Motivation
Our Results
Summary
Calculation of Highly Oscillatory Integrals by
Quadrature Methods
Krishna Thapa
Department of Physics & Astronomy
Texas A&M University
College Station, TX
May 18, 2012
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Outline
1
Motivation
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
2
Our Results
Main Results
Implementation
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Outline
1
Motivation
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
2
Our Results
Main Results
Implementation
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
A model for Vacuum Energy
Our model of quantum vacuum energy density near the
boundary has the form λz α .
Figure: Steeply rising potential near the boundary
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Why are we interested?
Spectral analysis of the rising potential gives Energy
momentum tensor:
Z ∞
Z 1 p
1
T (z) = 3
dρ
du 1 − u 2 cos(2zρu − 2δ(ρu))
π 0
0
(1)
where,
AiryAi(−u 2 ) δ(u) = ArcTan − u
AiryAi0 (−u 2 )
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
What does it look like?
1.0
1.0
0.5
0.5
100
100
100
100
100.01
-0.5
-0.5
-1.0
-1.0
100.02
100.03
100.04
Figure: The oscillatory cosine function
left:u goes from 100 to 100.0002
right: u goes from 100 to 100.004
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
What does it look like?
1.0
1.0
0.5
0.5
100
100
100
100
100.01
-0.5
-0.5
-1.0
-1.0
100.02
100.03
100.04
Figure: The oscillatory cosine function
left:u goes from 100 to 100.0002
right: u goes from 100 to 100.004
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Outline
1
Motivation
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
2
Our Results
Main Results
Implementation
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
T (z) is highly oscillatory.
Takes hours, if not days to calculate.
Only for α = 1.
We need to check for higher values of α.
Similar T (z) for higher α values are bound to give more
highly oscillatory integrals
We need systematic way to calculate these oscillatory
integrals.
Check whether our model for potential is plausible.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
T (z) is highly oscillatory.
Takes hours, if not days to calculate.
Only for α = 1.
We need to check for higher values of α.
Similar T (z) for higher α values are bound to give more
highly oscillatory integrals
We need systematic way to calculate these oscillatory
integrals.
Check whether our model for potential is plausible.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
T (z) is highly oscillatory.
Takes hours, if not days to calculate.
Only for α = 1.
We need to check for higher values of α.
Similar T (z) for higher α values are bound to give more
highly oscillatory integrals
We need systematic way to calculate these oscillatory
integrals.
Check whether our model for potential is plausible.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
T (z) is highly oscillatory.
Takes hours, if not days to calculate.
Only for α = 1.
We need to check for higher values of α.
Similar T (z) for higher α values are bound to give more
highly oscillatory integrals
We need systematic way to calculate these oscillatory
integrals.
Check whether our model for potential is plausible.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
T (z) is highly oscillatory.
Takes hours, if not days to calculate.
Only for α = 1.
We need to check for higher values of α.
Similar T (z) for higher α values are bound to give more
highly oscillatory integrals
We need systematic way to calculate these oscillatory
integrals.
Check whether our model for potential is plausible.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
T (z) is highly oscillatory.
Takes hours, if not days to calculate.
Only for α = 1.
We need to check for higher values of α.
Similar T (z) for higher α values are bound to give more
highly oscillatory integrals
We need systematic way to calculate these oscillatory
integrals.
Check whether our model for potential is plausible.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Outline
1
Motivation
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
2
Our Results
Main Results
Implementation
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Newton-Cotes Rule
Trapezoidal rule
Simpson’s rule
y
x1
h
x2
x
Figure: Plot showing integration by trapezoidal rule
Z
b
f (x)dx ≈ c1 f (a) + c2 f (b) =
a
(b − a)
(f (a) + f (b)).
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Newton-Cotes Rule
Trapezoidal rule
Simpson’s rule
y
x1
h
x2
x
Figure: Plot showing integration by trapezoidal rule
Z
b
f (x)dx ≈ c1 f (a) + c2 f (b) =
a
(b − a)
(f (a) + f (b)).
