5 Improved formatting edited Jun 13 '14 at 18:27 Michael E2 159k1313 gold badges217217 silver badges516516 bronze badges What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrateNIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  Edit 4: Since I was apparently confusing some people, here's roughly what I want to compute: NIntegrate[f[r1, r2, r3], {r1, 0, 1}, {r2, 0, 1 - r1}, {r3, 0, 1 - r1 - r2}]  This gives a result (2.8227403075197916*^-78)(2.8227403075197916*^-78) but also states: NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small  NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small My question is: What is the fastest and most robust NIntegrate Integration Strategy when I wand to do this with arbirtary $$1 and $$\left\{c_i\right\}$$? What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  Edit 4: Since I was apparently confusing some people, here's roughly what I want to compute: NIntegrate[f[r1, r2, r3], {r1, 0, 1}, {r2, 0, 1 - r1}, {r3, 0, 1 - r1 - r2}]  This gives a result (2.8227403075197916*^-78) but also states: NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small My question is: What is the fastest and most robust NIntegrate Integration Strategy when I wand to do this with arbirtary $$1 and $$\left\{c_i\right\}$$? What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  Edit 4: Since I was apparently confusing some people, here's roughly what I want to compute: NIntegrate[f[r1, r2, r3], {r1, 0, 1}, {r2, 0, 1 - r1}, {r3, 0, 1 - r1 - r2}]  This gives a result (2.8227403075197916*^-78) but also states:  NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small My question is: What is the fastest and most robust NIntegrate Integration Strategy when I wand to do this with arbirtary $$1 and $$\left\{c_i\right\}$$? 4 added 714 characters in body edited Mar 27 '13 at 16:49 user9886 11955 bronze badges What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  Edit 4: Since I was apparently confusing some people, here's roughly what I want to compute: NIntegrate[f[r1, r2, r3], {r1, 0, 1}, {r2, 0, 1 - r1}, {r3, 0, 1 - r1 - r2}]  This gives a result (2.8227403075197916*^-78) but also states: NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small My question is: What is the fastest and most robust NIntegrate Integration Strategy when I wand to do this with arbirtary $$1 and $$\left\{c_i\right\}$$? What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  Edit 4: Since I was apparently confusing some people, here's roughly what I want to compute: NIntegrate[f[r1, r2, r3], {r1, 0, 1}, {r2, 0, 1 - r1}, {r3, 0, 1 - r1 - r2}]  This gives a result (2.8227403075197916*^-78) but also states: NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small My question is: What is the fastest and most robust NIntegrate Integration Strategy when I wand to do this with arbirtary $$1 and $$\left\{c_i\right\}$$? 3 added 140 characters in body edited Mar 27 '13 at 16:07 user9886 11955 bronze badges What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. What would be the ideal integration strategy for a function like this: $$f(x_1,x_2,\dots,x_N)=\prod_{i=1}^Nx_i^{c_i}\Theta(x_i-p_i)$$ where $$x_i,p_i\in \mathbb{R}$$, $$\Theta$$ is the Heaviside function and $$c_i\in \mathbb{N}^+$$ when trying to integrate something like $$\int_0^1\mathrm{d}x_1\int_0^{1-x_1}\mathrm{d}x_2\dots\int_0^{1-x_1-x_2\dots-x_{N-1}}\mathrm{d}x_Nf(x_1,x_2,\dots,x_N)?$$ Edit: $$N$$ is usually rather small, namely $$N<10$$; the $$c_i$$, however, can be of the order of $$10^2$$. Edit 2: I am aware that this question is quite general, but maybe there are general rules how to NIntegrate monotonous functions like this. Edit 3: An example for $$f$$ would be f[x1_,x2_,x3_] = x1^23 x2^45 x3^123 HeavisideTheta[ x1-1/20 ] HeavisideTheta[ x2-1/20 ]  2 Added additional information edited Mar 27 '13 at 15:50 user9886 11955 bronze badges 1 asked Mar 27 '13 at 15:40 user9886 11955 bronze badges