7.2 Julia

1. see https://symbolicnumericintegration.sciml.ai/dev/

   It says:

   "Function integrate returns the integral of a univariate expression with constant real or complex coefficients. integrate returns a tuple with three values. The first one is the solved integral, the second one is the sum of the unsolved terms, and the third value is the residual error. If integrate is successful, the unsolved portion is reported as 0."

   see also the paper

   https://arxiv.org/abs/2201.12468

   import Pkg; Pkg.add("Symbolics")` to install the Symbolics package. 
    
   julia> using SymbolicNumericIntegration 
     Package SymbolicNumericIntegration not found, but a package named SymbolicNumericIntegration is available from a registry. 
     Install package? 
       (@v1.7) pkg> add SymbolicNumericIntegration 
     (y/n) [y]: 
    
   Precompiling project... 
     64 dependencies successfully precompiled in 539 seconds (110 already precompiled)
    
   

   To do integration, the commands are (where it is assumed the integrand say 3*x is in file called input.txt in same folder

   using Symbolics 
   using SymbolicNumericIntegration 
   using Dates 
   using CSV 
   using DataFrames 
   using Elliptic 
   using Elliptic.Jacobi 
   using SpecialFunctions 
    
   @syms x z y t a 
    
   integrand = include("input.txt") 
   integrate(integrand,x) 
    
   julia> @time integrate(3*sin(x)+x*cos(x),x) 
        0.138087 seconds (19.18 k allocations: 9.491 MiB) 
       (x*sin(x) - 2cos(x), 0, 0)
    
   

   To measure time used

    
   julia> time1 = now(); 
    
   julia> anti,n_unsolved,residual_error = integrate(3*sin(x)+x*cos(x),x) 
   (x*sin(x) - 2cos(x), 0, 0) 
    
   julia> time2 = now(); 
    
   julia> (Dates.DateTime(time2)-Dates.DateTime(time1))/ Millisecond(1) * (1 / 1000) 
   0.197 
    
   julia> anti 
   x*sin(x) - 2cos(x)
    
   

   To read the integrals, create text file like this (say input.txt)

   [[3*x,x,0,3/2*x^2], 
   [4*x,x,0,2*x^2]]
    
   

   Then do

   data=include("input.txt") 
    
   julia> length(data) 
   2 
    
   julia> data[1][1] 
   3x 
    
   julia> data[2][1] 
   4x
    
   

   Hence to loop over all integrands in file, do

   data=include("input.txt") 
    
   julia> for n in 1:length(data) 
          println(integrate(data[n][1],data[n][2])) 
          end 
   ((3//2)*(x^2), 0, 0) 
   (2(x^2), 0, 0)
    
   

   To read all lines as strings do

   data=readlines("input.txt") 
    
   julia> data[2] 
   "[x*sqrt(1 + 3*x), x, 2]," 
    
   julia> data[3] 
   "[x^2*sqrt(1 + x), x, 2],"
    
   

   This gives error

   expr=1/((1 + x)^2*sqrt(2)*sqrt(-im + x^2)) + 1/((1 + x)^2*sqrt(2)*sqrt(im + x^2)) 
   ERROR: TypeError: non-boolean (Num) used in boolean context 
   Stacktrace: 
    [1] sqrt(z::Complex{Num}) 
      @ Base ./complex.jl:501 
    [2] top-level scope 
      @ REPL[115]:1
    
   

   A better way to read the integrals from file is this. The file has to be in this format

   [x^2,x] 
   [x^3,x] 
   . 
   . 
   . 
   [sin(x),x]
    
   

   Then do

   using Symbolics 
   using SymbolicNumericIntegration 
   using Dates 
   using SpecialFunctions 
   using SymbolicUtils 
    
   @syms x z y t a 
   @syms _C0 _C1 _C2 _C3 _C4 _C5 _C6 _C7 _C8 _C9 
    
   #import Base: exp 
    
   #exp(x::Real) = Symbolics.Term(exp,[x]) 
   #exp(x::Int64) = Symbolics.Term(exp,[x]) 
    
    
   data=readlines("julia_input.txt"); 
    
   number_failed = 0 
   for counter in 1:10 #length(data) 
      #println("processing line ",counter) 
      try 
         item = eval(Meta.parse(data[counter])) 
         print("\n\n****calling integrate(",item[2],",",item[3],")") 
         anti,n_unsolved,residual_error = integrate(item[2],item[3]) 
         println("anti=",anti) 
         println("n_unsolved=",n_unsolved) 
         println("residual_error=",residual_error) 
    
      catch 
         number_failed = number_failed + 1 
         #println(">>>>>>>>>error on integral ",counter) 
         println(">>>>>>>>>error on integral ",Meta.parse(data[counter])) 
      end 
   end 
    
   println("Failed to read ", number_failed, " integrals")
    
   

   This way I know which line has an error when reading the interals.

   Find why this error

   julia> (-x)^(1//2) 
   ERROR: DomainError with -1.0: 
   Exponentiation yielding a complex result requires a complex argument. 
   Replace x^y with (x+0im)^y, Complex(x)^y, or similar. 
   Stacktrace: 
    [1] throw_exp_domainerror(x::Float64) 
      @ Base.Math ./math.jl:37 
    [2] ^(x::Float64, y::Float64) 
      @ Base.Math ./math.jl:1003 
    [3] ^ 
      @ ./promotion.jl:422 [inlined] 
    [4] ^ 
      @ ./rational.jl:481 [inlined] 
    [5] unstable_pow 
      @ ~/.julia/packages/SymbolicUtils/qulQp/src/types.jl:801 [inlined] 
    [6] ^(a::SymbolicUtils.Mul{Number, Int64, Dict{Any, Number}, Nothing}, b::Rational{Int64}) 
      @ SymbolicUtils ~/.julia/packages/SymbolicUtils/qulQp/src/types.jl:1047 
    [7] top-level scope 
      @ REPL[32]:1
    
   

   But the integrate command currently only supports univariate expression with constant real or complex coefficients. So it can integrate \(2*x\) but can not integrate \(a*x\) so not possible to use it in CAS integration tests.

2. to read symbolic expression from file see this post https://stackoverflow.com/questions/59543961/how-to-read-a-polynomial-from-a-text-file-in-julia