65021 292	×10⁻²
11	0.29154 738	×10¹⁰	0.37225 201	×10¹⁰	0.19952 026	×10⁻¹⁰	0.58373 946	×10⁻¹
12	0.63994 242	×10¹¹	0.37225 201	×10¹⁰	0.37946 279	×10⁻¹³	$0.44851\; 3\underline{87}$	×10⁻²
13	0.15329 463	×10¹³	0.19555 304	×10¹³	0.32057 909	×10⁻¹³	$0.49269\;\underline{383}$	×10⁻¹
14	0.39792 611	×10¹⁴	0.19555 304	×10¹³	0.44167 174	×10⁻¹⁶	$0.327\underline{92\; 861}$	×10⁻²
15	0.11126 602	×10¹⁶	0.14186 384	×10¹⁶	0.38242 250	×10⁻¹⁶	$0.425\underline{50\; 628}$	×10⁻¹
16	0.33340 012	×10¹⁷	0.14186 384	×10¹⁶	0.39924 861	×10⁻¹⁹	$0.\underline{00000\; 0 0 0}$

Symbols:: $\mathop{\mathbf{E}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Weber function, $w_{n}$ : sequence, $p_{n}$ : solutions and $e_{n}$ : sequence
Keywords:: Olver’s algorithm, difference equations
Referenced by:: ¶ ‣ §3.6(vi)
Permalink:: http://dlmf.nist.gov/3.6.T1

§3.6(vii) Linear Difference Equations of Other Orders

Keywords:: boundary-value methods or problems, difference equations
Referenced by:: §16.25, §3.6(i)
Permalink:: http://dlmf.nist.gov/3.6.vii

Similar considerations apply to the first-order equation

3.6.17 $a_{n}w_{{n+1}}-b_{n}w_{n}=d_{n}.$

Symbols:: $w_{n}$ : sequence, $a_{n}$ : coefficient, $b_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E17
Encodings:: TeX, pMML, png

Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. For analyses and examples see Gautschi (1997b).

For a difference equation of order $k$ ( $\geq 3$ ),

3.6.18 $a_{{n,k}}w_{{n+k}}+a_{{n,k-1}}w_{{n+k-1}}+\dots+a_{{n,0}}w_{n}=d_{n},$

Symbols:: $w_{n}$ : sequence, $a_{n}$ : coefficient and $d_{n}$ : coefficient
Permalink:: http://dlmf.nist.gov/3.6.E18
Encodings:: TeX, pMML, png

or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. Typically $k-\ell$ conditions are prescribed at the beginning of the range, and $\ell$ conditions at the end. Here $\ell\in[0,k]$ , and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions. Within this framework forward and backward recursion may be regarded as the special cases $\ell=0$ and $\ell=k$ , respectively.