Re: [R] efficiently finding the integrals of a sequence of functions
Thanks,Hans!
I agree that this is a good way of solving this problem.
Here is another way. Instead of defining a vector of uni-dimensional
functions and trying to integrating
each component (a uni-dimensional function), we can do something below
my.integrand-function(x,k)
{
return(f[x,k]) ## use matrix to represent the sequence of uni-dimensional
functions
}
my.integral-function(k) ## k=1,...,5000 denotes the function labels
{
return(integrate(my.integrand,lower=...,upper=...,k)$value)
}
When calculating the integrals, just perform
sapply(1:5000, my.integral)
This is a way of avoiding loops but the computing time needs to be carefully
examined.
Jeff
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[R] efficiently finding the integrals of a sequence of functions
Hi folks, I am having a question about efficiently finding the integrals of a list of functions. To be specific, here is a simple example showing my question. Suppose we have a function f defined by f-function(x,y,z) c(x,y^2,z^3) Thus, f is actually corresponding to three uni-dimensional functions f_1(x)=x, f_2(y)=y^2 and f_3(z)=z^3. What I am looking for are the integrals of these three functions f_1,f_2,f_3 over some interval, say, (0,1). More specifically, the integrals \int_0^1 f_1(x) dx, \int_0^1 f_2(y) dy and \int_0^1 f_3(z) dz. For this simple example, of course we can do these three integrals one by one using integrate (f_1, lower=0, upper=1) However, in practice, I have a sequence of 5000 uni-dimensional functions and hope to find all of their integrals (over some regions), so using loops to do this one by one is not efficient. A possible idea is to convert the sequence of functions to a list of objects, and use sapply() which allow us to find the integrals in a fashion of vectorizing. But I don't know how to convert the above example function f to a list. In my research problem, I can only define the 5000 functions in a vectorizing way like f-function(x1,x2,...,x5000) c(f1(x1),f2(x2),...,f5000(x5000)) So how to convert it to a list will be a crucial step to efficiently get the integrals. Thanks for all the suggestions! Jeff -- View this message in context: http://r.789695.n4.nabble.com/efficiently-finding-the-integrals-of-a-sequence-of-functions-tp4179452p4179452.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] efficiently finding the integrals of a sequence of functions
JeffND Zuofeng.Shang.5 at nd.edu writes: Hi folks, I am having a question about efficiently finding the integrals of a list of functions. We had the same discussion last month under the heading performance of adaptIntegrate vs. integrate, see https://stat.ethz.ch/pipermail/r-help/2011-November/295260.html It also depends on the accuracy you want to achieve. Adaptive methods will be slower as the adaptation will be different in each single step for each function, i.e. no vectorization here. Non-adaptive Gaussian quadrature appears to be a good candidate. Assume you have found grid points x_i and weights w_i for your interval [a, b], then if F is the matrix with F_ij = f_j(x_i) amd the integrals will be computed all at once with w %*% F . Example: A function that returns x, x^2, ..., x^5 in columns f - function(x) cbind(x, x^2, x^3, x^4, x^5) The grid points and weights for the interval [0, 1] are: x - c(0.025446, 0.129234, 0.297077, 0.50, 0.702923, 0.870766, 0.974554) w - c(0.064742, 0.139853, 0.190915, 0.208980, 0.190915, 0.139853, 0.064742) and the integrals for these five functions are w %*% f(x) # 0.5 0.334 0.25 0.2 0.167 Functions to calculate Gaussian points and weights are mentioned in the thread above. Hans Werner To be specific, here is a simple example showing my question. Suppose we have a function f defined by f-function(x,y,z) c(x,y^2,z^3) Thus, f is actually corresponding to three uni-dimensional functions f_1(x)=x, f_2(y)=y^2 and f_3(z)=z^3. What I am looking for are the integrals of these three functions f_1,f_2,f_3 over some interval, say, (0,1). More specifically, the integrals \int_0^1 f_1(x) dx, \int_0^1 f_2(y) dy and \int_0^1 f_3(z) dz. For this simple example, of course we can do these three integrals one by one using integrate (f_1, lower=0, upper=1) However, in practice, I have a sequence of 5000 uni-dimensional functions and hope to find all of their integrals (over some regions), so using loops to do this one by one is not efficient. A possible idea is to convert the sequence of functions to a list of objects, and use sapply() which allow us to find the integrals in a fashion of vectorizing. But I don't know how to convert the above example function f to a list. In my research problem, I can only define the 5000 functions in a vectorizing way like f-function(x1,x2,...,x5000) c(f1(x1),f2(x2),...,f5000(x5000)) So how to convert it to a list will be a crucial step to efficiently get the integrals. Thanks for all the suggestions! Jeff -- View this message in context: http://r.789695.n4.nabble.com/efficiently-finding-the-integrals-of-a- sequence-of-functions-tp4179452p4179452.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
