160
I. P. Markov and M. V. Markina
We notice that right-hand side in a complex-valued linear algebraic system (11.27)
is defined by values of amplitude function f (x). For our case, it means that to
calculate integrals I 1 [ϕ, τ ] and I 2 [ϕ, τ ], we need to solve three linear systems, one
for each phase function q m , m = 1, 2, 3, with 24 different right-hand sides each: six
different amplitude functions in I 1 and 18 different amplitude functions in I 2 .
To overcome this drawback, we will consider a variation of Levin’s method. Evans
and Webster (1997) proposed to form quadrature rule
I =
1
−1
f (x)e
iτ q(x) dx ≈
N
j=0
w j f (x j ),
(11.29)
where the weights w j are chosen, so that formula (11.29) is exact for the functions
h k (x) = iτ q
(x) p k (x) + p
k (x), k = 0, N .
(11.30)
Substituting the functions h k
x j
on a set of collocations points
x j
into the
quadrature formula, we obtain a complex-valued linear algebraic system for weights
w j
a k j w j = b k ,
(11.31)
a k j = iτ q
(x j ) p k (x j ) + p
k (x j ), k, j = 0, N ,
(11.32)
b k = p k (1)e
iτ q(1)
− p k (−1)e
iτ q(−1)
, k = 0, N .
(11.33)
This approach provides freedom of choice of collocation points
x j
and functions
p k . Originally Evans and Webster suggested the following
x j = cos( jπ/N ), p j (x) = T j (x), j = 0, N ,
(11.34)
where T j (x) is the j-th Chebyshev polynomial of the first kind.
In our case to compute integrals I 1 [ϕ, τ ] and I 2 [ϕ, τ ] with Evans-Webster method,
we need to solve only three complex-valued linear algebraic systems, one for each
phase function q m , m = 1, 2, 3.
We start with defining phase functions and obtaining their derivatives
q m (ψ) = −
cos ψ
√ λ m
, q
m (ψ) =
1
2
2λ m sin ψ + λ
m cos ψ
λ
3/2
m
,
(11.35)
λ
m (ψ) =
3
i=1
3
j=1
E im (n)E jm (n)) i j (n, β),
(11.36)
I. P. Markov and M. V. Markina
We notice that right-hand side in a complex-valued linear algebraic system (11.27)
is defined by values of amplitude function f (x). For our case, it means that to
calculate integrals I 1 [ϕ, τ ] and I 2 [ϕ, τ ], we need to solve three linear systems, one
for each phase function q m , m = 1, 2, 3, with 24 different right-hand sides each: six
different amplitude functions in I 1 and 18 different amplitude functions in I 2 .
To overcome this drawback, we will consider a variation of Levin’s method. Evans
and Webster (1997) proposed to form quadrature rule
I =
1
−1
f (x)e
iτ q(x) dx ≈
N
j=0
w j f (x j ),
(11.29)
where the weights w j are chosen, so that formula (11.29) is exact for the functions
h k (x) = iτ q
(x) p k (x) + p
k (x), k = 0, N .
(11.30)
Substituting the functions h k
x j
on a set of collocations points
x j
into the
quadrature formula, we obtain a complex-valued linear algebraic system for weights
w j
a k j w j = b k ,
(11.31)
a k j = iτ q
(x j ) p k (x j ) + p
k (x j ), k, j = 0, N ,
(11.32)
b k = p k (1)e
iτ q(1)
− p k (−1)e
iτ q(−1)
, k = 0, N .
(11.33)
This approach provides freedom of choice of collocation points
x j
and functions
p k . Originally Evans and Webster suggested the following
x j = cos( jπ/N ), p j (x) = T j (x), j = 0, N ,
(11.34)
where T j (x) is the j-th Chebyshev polynomial of the first kind.
In our case to compute integrals I 1 [ϕ, τ ] and I 2 [ϕ, τ ] with Evans-Webster method,
we need to solve only three complex-valued linear algebraic systems, one for each
phase function q m , m = 1, 2, 3.
We start with defining phase functions and obtaining their derivatives
q m (ψ) = −
cos ψ
√ λ m
, q
m (ψ) =
1
2
2λ m sin ψ + λ
m cos ψ
λ
3/2
m
,
(11.35)
λ
m (ψ) =
3
i=1
3
j=1
E im (n)E jm (n)) i j (n, β),
(11.36)