How Can We Numerically Integrate an Oscillatory Function Over a Large Interval?

[jeffer vitola]

hi all,,,, this is the integral that I would like you users work, I would like to know what methods and with all the steps as they arrive at the approach, I'm finishing a new numerical method to and I am doing some tests, to compare it with all methods made ​​by all users of this forum, and see if still worth the way as I pose and solve the exercise, there are no limitations can use your computer with the wolfram alpha or MATLAB to corroborate the numerical solution , but if the essence of the exercise is to see what methods and apply formulas to arrive at this result, Integrate[7*Pi^(13)*Sin[(x^4)], {x, 3*Pi*E, 73*Pi*E}] where E is number euler approximately 2.718281828 where Pi is number aproximately 3.141592654, or integral

,, \(\displaystyle \int_{3*Pi*e}^{73*Pi*e}\left(7*Pi^6*Pi^7 \right)\sin\left(x^4 \right)\,dx\) ,,

I hope your answers with the steps used to arrive at the approximation.

att
jefferson alexander vitola(Bigsmile)

 

[tkhunny]

First, what's the point of that mess of constants?
Second, WHY?!

 

[jeffer vitola]

First, what's the point of that mess of constants?
Second, WHY?!

hello,,,,,,does not understand the quiet exercise no problem,,,, with Respect to the arbitrary constants Are those I invent, I propose the integral and I think I can be say how I want my constant, if you want you can make a new topic and make comprehensive yours as you see fit and you can do the exercises Easiest, here I am asking a question on to advanced numerical integration, if you have to Contribute or comment on the issue your done them and resolve them, no need to ask questions out of place. clarify i use a translator because I do not speak English.

jefferson alexander vitola(Bigsmile)

 

[tkhunny]

0) My intent was to help you see that your proposed problem is a mess. If you really want to explore advanced numerical integration techniques, why cloud the experimentation with unnecessary elements?

1) $$\pi^{6}\cdot \pi^{7} = \pi^{13}$$ Why write them separately? Just to make it more complicated?

2) $$\int C\cdot f(x)\;dx = C\cdot \int f(x) \;dx$$ Why write the constant inside? Just to make it more complicated?

3) This is an amazingly intractable function over such a broad range. I might first think about Monte Carlo Techniques.

4) It is relatively well-behaved over much shorter periods. Some examination over the earlier "periods" may lead to reasonable conclusions for the rest. There may also be symmetries to exploit - some part of each "period" where we can conclude that portion is simply zero (0).

Lots to explore. How do you propose we go about it?

Suomen kieltä, vai?

 

[jeffer vitola]

0) My intent was to help you see that your proposed problem is a mess. If you really want to explore advanced numerical integration techniques, why cloud the experimentation with unnecessary elements?

1) $$\pi^{6}\cdot \pi^{7} = \pi^{13}$$ Why write them separately? Just to make it more complicated?

2) $$\int C\cdot f(x)\;dx = C\cdot \int f(x) \;dx$$ Why write the constant inside? Just to make it more complicated?

3) This is an amazingly intractable function over such a broad range. I might first think about Monte Carlo Techniques.

4) It is relatively well-behaved over much shorter periods. Some examination over the earlier "periods" may lead to reasonable conclusions for the rest. There may also be symmetries to exploit - some part of each "period" where we can conclude that portion is simply zero (0).

Lots to explore. How do you propose we go about it?

Suomen kieltä, vai?

hello,,,,,

I do not know or understand the language management of latex, so powers so wrote separate parts, so that's why I write the (PI raised to 6) (Pi raised to 7) ..,,

if you say that gives zero Prove numerical procedures,,,,,,,,,,,,,,,

if it is zero as you say,,,,,,,, then another error but in the MATLAB program and wolfram aplha program,,,,,,,,,,, these programs give non-zero solutions.

if your only contribution is to say you do not understand the exercise no problem, let other users trying to make it so,,,,,,, I see you master the power properties but this issue is advanced approximation oscillatory integrals, for children algebra,for algebra children I think on that topic you are very good,,, but I think in oscillatory integrals you know nothing,,,,

if you say that the integral is more difficult with the constant, you solve it without them, you solve the integral without the constant, not using them as an excuse to say that exercise can not be done,,,,,,,

you can not make the integral oscillatory, and so tries to discredit my Proposed, since You Can not do it, if your not quiet understand, but no more excuses.

you can not make the integral oscillatory,,, study and prepare better ,,, I see you lack enough on the issue of oscillatory integrals, ,,,,,, best of luck with easy exercises you if understand,,,,,,,, solve the procedures,,, your only contribution to my issue is that you can not do,,, that mathematical bad is your, I think you should get to study more and stop talking so much,,,

att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

...

if you say that gives zero Prove numerical procedures...

tkhunny did not say the entire integral is zero, but rather:

There may also be symmetries to exploit - some part of each "period" where we can conclude that portion is simply zero (0).

