Mixing
$int _ { 0 } ^ { L } left{ left( int _ { 0 } ^ { x } f ( y ) d y right) left( int _ { 0 } ^ { x } g ( z ) d z...
$begingroup$
As in the title,
What I wanna do is
$int _ { 0 } ^ { L } left{ left( int _ { 0 } ^ { x } f ( y ) d y right) left( int _ { 0 } ^ { x } g ( z ) d z right) right} d x$
Here, functions f and g are arbitrary ones, x is a variable, and L is a constant number.
I can do this numerically. But It takes so much computation costs.
Therefore, I'd like to convert this formula to a simpler form before doing numerical computation. (I think the integral by parts can be a key but I have no idea how/where I should apply it)
Are there any clever ways to do this?
integration
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
$endgroup$
add a comment |
$begingroup$
As in the title,
What I wanna do is
$int _ { 0 } ^ { L } left{ left( int _ { 0 } ^ { x } f ( y ) d y right) left( int _ { 0 } ^ { x } g ( z ) d z right) right} d x$
Here, functions f and g are arbitrary ones, x is a variable, and L is a constant number.
I can do this numerically. But It takes so much computation costs.
Therefore, I'd like to convert this formula to a simpler form before doing numerical computation. (I think the integral by parts can be a key but I have no idea how/where I should apply it)
Are there any clever ways to do this?
integration
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
$endgroup$
$begingroup$
What numerical algorithm are you using? The formula is already "simple".
$endgroup$
– user8469759
Jan 30 at 9:09
$begingroup$
I'm using the Gaussian quadrature. But, I need much more matrix computations than those I needed for solving just normal integral such as int(f(x),0,L).
$endgroup$
– Sinwoo Jeong
Jan 30 at 9:17
$begingroup$
I don't see the problem, It's a single for loop unless I'm missing something.
$endgroup$
– user8469759
Jan 30 at 9:21
$begingroup$
@SinwooJeong It shouldn't take meaningfully longer than a single integral. Going from $int_0^x ldots$ to $int_0^{x+dx} ldots$ just requires adding one evaluation of $ldots$ to the former...
$endgroup$
– Solomonoff's Secret
Jan 30 at 9:33
add a comment |
$begingroup$
As in the title,
What I wanna do is
$int _ { 0 } ^ { L } left{ left( int _ { 0 } ^ { x } f ( y ) d y right) left( int _ { 0 } ^ { x } g ( z ) d z right) right} d x$
Here, functions f and g are arbitrary ones, x is a variable, and L is a constant number.
I can do this numerically. But It takes so much computation costs.
Therefore, I'd like to convert this formula to a simpler form before doing numerical computation. (I think the integral by parts can be a key but I have no idea how/where I should apply it)
Are there any clever ways to do this?
integration
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
$endgroup$
As in the title,
What I wanna do is
$int _ { 0 } ^ { L } left{ left( int _ { 0 } ^ { x } f ( y ) d y right) left( int _ { 0 } ^ { x } g ( z ) d z right) right} d x$
Here, functions f and g are arbitrary ones, x is a variable, and L is a constant number.
I can do this numerically. But It takes so much computation costs.
Therefore, I'd like to convert this formula to a simpler form before doing numerical computation. (I think the integral by parts can be a key but I have no idea how/where I should apply it)
Are there any clever ways to do this?
integration
integration
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
edited Jan 30 at 10:11
YuiTo Cheng
2,1863937
2,1863937
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
asked Jan 30 at 8:30
Sinwoo JeongSinwoo Jeong
32
32
$begingroup$
What numerical algorithm are you using? The formula is already "simple".
$endgroup$
– user8469759
Jan 30 at 9:09
$begingroup$
I'm using the Gaussian quadrature. But, I need much more matrix computations than those I needed for solving just normal integral such as int(f(x),0,L).
$endgroup$
– Sinwoo Jeong
Jan 30 at 9:17
$begingroup$
I don't see the problem, It's a single for loop unless I'm missing something.
$endgroup$
– user8469759
Jan 30 at 9:21
$begingroup$
@SinwooJeong It shouldn't take meaningfully longer than a single integral. Going from $int_0^x ldots$ to $int_0^{x+dx} ldots$ just requires adding one evaluation of $ldots$ to the former...
$endgroup$
– Solomonoff's Secret
Jan 30 at 9:33
add a comment |
$begingroup$
What numerical algorithm are you using? The formula is already "simple".
$endgroup$
– user8469759
Jan 30 at 9:09
$begingroup$
I'm using the Gaussian quadrature. But, I need much more matrix computations than those I needed for solving just normal integral such as int(f(x),0,L).
$endgroup$
– Sinwoo Jeong
Jan 30 at 9:17
$begingroup$
I don't see the problem, It's a single for loop unless I'm missing something.
$endgroup$
– user8469759
Jan 30 at 9:21
$begingroup$
@SinwooJeong It shouldn't take meaningfully longer than a single integral. Going from $int_0^x ldots$ to $int_0^{x+dx} ldots$ just requires adding one evaluation of $ldots$ to the former...
