How to Symbolic and Numerical Integration

New to Julia
1

I have a problem in using SymPy and QuadGK packages as follows:
In a double integral, the inner integration has lower limit is x where x is the variable of the outer integral. Here is my code

using SymPy 
@syms y
f(z) = integrate(2*z*cos(y+z)/(1+y^2),(y,z,10))^2
t1 = integrate(f,0,1)
t1.evalf() # failed
# I also tried with QuadGK
using QuadGK
t2 = quadgk(f,0,1) # also failed

How could I obtain numerical value of this double integral.

Thank you.

2

Is this function really the one you want ? because i feel like you may be able to obtain the value analytically.

3

Do you want the answer symbolically or numerically?

4

Sorry for my late reply. The function here is just an example. In my own problem, the function f(z) is more complicated that I am not sure to have analytical form of t1. Anyway, I would like to have both analytical and numerical value of this example problem. Thank you @Irnv

5

Sorry for my late reply. I would like to have the answer both symbolically and numerically. In case there is no symbolic solution, I would like to have it numerically. Thank you @stevengj

6

In this case (likely for more complicated integrands), you shouldn’t use Symbolics at all. Either use nested calls to QuadGK or use a multi-dimensional integration package like HCubature.

4 Likes
7

Could you describe your problem, paste some tracelog? What does it mean failed?
I run your code into my Julia and works…

julia> @syms y
(y,)

julia> f(z) = integrate(2*z*cos(y+z)/(1+y^2),(y,z,10))^2
f (generic function with 1 method)

julia> t1 = integrate(f,0,1)

  1
  ⌠
  ⎮                      2
  ⎮    ⎛10              ⎞
  ⎮    ⎜⌠               ⎟
  ⎮  2 ⎜⎮  cos(x + y)   ⎟
4⋅⎮ x ⋅⎜⎮  ────────── dy⎟  dx
  ⎮    ⎜⎮     2         ⎟
  ⎮    ⎜⎮    y  + 1     ⎟
  ⎮    ⎜⌡               ⎟
  ⎮    ⎝x               ⎠
  ⌡
  0

julia>

julia> t1.evalf()
    1
    ⌠
    ⎮                      2
    ⎮    ⎛10              ⎞
    ⎮    ⎜⌠               ⎟
    ⎮  2 ⎜⎮  cos(x + y)   ⎟
4.0⋅⎮ x ⋅⎜⎮  ────────── dy⎟  dx
    ⎮    ⎜⎮     2         ⎟
    ⎮    ⎜⎮    y  + 1     ⎟
    ⎮    ⎜⌡               ⎟
    ⎮    ⎝x               ⎠
    ⌡
    0
8

From your suggestion, I came to the final workable solution below

using QuadGK
f(z) = quadgk(y->2*z*cos(y+z)/(1+y^2),z,10)[1]^2
t = quadgk(f,0,1)

Thank you @stevengj

9

By “failed”, I meant that no analytical nor numerical solution is found. But from suggestion of @stevengj, I could have numerical value of this double integral by using nested quadgk.