...but I think in oscillatory integrals you know nothing...I think you should get to study more and stop talking so much...

We at MHB do not allow such a rude tone to anyone, particularly to those who have taken the time and effort to offer help. I know you are using a translator, but it is clear these comments are not the result of loss of translation.

...if you say that the integral is more difficult with the constant, you solve it without them, you solve the integral without the constant, not using them as an excuse to say that exercise can not be done...

That is not what was said at all. One issue raised, and rightfully so in my opinion was writing:

\(\displaystyle \pi^6\cdot\pi^7\)

Why would you not combine these and then simply put the $7\pi^{13}$ out front as a constant factor which you could simply multiply the approximation by when you are done?

You are simply being asked reasonable and helpful questions, and to reply with an insulting tone is unacceptable. As I said, we do not allow this here, so please consider this a friendly advisement of our policy. You do not have to agree with what's being said, but you do have to be respectful when you voice your disagreement.

 

[jeffer vitola]

tkhunny did not say the entire integral is zero, but rather:We at MHB do not allow such a rude tone to anyone, particularly to those who have taken the time and effort to offer help. I know you are using a translator, but it is clear these comments are not the result of loss of translation.
That is not what was said at all. One issue raised, and rightfully so in my opinion was writing:

\(\displaystyle \pi^6\cdot\pi^7\)

Why would you not combine these and then simply put the $7\pi^{13}$ out front as a constant factor which you could simply multiply the approximation by when you are done?

You are simply being asked reasonable and helpful questions, and to reply with an insulting tone is unacceptable. As I said, we do not allow this here, so please consider this a friendly advisement of our policy. You do not have to agree with what's being said, but you do have to be respectful when you voice your disagreement.

Because I saw some mistook arbitrary constants,,,, User to a said my exercise was a disaster and That made ​​no sense,,,
I have taken the option of removing all constants,,,
to post it again,,,,,

I will come from time to time to the forum and see if your can someday understand and solve the integral oscillatory,
to Those Who said I was rude, I apologize because i did not Know That They felt unable to do exercise, i will be to see if I can compare my methods That Make users on this forum,
I mean That there are no more excuses not to solve the exercise hope their solutions,, thanks \(\displaystyle \int_{3*Pi*e}^{73*Pi*e}\sin\left(x^4 \right)\,dx\)
bye
att
jefferson alexander vitola(Bigsmile)

 

[tkhunny]

Trying to get something tractable out of these two hints:

$$\int \sin(x^{4}) + \dfrac{16}{5}x^{8}\sin(x^{4})\;dx = x\cdot\sin(x^{4})-\dfrac{4}{5}x^{5}\cos(x^{4})+C$$

and the relatively obvious

$$\int 4x^{3}\sin(x^{4})\;dx = -\cos(x^{4}) + C$$

Next on my list is to wonder just how far we have to go. This thing is oscillating pretty quickly by x = 30. Do we REALLY need anything past x = 100? There are 6 "periods" between x = 100 and x = 100.00001. Likewise, there are 5 "periods" between x = 200 and x = 200.000001. "Periods" are only about 0.00000001 wide around x = 600.

Mitä sanotte?

 

[jeffer vitola]

Trying to get something tractable out of these two hints:

$$\int \sin(x^{4}) + \dfrac{16}{5}x^{8}\sin(x^{4})\;dx = x\cdot\sin(x^{4})-\dfrac{4}{5}x^{5}\cos(x^{4})+C$$

and the relatively obvious

$$\int 4x^{3}\sin(x^{4})\;dx = -\cos(x^{4}) + C$$

Next on my list is to wonder just how far we have to go. This thing is oscillating pretty quickly by x = 30. Do we REALLY need anything past x = 100? There are 6 "periods" between x = 100 and x = 100.00001. Likewise, there are 5 "periods" between x = 200 and x = 200.000001. "Periods" are only about 0.00000001 wide around x = 600.