$endgroup$
– Solomonoff's Secret
Jan 30 at 9:33
$begingroup$
What numerical algorithm are you using? The formula is already "simple".
$endgroup$
– user8469759
Jan 30 at 9:09
$begingroup$
What numerical algorithm are you using? The formula is already "simple".
$endgroup$
– user8469759
Jan 30 at 9:09
$begingroup$
I'm using the Gaussian quadrature. But, I need much more matrix computations than those I needed for solving just normal integral such as int(f(x),0,L).
$endgroup$
– Sinwoo Jeong
Jan 30 at 9:17
$begingroup$
I'm using the Gaussian quadrature. But, I need much more matrix computations than those I needed for solving just normal integral such as int(f(x),0,L).
$endgroup$
– Sinwoo Jeong
Jan 30 at 9:17
$begingroup$
I don't see the problem, It's a single for loop unless I'm missing something.
$endgroup$
– user8469759
Jan 30 at 9:21
$begingroup$
I don't see the problem, It's a single for loop unless I'm missing something.
$endgroup$
– user8469759
Jan 30 at 9:21
$begingroup$
@SinwooJeong It shouldn't take meaningfully longer than a single integral. Going from $int_0^x ldots$ to $int_0^{x+dx} ldots$ just requires adding one evaluation of $ldots$ to the former...
$endgroup$
– Solomonoff's Secret
Jan 30 at 9:33
$begingroup$
@SinwooJeong It shouldn't take meaningfully longer than a single integral. Going from $int_0^x ldots$ to $int_0^{x+dx} ldots$ just requires adding one evaluation of $ldots$ to the former...
$endgroup$
– Solomonoff's Secret
Jan 30 at 9:33
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It seems very straightforward as it is... For general functions $f,g$ there is no way of simplifying the expression. From the numerical point of view, taking a regular decomposition of $[0,L]$ and some numerical quadrature $int_0^L h(x) dx approx sum_{i=1}^n w_i f_i$, you just have to implement the expression
$$
sum_{i=1}^n left(w_i left(sum_{j=1}^{i} w_j f_jright) left(sum_{j=1}^{i} w_j g_jright) right)
$$
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
It seems very straightforward as it is... For general functions $f,g$ there is no way of simplifying the expression. From the numerical point of view, taking a regular decomposition of $[0,L]$ and some numerical quadrature $int_0^L h(x) dx approx sum_{i=1}^n w_i f_i$, you just have to implement the expression
$$
sum_{i=1}^n left(w_i left(sum_{j=1}^{i} w_j f_jright) left(sum_{j=1}^{i} w_j g_jright) right)
$$
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
$endgroup$
add a comment |
$begingroup$
It seems very straightforward as it is... For general functions $f,g$ there is no way of simplifying the expression. From the numerical point of view, taking a regular decomposition of $[0,L]$ and some numerical quadrature $int_0^L h(x) dx approx sum_{i=1}^n w_i f_i$, you just have to implement the expression
$$
sum_{i=1}^n left(w_i left(sum_{j=1}^{i} w_j f_jright) left(sum_{j=1}^{i} w_j g_jright) right)
$$
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
$endgroup$
add a comment |
$begingroup$
It seems very straightforward as it is... For general functions $f,g$ there is no way of simplifying the expression. From the numerical point of view, taking a regular decomposition of $[0,L]$ and some numerical quadrature $int_0^L h(x) dx approx sum_{i=1}^n w_i f_i$, you just have to implement the expression
$$
sum_{i=1}^n left(w_i left(sum_{j=1}^{i} w_j f_jright) left(sum_{j=1}^{i} w_j g_jright) right)
$$
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
$endgroup$
It seems very straightforward as it is... For general functions $f,g$ there is no way of simplifying the expression. From the numerical point of view, taking a regular decomposition of $[0,L]$ and some numerical quadrature $int_0^L h(x) dx approx sum_{i=1}^n w_i f_i$, you just have to implement the expression
$$
sum_{i=1}^n left(w_i left(sum_{j=1}^{i} w_j f_jright) left(sum_{j=1}^{i} w_j g_jright) right)
$$
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
answered Jan 30 at 9:18
PierreCarrePierreCarre
1,690212
1,690212
add a comment |
add a comment |
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$begingroup$
What numerical algorithm are you using? The formula is already "simple".
$endgroup$
– user8469759
Jan 30 at 9:09
$begingroup$
I'm using the Gaussian quadrature. But, I need much more matrix computations than those I needed for solving just normal integral such as int(f(x),0,L).
$endgroup$
– Sinwoo Jeong
Jan 30 at 9:17
$begingroup$
I don't see the problem, It's a single for loop unless I'm missing something.
$endgroup$
– user8469759
Jan 30 at 9:21
$begingroup$
@SinwooJeong It shouldn't take meaningfully longer than a single integral. Going from $int_0^x ldots$ to $int_0^{x+dx} ldots$ just requires adding one evaluation of $ldots$ to the former...
$endgroup$
– Solomonoff's Secret
Jan 30 at 9:33