Mixing
Bounds for the error of this approximation to the Bessel function
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I found a nice explicit approximation to the Bessel function today, using the integral:
$$J_0(x)=frac{2}{pi} int_0^1 frac{cos x u}{sqrt{1-u^2}}du$$
With Chebyshev-Gauss quadrature we can see that the following approximation works:
$$J_0(x) approx frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
Here's an example for $n=20$, which shows that the approximation is very good for small $x$.
However, for large arguments the quadrature quickly loses the accuracy, because of strong oscillatory behavior of the integrand.
Numerically for $ x gg n$ the error quickly blows up and starts oscillating.
Because I want to implement this approximation in an algorithm, I would like to know if we can derive explicit bounds for the error:
$$E(n,x)=J_0(x)- frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
integration numerical-methods approximation bessel-functions
asked Nov 16 at 21:21
Yuriy S
15.5k433115
add a comment |
up vote
1
down vote
favorite
I found a nice explicit approximation to the Bessel function today, using the integral:
$$J_0(x)=frac{2}{pi} int_0^1 frac{cos x u}{sqrt{1-u^2}}du$$
With Chebyshev-Gauss quadrature we can see that the following approximation works:
$$J_0(x) approx frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
Here's an example for $n=20$, which shows that the approximation is very good for small $x$.
However, for large arguments the quadrature quickly loses the accuracy, because of strong oscillatory behavior of the integrand.
Numerically for $ x gg n$ the error quickly blows up and starts oscillating.
Because I want to implement this approximation in an algorithm, I would like to know if we can derive explicit bounds for the error:
$$E(n,x)=J_0(x)- frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
integration numerical-methods approximation bessel-functions
asked Nov 16 at 21:21
Yuriy S
15.5k433115
One more interesting question from you ! Cheers.
– Claude Leibovici
Nov 19 at 9:53
@ClaudeLeibovici, thank you! Your expertise is always appreciated. I never had a numerical methods course, so I learn them as I go along. As for this, I already found the explicit expression for the remainder term, so I will probably answer the question myself
– Yuriy S
Nov 19 at 10:05
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I found a nice explicit approximation to the Bessel function today, using the integral:
$$J_0(x)=frac{2}{pi} int_0^1 frac{cos x u}{sqrt{1-u^2}}du$$
With Chebyshev-Gauss quadrature we can see that the following approximation works:
$$J_0(x) approx frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
Here's an example for $n=20$, which shows that the approximation is very good for small $x$.
However, for large arguments the quadrature quickly loses the accuracy, because of strong oscillatory behavior of the integrand.
Numerically for $ x gg n$ the error quickly blows up and starts oscillating.
Because I want to implement this approximation in an algorithm, I would like to know if we can derive explicit bounds for the error:
$$E(n,x)=J_0(x)- frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
integration numerical-methods approximation bessel-functions
asked Nov 16 at 21:21
Yuriy S
15.5k433115
I found a nice explicit approximation to the Bessel function today, using the integral:
$$J_0(x)=frac{2}{pi} int_0^1 frac{cos x u}{sqrt{1-u^2}}du$$
With Chebyshev-Gauss quadrature we can see that the following approximation works:
$$J_0(x) approx frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
Here's an example for $n=20$, which shows that the approximation is very good for small $x$.
However, for large arguments the quadrature quickly loses the accuracy, because of strong oscillatory behavior of the integrand.
Numerically for $ x gg n$ the error quickly blows up and starts oscillating.
Because I want to implement this approximation in an algorithm, I would like to know if we can derive explicit bounds for the error:
$$E(n,x)=J_0(x)- frac{1}{n} sum_{k=1}^n cos left(x cos left( frac{2k-1}{2n} pi right) right)$$
integration numerical-methods approximation bessel-functions
integration numerical-methods approximation bessel-functions
asked Nov 16 at 21:21
Yuriy S
15.5k433115
asked Nov 16 at 21:21
Yuriy S
15.5k433115
edited 2 days ago
asked Nov 16 at 21:21
Yuriy S
15.5k433115
asked Nov 16 at 21:21
Yuriy S
15.5k433115
asked Nov 16 at 21:21
Yuriy S
15.5k433115
15.5k433115
One more interesting question from you ! Cheers.
– Claude Leibovici
Nov 19 at 9:53
@ClaudeLeibovici, thank you! Your expertise is always appreciated. I never had a numerical methods course, so I learn them as I go along. As for this, I already found the explicit expression for the remainder term, so I will probably answer the question myself
– Yuriy S
Nov 19 at 10:05
add a comment |
One more interesting question from you ! Cheers.
– Claude Leibovici
Nov 19 at 9:53
@ClaudeLeibovici, thank you! Your expertise is always appreciated. I never had a numerical methods course, so I learn them as I go along. As for this, I already found the explicit expression for the remainder term, so I will probably answer the question myself
– Yuriy S
Nov 19 at 10:05
One more interesting question from you ! Cheers.
– Claude Leibovici
Nov 19 at 9:53
One more interesting question from you ! Cheers.
– Claude Leibovici
Nov 19 at 9:53
@ClaudeLeibovici, thank you! Your expertise is always appreciated. I never had a numerical methods course, so I learn them as I go along. As for this, I already found the explicit expression for the remainder term, so I will probably answer the question myself
– Yuriy S
Nov 19 at 10:05
@ClaudeLeibovici, thank you! Your expertise is always appreciated. I never had a numerical methods course, so I learn them as I go along. As for this, I already found the explicit expression for the remainder term, so I will probably answer the question myself
– Yuriy S
Nov 19 at 10:05
add a comment |
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One more interesting question from you ! Cheers.
– Claude Leibovici
Nov 19 at 9:53
@ClaudeLeibovici, thank you! Your expertise is always appreciated. I never had a numerical methods course, so I learn them as I go along. As for this, I already found the explicit expression for the remainder term, so I will probably answer the question myself
– Yuriy S
Nov 19 at 10:05