Mitä sanotte?

hello,,,,,,,,,,,,,,,,,,,,,,,, if this is the real challenge or the million dollar question as to calculate the numerical value of the whole from a very large and comprehensive range,,, , or from an interval That is too small,,,,,, this is what raised important exercise for me,,,,,, this is where I want to see as everyone you work and resolve,,,, as you approach and pose exercise to fix it,,,,,, I can listen and watch your ideas, Their Routines and you have tried to work as an integral oscillatory,,,, this is my objective notion of this exercise, I hope as we address and raise you exercise to work it though this oscillatory very quickly, I expect to see his ideas,,,,att
jefferson alexander vitola(Bigsmile)

 

[tkhunny]

Alas, we've seen none of your ideas.

 

[tkhunny]

Okay, my official first estimate:

$$\int\limits_{3\cdot\pi\cdot e}^{73\cdot\pi\cdot e}\sin(x^4)\;dx \approx \int\limits_{3\cdot\pi\cdot e}^{3\cdot\pi\cdot e + 0.000013403420002}\sin(x^4)\;dx$$

That mysterious number represents the size of the first piece of the integral Domain. Once the full "Period"s start, I have assumed it essentially zero. Every "period" is so close to zero, and so narrow. Although, as each second half "period" is about 0.0004% shorter than each first half period, my approximation is a lower bound. An adequate upper bound could be created simply by including the first half "period".

There you go. I'm tired of thinking about it.

 

[jeffer vitola]

Okay, my official first estimate:

$$\int\limits_{3\cdot\pi\cdot e}^{73\cdot\pi\cdot e}\sin(x^4)\;dx \approx \int\limits_{3\cdot\pi\cdot e}^{3\cdot\pi\cdot e + 0.000013403420002}\sin(x^4)\;dx$$

That mysterious number represents the size of the first piece of the integral Domain. Once the full "Period"s start, I have assumed it essentially zero. Every "period" is so close to zero, and so narrow. Although, as each second half "period" is about 0.0004% shorter than each first half period, my approximation is a lower bound. An adequate upper bound could be created simply by including the first half "period".

There you go. I'm tired of thinking about it.

hello,,,,,,,,,,,,,,,,,,,,, you tell me you're tired of thinking about this exercise,,,, it Seems that you are the only one around this forum who wants to solve, the others only know how to thank in your post, the other users will give thanks to your post When the Contributions are only say and say That my exercise is a mess,,,, I think I explain at the beginning of the post I wanted to see the methods and Routines for comparison with mine to see if my Routines are good or not good compared to what you did, so far I see You have solved nothing,,,, invite you to tell all your friends on the forum to help you think about the solution of the integrals, for you to you not get tired.

you approach you make of the integrals gives answers and different solutions, you have to rethink how you're evaluating,,,,
$$\int\limits_{3\cdot\pi\cdot e}^{73\cdot\pi\cdot e}\sin(x^4)\;dx \approx \int\limits_{3\cdot\pi\cdot e}^{3\cdot\pi\cdot e + 0.000013403420002}\sin(x^4)\;dx$$
You have said this,,,,,,,,and this is not true,,,,,,,,,, use the wolfram alpha and you looked that the solutions are different and not equal nor are approximate,,,
ask Him to help all your forum friends who always agree with you and so far always give thanks every thing you say,,,,,

I'll wait few new solutions make of all user the forum,,,,

att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

I've got an idea...why don't you show us what you have done? (Nod)

 

[jeffer vitola]

Alas, we've seen none of your ideas.

hello,,,,,,,,,,,,,,,,,,,,, I think I explain at the beginning of the post I wanted to see the methods and Routines for comparison with mine to see if my Routines are good or not good compared to what you did, so far I see You have solved nothing,,,, invite you to tell all your friends on the forum to help you think about the solution of the integrals, for you to you not get tired ,,,

Of Members are to all the forum,,, you are the only one who At least tried to work the integrals,,, no one has provided any solution to exercise Proposed by me, i see all around the forum solve the integrals easy exercises for more than 2 different methods and Provide interesting things in other subjects of integrals of other users,,,, in my post nobody even been actual Able to give any contribution to the numerical solution of oscillatory integrals,,,

a pity that no contribution in this issue of oscillatory integrals,,,,,,, only all can do the exercises of integrals of other users, but no one understand my topic nor my exercise of advanced numerical approximation?

att
jefferson alexander vitola(Bigsmile)

 

[jeffer vitola]

