Added Gauss-Turán quadrature implementation

ext/IntegralsGaussTuran.jl

@@ -0,0 +1,61 @@
+module IntegralsGaussTuran
+
+using ForwardDiff
+using Integrals
+# Defining the GaussTuran struct
+struct GaussTuran{B} <: SciMLBase.AbstractIntegralAlgorithm
+    n::Int # number of points
+    s::Int # order of derivative
+    ad_backend::B # for now ForwardDiff
+end
+
+const xgt51 = [0.171573532582957e-02,
+               0.516674690067835e-01,
+               0.256596242621252e+00,
+               0.614259881077552e+00,
+               0.929575800557533e+00]
+
+const agt51 = [(0.121637123277610E-01, 0.283102654629310E-04, 0.214239866660517E-07),
+               (0.114788544658756E+00, 0.141096832290629E-02, 0.357587075122775E-04),
+               (0.296358604286158E+00, 0.391442503354071E-02, 0.677935112926019E-03),
+               (0.373459975331693E+00, -0.111299945126195E-02, 0.139576858045244E-02),
+               (0.203229163395632E+00, -0.455530407798230E-02, 0.226019273068948E-03)]
+
+# Gauss-Turán quadrature (gt51) function
+function gt51(f, a, b)
+    res = zero(eltype(f(a)))  # Initializing result
+
+    # Transformation factor
+    factor = b - a
+
+    for i in 1:5
+        # Map the nodes to the interval [a, b]
+        xi = xgt51[i] * factor + a
+
+        # Compute function value and derivatives at xi using ForwardDiff
+        fi = f(xi)
+        dfi = ForwardDiff.derivative(f, xi) * factor
+        d2fi = ForwardDiff.derivative(x -> ForwardDiff.derivative(f, x), xi) * factor^2
+
+        # Get the weights
+        Ai1, Ai2, Ai3 = agt51[i]
+
+        # Accumulate the result
+        res += Ai1 * fi + Ai2 * dfi + Ai3 * d2fi
+    end
+
+    return res * factor
+end
+
+# Integrals.__solvebp_call for GaussTuran
+function Integrals.__solvebp_call(prob::IntegralProblem, 
+    alg::GaussTuran, 
+    sensealg, domain, p; 
+    reltol=nothing, abstol=nothing, 
+    maxiters=nothing)
+    integrand = prob.f
+    a, b = domain
+    return gt51((x) -> integrand(x, p), a, b)
+end
+
+end # module

src/algorithms.jl

@@ -162,3 +162,13 @@ struct QuadratureRule{Q} <: SciMLBase.AbstractIntegralAlgorithm
     end
 end
 QuadratureRule(q; n = 250) = QuadratureRule(q, n)
+
+
+"""
+Gauss Turan Quadrature using xgt51 and agt51
+"""
+struct GaussTuran{B} <: SciMLBase.AbstractIntegralAlgorithm
+    n::Int # number of points
+    s::Int # order of derivative
+    ad_backend::B # for now ForwardDiff
+end

test/gaussturan_tests.jl

@@ -0,0 +1,13 @@
+using Test
+using ForwardDiff
+using Integrals
+
+a, b = 0.0, π
+
+result = solve(IntegralProblem((x, p) -> sin(x), (0, π)), GaussTuran(5, 2, ForwardDiff))
+
+expected_result = 2.0
+
+@test isapprox(result, expected_result, atol=1e-6)
+
+println("Test passed, result of integration: ", result)