RcppNumerical: Rcpp Integration for Numerical Computing Libraries
Author: Yixuan Qiu
Date: 2026-02-28

## Introduction

Rcpp is a powerful tool to write fast C++ code to speed up R programs. However, it is not easy, or at least not straightforward, to compute numerical integration or do optimization using pure C++ code inside Rcpp.

RcppNumerical integrates a number of open source numerical computing libraries into Rcpp, so that users can call convenient functions to accomplish such tasks.

- To use RcppNumerical with `Rcpp::sourceCpp()`, add

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]

in the C++ source file. - To use RcppNumerical in your package, add `Imports: RcppNumerical` and `LinkingTo: Rcpp, RcppEigen, RcppNumerical` to the `DESCRIPTION` file, and `import(RcppNumerical)` to the `NAMESPACE` file.

## Numerical Integration

### One-dimensional

The one-dimensional numerical integration code contained in RcppNumerical is based on the NumericalIntegration library developed by Sreekumar Thaithara Balan, Mark Sauder, and Matt Beall.

To compute integration of a function, first define a functor derived from the `Func` class (under the namespace `Numer`):

    class Func
    {
    public:
        virtual double operator()(const double& x) const = 0;
        virtual void eval(double* x, const int n) const
        {
            for(int i = 0; i < n; i++)
                x[i] = this->operator()(x[i]);
        }

        virtual ~Func() {}
    };

The first function evaluates one point at a time, and the second version overwrites each point in the array by the corresponding function values. Only the second function will be used by the integration code, but usually it is easier to implement the first one.

RcppNumerical provides a wrapper function for the NumericalIntegration library with the following interface:

    inline double integrate(
        const Func& f, const double& lower, const double& upper,
        double& err_est, int& err_code,
        const int subdiv = 100, const double& eps_abs = 1e-8, const double& eps_rel = 1e-6,
        const Integrator<double>::QuadratureRule rule = Integrator<double>::GaussKronrod41
    )

- `f`: The functor of integrand.
- `lower`, `upper`: Limits of integral.
- `err_est`: Estimate of the error (output).
- `err_code`: Error code (output). See `inst/include/integration/Integrator.h` Line 676-704.
- `subdiv`: Maximum number of subintervals.
- `eps_abs`, `eps_rel`: Absolute and relative tolerance.
- `rule`: Integration rule. Possible values are `GaussKronrod{15, 21, 31, 41, 51, 61, 71, 81, 91, 101, 121, 201}`. Rules with larger values have better accuracy, but may involve more function calls.
- Return value: The final estimate of the integral.

See a full example below, which can be compiled using the `Rcpp::sourceCpp` function in Rcpp.

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]
    #include <RcppNumerical.h>
    using namespace Numer;

    // P(0.3 < X < 0.8), X ~ Beta(a, b)
    class BetaPDF: public Func
    {
    private:
        double a;
        double b;
    public:
        BetaPDF(double a_, double b_) : a(a_), b(b_) {}

        double operator()(const double& x) const
        {
            return R::dbeta(x, a, b, 0);
        }
    };

    // [[Rcpp::export]]
    Rcpp::List integrate_1d_test()
    {
        const double a = 3, b = 10;
        const double lower = 0.3, upper = 0.8;
        const double true_val = R::pbeta(upper, a, b, 1, 0) -
                                R::pbeta(lower, a, b, 1, 0);

        BetaPDF f(a, b);
        double err_est;
        int err_code;
        const double res = integrate(f, lower, upper, err_est, err_code);
        return Rcpp::List::create(
            Rcpp::Named("true") = true_val,
            Rcpp::Named("approximate") = res,
            Rcpp::Named("error_estimate") = err_est,
            Rcpp::Named("error_code") = err_code
        );
    }

Runing the `integrate_1d_test()` function in R gives

    integrate_1d_test()
    # $true
    # [1] 0.2528108
    #
    # $approximate
    # [1] 0.2528108
    #
    # $error_estimate
    # [1] 2.806764e-15
    #
    # $error_code
    # [1] 0

### Multi-dimensional

Multi-dimensional integration in RcppNumerical is done by the Cuba library developed by Thomas Hahn.