I've got an idea...why don't you show us what you have done? (Nod)

hello,,,,,,,,,,,,,Never said I'll post my methods and procedures here in this forum,,, I always said that I would like to see how they work this integral, to compare my results fact ,,,,,, on the other topics of this forum from other users of integral calculus see you all solve the exercises for more than 2 different methods and do not ask questions, just solve the exercise and ready,,, I think the problem with my integrated oscillatory is that they can not solve, so make so many questions and their contributions are not the solution of the integral,,, for this so you ask me as I resolve exercises, that you and all have doubts about how to solve it,,,
quiet I leave the problem to see if someone around the forum you can fix it and all the steps to reach the numerical response,,,,
att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

hello,,,,,,,,,,,,,Never said I'll post my methods and procedures here in this forum,,, I always said that I would like to see how they work this integral, to compare my results fact ,,,,,, on the other topics of this forum from other users of integral calculus see you all solve the exercises for more than 2 different methods and do not ask questions, just solve the exercise and ready,,, I think the problem with my integrated oscillatory is that they can not solve, so make so many questions and their contributions are not the solution of the integral,,, for this so you ask me as I resolve exercises, that you and all have doubts about how to solve it,,,
quiet I leave the problem to see if someone around the forum you can fix it and all the steps to reach the numerical response,,,,
att
jefferson alexander vitola(Bigsmile)

No one here ever said they were going to submit their work for your approval. You want to sit back and let the work of others pour in, so you can compare it to something you may or may not have? In the meantime, you chide our forum members as a whole for not providing their work to you...people in general do not respond well to this. (Wink)

I suggest, as a show of good faith, that you provide what you have, and then you may stand a better chance of others here providing suggestion on how you might improve it, although I make no promises. People are free to chose which topics they wish to participate in.

 

[jeffer vitola]

No one here ever said they were going to submit their work for your approval. You want to sit back and let the work of others pour in, so you can compare it to something you may or may not have? In the meantime, you chide our forum members as a whole for not providing their work to you...people in general do not respond well to this. (Wink)

I suggest, as a show of good faith, that you provide what you have, and then you may stand a better chance of others here providing suggestion on how you might improve it, although I make no promises. People are free to chose which topics they wish to participate in.

hello,,,,,,,,,,,,,,,

with Respect to That first I must show my Routines and solutions, I think I am the only user who asks a question about an exercise and they said Can not resolve,and they said what first i be must show me solved what make,

What you'll wait to solve the exercise and not take out excuses, like in the other integral calculus topics where nobody asks questions and solve all the exercises,,,,

I hope as solving the exercise of some of you,,,att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

In general we expect people posting questions to show some effort, as specified in our forum rules.

The question in which full solutions are provided are found in our Questions From Other Sites sub-forum, and the reason for this is given there. Otherwise, you will notice that people are encouraged to show what they have done.

I can't force you to show your work, but only advise you that you probably are not going to get many responses approaching it in the manner you have so far.

 

[zzephod]

Basic pseudo-bootstraped MC (10 by 100,000 sample MC to give estimate based on 1,000,000 points with estimate of SE) estimate of

$$\phantom{-------------} I=\int_{3.\pi.e}^{73.\pi.e} \sin(x^4) dx$$

>>> def Intg(N,M):
        ss=sin( (rand(M,N)*70*pi*exp(1)+3*pi*exp(1))**4.0 )
        return sum(ss,1)/N
>>>
>>> ii=Intg(100000,10)
>>>
>>> mean(ii)
-0.00010704988066875154
>>>
>>> std(ii)/sqrt(size(ii))
0.00088996139561893495
>>>

Execution time in PyLab, on a ~2.7Gz Intel i5 under WinXp, for Intg(100000,10) is ~0.14 s

Now show us yours...

.

 

[jeffer vitola]

Basic pseudo-bootstraped MC (10 by 100,000 sample MC to give estimate based on 1,000,000 points with estimate of SE) estimate of

$$\phantom{-------------} I=\int_{3.\pi.e}^{73.\pi.e} \sin(x^4) dx$$

>>> def Intg(N,M):
        ss=sin( (rand(M,N)*70*pi*exp(1)+3*pi*exp(1))**4.0 )
        return sum(ss,1)/N
>>>
>>> ii=Intg(100000,10)
>>>
>>> mean(ii)
-0.00010704988066875154
>>>
>>> std(ii)/sqrt(size(ii))
0.00088996139561893495
>>>

Execution time in PyLab, on a ~2.7Gz Intel i5 under WinXp, for Intg(100000,10) is ~0.14 s

Now show us yours...

