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# Resources

# Integration over user defined function in GAUSS

Hi,

I wanted to know whether there is a way to calculate one dimensional integral over an user defined function in GAUSS. I see than intquad2 and intquad3 can respectively be used for two and three dimensional integral, is there an option for one dimensional integration (not over a rectangular distribution, but over an user defined distribution).

I would appreciate your response in this regard.

Thanks and Regards

Annesha

## 5 Answers

The simplest method would be to use function `intquad1` to calculate a one dimensional integral for example:

range_end = 1; range_start = 0; area = intquad1(&myProc, range_end|range_start); proc (1) = myProc(x); retp(x .* sin(x)); endp;

The other option is to use `inthp4`. Here is an example. You can keep the wrapper function `simpleInthp41d` if you want and just call that function to make it simpler to use.

area = simpleInthp1D(&myProc, 0, 1); proc (1) = myProc(struct DS *data, x); //If your function needs data other than from 'x' //you can pull it from the 'data' struct like this //new_var = data->dataMatrix; retp(x .* sin(x)); endp; proc (1) = simpleInthp1D(fct, range_start, range_end); local fct:proc, area_; //Declare control structure and fill //in defaults struct inthpControl ctl; ctl = inthpControlCreate(); //Make any control structure changes here //For example: //ctl.eps = 1e-5; //Declare and DS structure for any extra data struct DS d0; d0 = dsCreate(); lim = range_end|range_start; //Call integration routine area_ = inthp4(&fct, &d0, ctl, lim); retp(area_); endp;

Hello Please correct me if I am wrong, but in your examples isn't the integration is being done over a rectangular region. For example in your first example isn't it the assumption that x is uniformly distributed between 0 and 1?

What I want to do is I want to define a distribution for x other than rectangular, there are are options to do that for two/three dimensional integrals using intquad2 and intquad3, but I do not know how to use that in one dimenssinal case.

Look forward to your help.

Regards

Annesha

I am confused. When I first read your question, I was assuming that you simply wanted to calculate the area under the curve of a user defined function. For example, let's say the area under the curve of the sin function between 0 and 1.

Are you wanting to perform a double integral over a non-rectangular region as illustrated in slide number 2 at this link? http://www.math.ust.hk/~malwu/math100/lecture15.pdf

If not, could you point me to something that explains what you are trying to do?

Hello,

I do not want to do a double integral.

Let me try to explain what I want to do:

If g(.) is a function of x and f(.) is the pdf (probability density function of x), then all I want to do is to calculate the expected value of the function g(.) with respect to x, and the formula for doing this is

And also as I mentioned previously, I want to define both g(x) and the pdf, f(x). Hope its a little less ambiguous now.

I would appreciate your help in this regard.

Regards

Annesha

**Finding expected value of a continuous function of a random variable**

ExpectedValueRVFunc that works through an example analytically using the a uniform distribution function explicitly. Below is the code that corresponds to the analytical solution from the attachment.

//Main code area = inthp4Simple(&myProc, 0, 10); print area; //User define procedures proc (1) = pdfu(x, range_start, range_end); retp(1./(range_end - range_start)); endp; proc (1) = myProc(struct DS *data, x); local range_end, range_start; //If your function needs data other than from 'x' //you can pull it from the 'data' struct like this range_start = data->dataMatrix[1]; range_end = data->dataMatrix[2]; retp(x .* x .* pdfu(x, range_start, range_end)); endp; //Main helper procedure for reuse proc (1) = inthp4Simple(fct, range_start, range_end); local fct:proc, area_, lim; //Declare control structure and fill //in defaults struct inthpControl ctl; ctl = inthpControlCreate(); //Make any control structure changes here //For example: //ctl.eps = 1e-5; //Declare and DS structure which will //pass extra data to 'pdfu' function struct DS d0; d0 = dsCreate(); //Passing range so it can be sent to 'pdfu' d0.dataMatrix = range_start|range_end; lim = range_end|range_start; //Call integration routine area_ = inthp4(&fct, &d0, ctl, lim); retp(area_); endp;

If we make a couple of changes to the code above, we can make it work for an infinite integral and a non-uniform distribution. Here is an example using the normal distribution. You would replace `pdfn` with your probability density function:

//Main code area = inthp4Simple(&myProc); print area; //User define procedure proc (1) = myProc(struct DS *data, x); retp(.*pdfn(x)); endp; //Main helper procedure for reuse proc (1) = inthp4Simple(fct); local fct:proc, area_, lim; //Declare control structure and fill //in defaults struct inthpControl ctl; ctl = inthpControlCreate(); //Declare and DS structure which will //pass extra data to 'pdfu' function struct DS d0; d0 = dsCreate(); //Call integration routine //'inthp1' calculates an integral over //an infinite interval area_ = inthp1(&fct, &d0, ctl); retp(area_); endp;

