Three-term recurrence relation explained
In mathematics, and especially in numerical analysis, a homogeneous linear three-term recurrence relation (TTRR, the qualifiers "homogeneous linear" are usually taken for granted)[1] is a recurrence relation of the form
for
where the sequences
and
, together with the initial values
govern the evolution of the sequence
.
Applications
If the
and
are constant and independent of the step index
n, then the TTRR is a Linear recurrence with constant coefficients of order 2. Arguably the simplest, and most prominent, example for this case is the
Fibonacci sequence, which has constant coefficients
.
Orthogonal polynomials Pn all have a TTRR with respect to degree n,
Pn(x)=(Anx+Bn)Pn-1(x)+CnPn-2(x)
where
An is not 0. Conversely,
Favard's theorem states that a sequence of polynomials satisfying a TTRR is a sequence of orthogonal polynomials.
Also many other special functions have TTRRs. For example, the solution to
is given by the
Bessel function
. TTRRs are an important tool for the numeric computation of special functions.
TTRRs are closely related to continuous fractions.
Solution
Solutions of a TTRR, like those of a linear ordinary differential equation, form a two-dimensional vector space: any solution can be written as the linear combination of any two linear independent solutions. A unique solution is specified through the initial values
.
[2] See also
Literature
- Walter Gautschi. Computational Aspects of Three-Term Recurrence Relations. SIAM Review, 9:24–80 (1967).
- Walter Gautschi. Minimal Solutions of Three-Term Recurrence Relation and Orthogonal Polynomials. Mathematics of Computation, 36:547–554 (1981).
- Amparo Gil, Javier Segura, and Nico M. Temme. Numerical Methods for Special Functions. siam (2007)
- J. Wimp, Computation with recurrence relations, London: Pitman (1984)
Notes and References
- Gi, Segura, Temme (2007), Chapter 4.1
- Gi, Segura, Temme (2007), Chapter 4.1