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Newton-Cotes Rule
Trapezoidal rule
Simpson’s rule
y
x1
h
x2
x
Figure: Plot showing integration by trapezoidal rule
Z
b
f (x)dx ≈ c1 f (a) + c2 f (b) =
a
(b − a)
(f (a) + f (b)).
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Newton-Cotes Rule
Trapezoidal rule
Simpson’s rule
y
x1
h
x2
x
Figure: Plot showing integration by trapezoidal rule
Z
b
f (x)dx ≈ c1 f (a) + c2 f (b) =
a
(b − a)
(f (a) + f (b)).
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Newton-Cotes Rule
Trapezoidal rule
Simpson’s rule
y
x1
h
x2
x
Figure: Plot showing integration by trapezoidal rule
Z
b
f (x)dx ≈ c1 f (a) + c2 f (b) =
a
(b − a)
(f (a) + f (b)).
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Gauss-Quadrature
Rb
a
f (x)dx ≈ c1 f (x1 ) + c2 f (x2 ).
Here, c1 , c2 , x1 , and x2 are all unknowns.
In this case, these four constants are found by integrating
third order polynomials and equating the coefficients.
x1 =
b − a −1 b + a
√ +
,
2
2
3
x2 =
b−a 1
b+a
√ +
,
2
2
3
c1 =
b−a
b−a
, and c2 =
.
2
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Gauss-Quadrature
Rb
a
f (x)dx ≈ c1 f (x1 ) + c2 f (x2 ).
Here, c1 , c2 , x1 , and x2 are all unknowns.
In this case, these four constants are found by integrating
third order polynomials and equating the coefficients.
x1 =
b − a −1 b + a
√ +
,
2
2
3
x2 =
b−a 1
b+a
√ +
,
2
2
3
c1 =
b−a
b−a
, and c2 =
.
2
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Gauss-Quadrature
Rb
a
f (x)dx ≈ c1 f (x1 ) + c2 f (x2 ).
Here, c1 , c2 , x1 , and x2 are all unknowns.
In this case, these four constants are found by integrating
third order polynomials and equating the coefficients.
x1 =
b − a −1 b + a
√ +
,
2
2
3
x2 =
b−a 1
b+a
√ +
,
2
2
3
c1 =
b−a
b−a
, and c2 =
.
2
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Gauss-Quadrature
Rb
a
f (x)dx ≈ c1 f (x1 ) + c2 f (x2 ).
Here, c1 , c2 , x1 , and x2 are all unknowns.
In this case, these four constants are found by integrating
third order polynomials and equating the coefficients.
x1 =
b − a −1 b + a
√ +
,
2
2
3
x2 =
b−a 1
b+a
√ +
,
2
2
3
c1 =
b−a
b−a
, and c2 =
.
2
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Gauss-Quadrature
Rb
a
f (x)dx ≈ c1 f (x1 ) + c2 f (x2 ).
Here, c1 , c2 , x1 , and x2 are all unknowns.
In this case, these four constants are found by integrating
third order polynomials and equating the coefficients.
x1 =
b − a −1 b + a
√ +
,
2
2
3
x2 =
b−a 1
b+a
√ +
,
2
2
3
c1 =
b−a
b−a
, and c2 =
.
2
2
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Filon’s method
Z
b
Z
∞
f (x)sinωxdx and
a
0
Z
f (x) sin(ωx)dx =
2µ+2
X
f (x)
sinωx
x
f (x) sin(ωx)
m=µ
mµ (xµ ) = f (xµ ), mµ+1 (xµ+1 ) = f (xµ+1 ), and mµ+2 (xµ+2 ) = f (xµ+2 ).
Z
b
f (x)sinωxdx ≈
a
n−1 Z
X
x2µ+2
mµ (x)sinωxdx
µ=0 x2µ
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Filon’s method
Z
b
Z
∞
f (x)sinωxdx and
a
0
Z
f (x) sin(ωx)dx =
2µ+2
X
f (x)
sinωx
x
f (x) sin(ωx)
m=µ
mµ (xµ ) = f (xµ ), mµ+1 (xµ+1 ) = f (xµ+1 ), and mµ+2 (xµ+2 ) = f (xµ+2 ).