## Your Answer

## 5 Answers

The simplest method would be to use function `intquad1` to calculate a one dimensional integral for example:

range_end = 1; range_start = 0; area = intquad1(&myProc, range_end|range_start); proc (1) = myProc(x); retp(x .* sin(x)); endp;

The other option is to use `inthp4`. Here is an example. You can keep the wrapper function `simpleInthp41d` if you want and just call that function to make it simpler to use.

area = simpleInthp1D(&myProc, 0, 1); proc (1) = myProc(struct DS *data, x); //If your function needs data other than from 'x' //you can pull it from the 'data' struct like this //new_var = data->dataMatrix; retp(x .* sin(x)); endp; proc (1) = simpleInthp1D(fct, range_start, range_end); local fct:proc, area_; //Declare control structure and fill //in defaults struct inthpControl ctl; ctl = inthpControlCreate(); //Make any control structure changes here //For example: //ctl.eps = 1e-5; //Declare and DS structure for any extra data struct DS d0; d0 = dsCreate(); lim = range_end|range_start; //Call integration routine area_ = inthp4(&fct, &d0, ctl, lim); retp(area_); endp;

Hello Please correct me if I am wrong, but in your examples isn't the integration is being done over a rectangular region. For example in your first example isn't it the assumption that x is uniformly distributed between 0 and 1?

What I want to do is I want to define a distribution for x other than rectangular, there are are options to do that for two/three dimensional integrals using intquad2 and intquad3, but I do not know how to use that in one dimenssinal case.

Look forward to your help.

Regards

Annesha

I am confused. When I first read your question, I was assuming that you simply wanted to calculate the area under the curve of a user defined function. For example, let's say the area under the curve of the sin function between 0 and 1.

Are you wanting to perform a double integral over a non-rectangular region as illustrated in slide number 2 at this link? http://www.math.ust.hk/~malwu/math100/lecture15.pdf

If not, could you point me to something that explains what you are trying to do?

Hello,

I do not want to do a double integral.

Let me try to explain what I want to do:

If g(.) is a function of x and f(.) is the pdf (probability density function of x), then all I want to do is to calculate the expected value of the function g(.) with respect to x, and the formula for doing this is

And also as I mentioned previously, I want to define both g(x) and the pdf, f(x). Hope its a little less ambiguous now.

I would appreciate your help in this regard.

Regards

Annesha

**Finding expected value of a continuous function of a random variable**

ExpectedValueRVFunc that works through an example analytically using the a uniform distribution function explicitly. Below is the code that corresponds to the analytical solution from the attachment.

//Main code area = inthp4Simple(&myProc, 0, 10); print area; //User define procedures proc (1) = pdfu(x, range_start, range_end); retp(1./(range_end - range_start)); endp; proc (1) = myProc(struct DS *data, x); local range_end, range_start; //If your function needs data other than from 'x' //you can pull it from the 'data' struct like this range_start = data->dataMatrix[1]; range_end = data->dataMatrix[2]; retp(x .* x .* pdfu(x, range_start, range_end)); endp; //Main helper procedure for reuse proc (1) = inthp4Simple(fct, range_start, range_end); local fct:proc, area_, lim; //Declare control structure and fill //in defaults struct inthpControl ctl; ctl = inthpControlCreate(); //Make any control structure changes here //For example: //ctl.eps = 1e-5; //Declare and DS structure which will //pass extra data to 'pdfu' function struct DS d0; d0 = dsCreate(); //Passing range so it can be sent to 'pdfu' d0.dataMatrix = range_start|range_end; lim = range_end|range_start; //Call integration routine area_ = inthp4(&fct, &d0, ctl, lim); retp(area_); endp;

If we make a couple of changes to the code above, we can make it work for an infinite integral and a non-uniform distribution. Here is an example using the normal distribution. You would replace `pdfn` with your probability density function:

//Main code area = inthp4Simple(&myProc); print area; //User define procedure proc (1) = myProc(struct DS *data, x); retp(.*pdfn(x)); endp; //Main helper procedure for reuse proc (1) = inthp4Simple(fct); local fct:proc, area_, lim; //Declare control structure and fill //in defaults struct inthpControl ctl; ctl = inthpControlCreate(); //Declare and DS structure which will //pass extra data to 'pdfu' function struct DS d0; d0 = dsCreate(); //Call integration routine //'inthp1' calculates an integral over //an infinite interval area_ = inthp1(&fct, &d0, ctl); retp(area_); endp;