.

hello,,,,,,,,,,,,,,,,,,,,,,
I do not know the program that you use to approximate the integral,,,,,,,which program is using?,,,,,you wrote in your code 70 and it was 73,,,,, I think it does not affect much the approximate solution but then only with my method,,,,, with your method suddenly yours if enough influence,,,,,,,,

Might be you instead of giving you could ask the 100,000 partitions that are 100 million program Partitions to be more accurate the approximation,,,,,,,,

if there is no limit on partitions that handles your program that I would like to bring it to 1000 million partitions to see how accurate the approximation and gives correct how decimal places,,,,,

I think it's too early or too fast show my procedures and oscillatory procedures development integrals, and only you and another user are the ones Who Have tried to solve the exercise Throughout the forum,,,,,,, I wait will more users to try to solve the exercise proposed by me,,,,

I clarify that I never said I'll post my results here in this forum,,,,,, I do not want and is not my intention to show that procedures and methods I solve the exercise,,,,,,,,, but then,,,, not sure yet,,,, I'm going to think,,,,,,,, while I I think, you could send more procedures and forms and methods to solve the exercise,,,,,,thanks for trying to solve my exercise oscillatory integrals,,,,att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

Using Online Integral Calculator I find:

\(\displaystyle \int_{3\pi e}^{73\pi e}\sin\left(x^4 \right)\,dx\approx9.223830260319849\,\times\,10^{-6}\)

Using W|A, I find:

\(\displaystyle \int_{3\pi e}^{73\pi e}\sin\left(x^4 \right)\,dx\approx\)

\(\displaystyle 9.2238302414744885138740661860325989095177512372824901723319193617115296332\,\times\,10^{-6}\)

 

[zzephod]

hello,,,,,,,,,,,,,,,,,,,,,,
I do not know the program that you use to approximate the integral,,,,,,,which program is using?,,,,,you wrote in your code 70 and it was 73,,,,, I think it does not affect much the approximate solution but then only with my method,,,,, with your method suddenly yours if enough influence,,,,,,,,

Might be you instead of giving you could ask the 100,000 partitions that are 100 million program Partitions to be more accurate the approximation,,,,,,,,

if there is no limit on partitions that handles your program that I would like to bring it to 1000 million partitions to see how accurate the approximation and gives correct how decimal places,,,,,

I think it's too early or too fast show my procedures and oscillatory procedures development integrals, and only you and another user are the ones Who Have tried to solve the exercise Throughout the forum,,,,,,, I wait will more users to try to solve the exercise proposed by me,,,,

I clarify that I never said I'll post my results here in this forum,,,,,, I do not want and is not my intention to show that procedures and methods I solve the exercise,,,,,,,,, but then,,,, not sure yet,,,, I'm going to think,,,,,,,, while I I think, you could send more procedures and forms and methods to solve the exercise,,,,,,thanks for trying to solve my exercise oscillatory integrals,,,,att
jefferson alexander vitola(Bigsmile)

This is a MC integral, so the SE of the estimate scales as the square root of the number of samples.

I will not do any more until you show something on your side, since at present you in breach of the implied social contract on this site.

.

 

[jeffer vitola]

Using Online Integral Calculator I find:

\(\displaystyle \int_{3\pi e}^{73\pi e}\sin\left(x^4 \right)\,dx\approx9.223830260319849\,\times\,10^{-6}\)

Using W|A, I find:

\(\displaystyle \int_{3\pi e}^{73\pi e}\sin\left(x^4 \right)\,dx\approx\)

\(\displaystyle 9.2238302414744885138740661860325989095177512372824901723319193617115296332\,\times\,10^{-6}\)

hello,,,,,,,,,,,,,,,,,,,

,,,,,,,this is correct,,,,,,,,,,,,,,,,,now the important is how you what your procedures and methods you use to reach this approximation, this is true essence and the importance of my proposed,,,,,att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

My method? Well, I went clackety-clack at my keyboard, et voila! (Poolparty)

 

[jeffer vitola]

This is a MC integral, so the SE of the estimate scales as the square root of the number of samples.

I will not do any more until you show something on your side, since at present you in breach of the implied social contract on this site.