Z
b
f (x)sinωxdx ≈
a
n−1 Z
X
x2µ+2
mµ (x)sinωxdx
µ=0 x2µ
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Filon’s method
Z
b
Z
∞
f (x)sinωxdx and
a
0
Z
f (x) sin(ωx)dx =
2µ+2
X
f (x)
sinωx
x
f (x) sin(ωx)
m=µ
mµ (xµ ) = f (xµ ), mµ+1 (xµ+1 ) = f (xµ+1 ), and mµ+2 (xµ+2 ) = f (xµ+2 ).
Z
b
f (x)sinωxdx ≈
a
n−1 Z
X
x2µ+2
mµ (x)sinωxdx
µ=0 x2µ
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Filon’s method
Z
b
Z
∞
f (x)sinωxdx and
a
0
Z
f (x) sin(ωx)dx =
2µ+2
X
f (x)
sinωx
x
f (x) sin(ωx)
m=µ
mµ (xµ ) = f (xµ ), mµ+1 (xµ+1 ) = f (xµ+1 ), and mµ+2 (xµ+2 ) = f (xµ+2 ).
Z
b
f (x)sinωxdx ≈
a
n−1 Z
X
x2µ+2
mµ (x)sinωxdx
µ=0 x2µ
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Filon’s method
Z
b
Z
∞
f (x)sinωxdx and
a
0
Z
f (x) sin(ωx)dx =
2µ+2
X
f (x)
sinωx
x
f (x) sin(ωx)
m=µ
mµ (xµ ) = f (xµ ), mµ+1 (xµ+1 ) = f (xµ+1 ), and mµ+2 (xµ+2 ) = f (xµ+2 ).
Z
b
f (x)sinωxdx ≈
a
n−1 Z
X
x2µ+2
mµ (x)sinωxdx
µ=0 x2µ
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Filon’s method
Z
b
Z
∞
f (x)sinωxdx and
a
0
Z
f (x) sin(ωx)dx =
2µ+2
X
f (x)
sinωx
x
f (x) sin(ωx)
m=µ
mµ (xµ ) = f (xµ ), mµ+1 (xµ+1 ) = f (xµ+1 ), and mµ+2 (xµ+2 ) = f (xµ+2 ).
Z
b
f (x)sinωxdx ≈
a
n−1 Z
X
x2µ+2
mµ (x)sinωxdx
µ=0 x2µ
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
ClenshawCurtis Method
C.W.Clenshaw and A.R. Curtis in 1960. Expand f(x) in
Chebyshev polynomials.
f (x) = F (t) =
1
1
a0 +a1 T1 (t)+a2 T2 (t)+...+ an Tn (t), (a ≤ x ≤ b)
2
2
(2)
where,
Tn (t) = cos(n cos−1 (t), t =
2x − (b + a)
b−a
(3)
and this eventually reduces to
∞
X
a0
nπ
f (x) =
T0 (x) +
an Tn (x), xn = cos( ).
2
N
(4)
n=1
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
ClenshawCurtis Method
C.W.Clenshaw and A.R. Curtis in 1960. Expand f(x) in
Chebyshev polynomials.
f (x) = F (t) =
1
1
a0 +a1 T1 (t)+a2 T2 (t)+...+ an Tn (t), (a ≤ x ≤ b)
2
2
(2)
where,
Tn (t) = cos(n cos−1 (t), t =
2x − (b + a)
b−a
(3)
and this eventually reduces to
∞
X
a0
nπ
f (x) =
T0 (x) +
an Tn (x), xn = cos( ).
