Page 77 - Applied Numerical Methods Using MATLAB

66 MATLAB USAGE AND COMPUTATIONAL ERRORS
(f) Make a routine which computes the sum

K k
λ −λ
S(K) = e for λ = 100 and an integer K (P1.20.6)
k!
k=0
and then, use the routine to find the value of S(155).

1.21 Recursive Routines for Efficient Computation
(a) The Hermite Polynomial [K-1]
Consider the Hermite polynomial defined as

2 d N −x 2
N x
H 0 (x) = 1, H N (x) = (−1) e e (P1.21.1)
dx N
(i) Show that the derivative of this polynomial function can be writ-
ten as
2 d N −x 2 N x 2 d N+1 −x 2
N
x

H (x) = (−1) 2xe e + (−1) e e
N
dx N dx N+1
= 2xH N (x) − H N+1 (x) (P1.21.2)
and so the (N + 1)th-degree Hermite polynomial can be obtained
recursively from the Nth-degree Hermite polynomial as

H N+1 (x) = 2xH N (x) − H (x) (P1.21.3)
N
(ii) Make a MATLAB routine “Hermitp(N)” which uses Eq. (P1.21.3)
to generate the Nth-degree Hermite polynomial H N (x).
(b) The Bessel Function of the First Kind [K-1]
Consider the Bessel function of the first kind of order k defined as
1 π
J k (β) = cos(kδ − β sin δ)dδ (P1.21.4a)
π 0
k ∞ m 2m
β (−1) β k
= ≡ (−1) J −k (β) (P1.21.4b)
m
2 4 m!(m + k)!
m=0
(i) Define the integrand of (P1.21.4a) in the name of ‘Bessel_inte-
grand(x,beta,k)’ and store it in an M-file named “Bessel_
integrand.m”.
(ii) Complete the following routine “Jkb(K,beta)”, which uses
(P1.21.4b) in a recursive way to compute J k (β) of order k =
1:K for given K and β (beta).
(iii) Run the following program nm1p21b which uses Eqs. (P1.21.4a)
and (P1.21.4b) to get J 15 (β) for β = 0:0.05:15. What is the norm