Error propagation of a variable for an integral

[Arman777]

I have an integral that depends on two parameters $a\pm\delta a$ and $b\pm \delta b$. I am doing this integral numerically and no python function can calculate the integral with uncertainties.

So I have calculated the integral for each min, max values of a and b.
As a result I have obtained 4 values, such that;

$$(a + \delta a, b + \delta b) = 13827.450210 \pm 0.000015~~(1)$$
$$(a + \delta a, b - \delta b) = 13827.354688 \pm 0.000015~~(2)$$
$$(a - \delta a, b + \delta b) = 13912.521548 \pm 0.000010~~(3)$$
$$(a - \delta a, b - \delta b) = 13912.425467 \pm 0.000010~~(4)$$

So it is clear that $(2)$ gives the min and $(3)$ gives the max. Let us show the result of the integral as $c \pm \delta c$. So my problem is what is $c$ and $\delta c$ here?

The integral is something like this

$$I(a,b,x) =C\int_0^b \frac{dx}{\sqrt{a(1+x)^3 + \eta(1+x)^4 + (\gamma^2 - a - \eta)}}$$

where $\eta$ and $\gamma$ are constant.

Note: You guys can also generalize it by taking $\eta \pm \delta \eta$ but it is not necessary for now.

I have to take derivatives or integrals numerically. There's no known analytical solution for the integral.

$\eta = 4.177 \times 10^{-5}$, $a = 0.1430 \pm 0.0011$, $b = 1089.92 \pm 0.25$, $\gamma = 0.6736 \pm 0.0054$, $C = 2997.92458$

 

[pasmith]

Generally, if your intergal is $I(a,b) = \int_{x_0}^{x_1} F(a,b,x)\,dx$, then uncertainties in $a$ and $b$ would lead to $$I(a,b) \pm\frac{\partial I}{\partial a}\delta a \pm \frac{\partial I}{\partial b}\delta b$$ with the partial derivatives evaluated at $a$ and $b$. You can evaluate these derivatives by differentiating under the integral:
$$\frac{\partial I}{\partial a} = \int_{x_0}^{x_1} \frac{\partial F}{\partial a}(a, b, x)\,dx$$ etc.

 

[Arman777]

The integral is something like this

$$I(a,b,x) =\int_0^b \frac{dx}{\sqrt{a(1+x)^3 + \eta(1+x)^4 + (\gamma^2 - a - \eta)}}$$

where $\eta$ and $\gamma$ are constant.

 

[mfb]

Assuming a and b are uncorrelated: Calculate the integral at (a,b+delta_b), (a,b-delta_b) and analogous for delta_a. Then use the deviations as approximation for $\frac{\partial I}{\partial b}\delta b$ and add both uncertainties in quadrature.

 

[Arman777]

Assuming a and b are uncorrelated: Calculate the integral at (a,b+delta_b), (a,b-delta_b) and analogous for delta_a. Then use the deviations as approximation for $\frac{\partial I}{\partial b}\delta b$ and add both uncertainties in quadrature.

I did not quite understand it..Can you maybe put it in a more mathematical way

 

[Arman777]

  from numpy import sqrt
    from scipy import integrate
    import uncertainties as u
    from uncertainties.umath import *

    #Important Parameters
    C = 2997.92458  # speed of light in [km/s]
    eta = 4.177 * 10**(-5)
    a = u.ufloat(0.1430, 0.0011)
    b = u.ufloat(1089.92, 0.25)
    gama = u.ufloat(0.6736, 0.0054)

    @u.wrap
    def D_zrec_finder(gama, a, b):
        def D_zrec(z):
            return C / sqrt(a * (1+z)**3 + eta * (1+z)**4 + (gama**2 - a - eta))
        result, error = integrate.quad(D_zrec, 0, b)
        return result


    print((D_zrec_finder(gama, a, b)).n)
    print((D_zrec_finder(gama, a, b)).s)

This works

 

[mfb]

I did not quite understand it..Can you maybe put it in a more mathematical way

$$\frac{\partial I}{\partial b}\delta b \approx \frac 1 2 (I(a,b+\delta b)-I(a,b-\delta b))$$
$$\frac{\partial I}{\partial a}\delta a \approx \frac 1 2 (I(a+\delta a,b)-I(a-\delta a,b))$$

As b is your integration border you can simplify this one: $\frac{\partial I}{\partial b}$ is simply the function value at x=b.