2
N
(4)
n=1
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
ClenshawCurtis Method
C.W.Clenshaw and A.R. Curtis in 1960. Expand f(x) in
Chebyshev polynomials.
f (x) = F (t) =
1
1
a0 +a1 T1 (t)+a2 T2 (t)+...+ an Tn (t), (a ≤ x ≤ b)
2
2
(2)
where,
Tn (t) = cos(n cos−1 (t), t =
2x − (b + a)
b−a
(3)
and this eventually reduces to
∞
X
a0
nπ
f (x) =
T0 (x) +
an Tn (x), xn = cos( ).
2
N
(4)
n=1
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Levin-Iserles0 Method
Improvement over Filon’s method
1 − eiω 1
1 + eiω
+
12
f (0)
−6
iω
iω 3
ω4
eiω
1 + eiω
1 − eiω +6
−
12
f (1)
+
iω
iω 3
ω4
1
2 + eiω
1 − eiω 0
+ − 2 −2
+
6
f (0)
ω
iω 3
ω4
eiω
1 + eiω
1 − eiω 0
+
−
2
+
6
f (1)
ω2
iω 3
ω4
Q2F [f ] = −
(5)
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
Levin-Iserles0 Method
Improvement over Filon’s method
1
1 − eiω 1 + eiω
+
12
f (0)
−6
iω
iω 3
ω4
eiω
1 + eiω
1 − eiω +6
−
12
f (1)
+
iω
iω 3
ω4
1
2 + eiω
1 − eiω 0
+ − 2 −2
+
6
f (0)
ω
iω 3
ω4
eiω
1 + eiω
1 − eiω 0
+
−
2
+
6
f (1)
ω2
iω 3
ω4
Q2F [f ] = −
(5)
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
Outline
1
Motivation
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
2
Our Results
Main Results
Implementation
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
Was it worth the time?
Yes, and No.
Iserles’ method did not work for our T (z) integral.
Were able to calculate integrals much faster.
Not very consistent.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
Was it worth the time?
Yes, and No.
Iserles’ method did not work for our T (z) integral.
Were able to calculate integrals much faster.
Not very consistent.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
Was it worth the time?
Yes, and No.
Iserles’ method did not work for our T (z) integral.
Were able to calculate integrals much faster.
Not very consistent.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
Was it worth the time?
Yes, and No.
Iserles’ method did not work for our T (z) integral.
Were able to calculate integrals much faster.
Not very consistent.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
Outline
1
Motivation
Study of Vacuum Energy
Oscillatory Integrals
Earlier Literature
2
Our Results
Main Results
Implementation
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Main Results
Implementation
What quadrature method to choose?
Levin-Iserles0 method seems more promising.
Clenshawcurtis’ qudrature method also works well.
-2.0
-1.5
-1.0
-0.5
0.008
0.008
0.006
0.006
0.004
0.004
0.002
0.002
-2.0
-1.5
-1.0
-0.5
Figure: T (z)using Levin and Clenshawcurtis method
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Summary
Highly oscillatory integrals can be calculatedmuch faster
than by conventional methods.
choose methods judiciously.
Reduce error for integrands with large frequency.
Outlook
Not enough data for conclusion.
Check for higher values of α.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Motivation
Our Results
Summary
Summary
Highly oscillatory integrals can be calculatedmuch faster
than by conventional methods.
choose methods judiciously.
Reduce error for integrands with large frequency.
Outlook
Not enough data for conclusion.
Check for higher values of α.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Appendix
Support
NSF−0554849 and PHY−0968269.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods
Appendix
Support
For Further Reading I
J. D. Bouas and S. A. Fulling and F. D. Mera and K. Thapa
and C. S. Trendafilova and and J. Wagner
Investigating the Spectral Geometry of a Soft Wall
Proceedings of Symposia in Pure Mathematics, 2011.
Arieh Iserles and Syvert P. Norsett
Quadrature methods for multivariate highly oscillatory
integrals using derivatives.
Mathematics of Computation, 2006.
Louis Napoleon George Filon
On a quadrature formula for trigonometric integrals
Proc. Roy soc. Edinburgh, 1928.
Krishna Thapa Department of Physics & Astronomy Texas A&M University
Calculation
College
of Oscillatory
Station, TX
Integrals by Quadrature Methods