.

hello,,,,,,,,,,,,,,,,

you did not answer me about that in the code you wrote was not 70 it was 73,,,,,,, and how this would have involved influence and the approximate solution of the integral oscillatory,,,,,,

I think until I see real contributions made by you, with your procedures and steps to reach the numerical approximation,,,,,,,,,,,,,,,,,, you do not require me to me that you could show the steps and procedures I do to solve it,,,,,, first you would have to solve all the steps and procedures to get to the approximation, and after that there if they could require me to show my steps, but before that no I see fair,,,,,

assured that I am very patient and calm, I can wait until they start to solve it,,,,att
jefferson alexander vitola(Bigsmile)

 

[jeffer vitola]

My method? Well, I went clackety-clack at my keyboard, et voila! (Poolparty)

jajajajajjaja the joke is good,,,,,, just when you want to try to solve it, I'll wait,,,,,,
My main purpose is to know you how you would address the problem and apply methods to solve oscillatory integral,,,,,,,
att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

...

I think until I see real contributions made by you, with your procedures and steps to reach the numerical approximation,,,,,,,,,,,,,,,,,, you do not require me to me that you could show the steps and procedures I do to solve it...

Let me get this straight...

You get to require that we show what we have done (which in my case is to use a computer), but you are above such requirements and get to choose if and when you are going to contribute anything? (Smirk)

I have developed an approximation scheme based on sinusoidal curves, but I think I shall wait until I see something from you before divulging anything. In the end though, my method still requires the brute force of a computer to make it practical.

 

[jeffer vitola]

Let me get this straight...

You get to require that we show what we have done (which in my case is to use a computer), but you are above such requirements and get to choose if and when you are going to contribute anything? (Smirk)

I have developed an approximation scheme based on sinusoidal curves, but I think I shall wait until I see something from you before divulging anything. In the end though, my method still requires the brute force of a computer to make it practical.

hello,,,,,,,
fantastic,,,,, what your method is interesting,,,,,,,,

my intention is not to force you or force you to show your procedures, you are all free not to do it,,,, what I have said is that I find it strange that you ask me first see my results and methods developed by me, and if after that you would bring your methods, and I've seen most of the forum and in the other issues resolved all questions without a problem and without inconvenience and anyone you nobody do not asks questions ,,,,, I see it's only me who ask you first show I've done and use procedures to solve the question that I have posed at the beginning of the forum,,, as I hope to show their progress and methods if you want, if you do not want there is problem nothing happens ,,, I'm used to no one solves the steps to get to the approximation,,,let me topic open who want or for users who wish to propose theorems or methods and procedures to achieve oscillatory approximate the integrals,,, I am open mind to you would use procedures and new ideas to address the issue,,,,att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

Take a look around the forums again, you will see that the vast majority of people who ask questions show what they have tried so far, and those that don't are encouraged to do so.

Your case is different though...you are (presumably) not asking for help with a homework problem, you are wanting to see how others would approach a problem in elementary numerical analysis.

I am assuming you are perhaps looking for ways to improve your own method. It would make more sense for you to demonstrate your approach so that others might offer you insights into how you might accomplish this.

 

[Jameson]

We require everyone to show some effort and you are the first person on this forum who claims to have a method for solving something but won't show anything. This game is getting tiring. Either show us what you've done or the thread will be closed. MarkFL and others have been extremely patient with you but this thread is way too long already. Please think carefully about your reply. I hope that something can be salvaged from this discussion.

 

[jeffer vitola]

Take a look around the forums again, you will see that the vast majority of people who ask questions show what they have tried so far, and those that don't are encouraged to do so.

Your case is different though...you are (presumably) not asking for help with a homework problem, you are wanting to see how others would approach a problem in elementary numerical analysis.

I am assuming you are perhaps looking for ways to improve your own method. It would make more sense for you to demonstrate your approach so that others might offer you insights into how you might accomplish this.

hello,,,,,,,,,,,,,,,,,

I want to compare my method and style,,,, if I Realize That the numerical method of you is better than mine, I simply do I give up and will not work more,,, so I wanted to know how you solved it, ,,, the question for me is not a homework task is not,,,,,

so far it Appears That the numerical method what i designed and created by me is the only and the best so far yet,,,,,,, since no one else has been Able to make any approximation methods using own or of other mathematicians,,,,,, i will wait to see That goes with This Proposed of exercise,,,,,,,,,,,,att
jefferson alexander vitola(Bigsmile)

 

[MarkFL]

We are getting nowhere fast here. I am closing this topic. If you wish to discuss this further, send your work via PM to one of the administrators or global moderators, and we will consider reopening the topic for further discussion.

edit: There is a substantial literature on numerical methods for oscillatory integrals, so you might want to try google.