To calculate the integration of a multivariate function, one needs to define a functor that inherits from the `MFunc` class:

    class MFunc
    {
    public:
        virtual double operator()(Constvec& x) = 0;

        virtual ~MFunc() {}
    };

Here `Constvec` represents a read-only vector with the definition

    // Constant reference to a vector
    using Constvec = const Eigen::Ref<const Eigen::VectorXd>;

(Basically you can treat `Constvec` as a `const Eigen::VectorXd`. Using `Eigen::Ref` is mainly to avoid memory copy. See the explanation here.)

The function provided by RcppNumerical for multi-dimensional integration is

    inline double integrate(
        MFunc& f, Constvec& lower, Constvec& upper,
        double& err_est, int& err_code,
        const int maxeval = 1000,
        const double& eps_abs = 1e-6, const double& eps_rel = 1e-6
    )

- `f`: The functor of integrand.
- `lower`, `upper`: Limits of integral. Both are vectors of the same dimension of `f`.
- `err_est`: Estimate of the error (output).
- `err_code`: Error code (output). Non-zero values indicate failure of convergence.
- `maxeval`: Maximum number of function evaluations.
- `eps_abs`, `eps_rel`: Absolute and relative tolerance.
- Return value: The final estimate of the integral.

See the example below:

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]
    #include <RcppNumerical.h>
    using namespace Numer;

    // P(a1 < X1 < b1, a2 < X2 < b2), (X1, X2) ~ N([0], [1   rho])
    //                                            ([0], [rho   1])
    class BiNormal: public MFunc
    {
    private:
        const double rho;
        double const1;  // 2 * (1 - rho^2)
        double const2;  // 1 / (2 * PI) / sqrt(1 - rho^2)
    public:
        BiNormal(const double& rho_) : rho(rho_)
        {
            const1 = 2.0 * (1.0 - rho * rho);
            const2 = 1.0 / (2 * M_PI) / std::sqrt(1.0 - rho * rho);
        }

        // PDF of bivariate normal
        double operator()(Constvec& x)
        {
            double z = x[0] * x[0] - 2 * rho * x[0] * x[1] + x[1] * x[1];
            return const2 * std::exp(-z / const1);
        }
    };

    // [[Rcpp::export]]
    Rcpp::List integrate_md_test()
    {
        BiNormal f(0.5);  // rho = 0.5
        Eigen::VectorXd lower(2);
        lower << -1, -1;
        Eigen::VectorXd upper(2);
        upper << 1, 1;
        double err_est;
        int err_code;
        const double res = integrate(f, lower, upper, err_est, err_code);
        return Rcpp::List::create(
            Rcpp::Named("approximate") = res,
            Rcpp::Named("error_estimate") = err_est,
            Rcpp::Named("error_code") = err_code
        );
    }

We can test the result in R:

    library(mvtnorm)
    trueval = pmvnorm(c(-1, -1), c(1, 1), sigma = matrix(c(1, 0.5, 0.5, 1), 2))
    integrate_md_test()
    # $approximate
    # [1] 0.4979718
    #
    # $error_estimate
    # [1] 4.612333e-09
    #
    # $error_code
    # [1] 0
    as.numeric(trueval) - integrate_md_test()$approximate
    # [1] 2.893336e-11

### Handling Infinite Limits

Infinite intagral limits are also supported. In the case of one-dimensional integration:

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]
    #include <RcppNumerical.h>
    using namespace Numer;

    class TestInf: public Func
    {
    public:
        double operator()(const double& x) const
        {
            return x * x * R::dnorm(x, 0.0, 1.0, 0);
        }
    };

    // [[Rcpp::export]]
    Rcpp::List integrate_1d_inf_test(const double& lower, const double& upper)
    {
        TestInf f;
        double err_est;
        int err_code;
        const double res = integrate(f, lower, upper, err_est, err_code);
        return Rcpp::List::create(
            Rcpp::Named("approximate") = res,
            Rcpp::Named("error_estimate") = err_est,
            Rcpp::Named("error_code") = err_code
        );
    }

    # integrate() in R
    integrate(function(x) x^2 * dnorm(x), 0.5, Inf)
    # 0.4845702 with absolute error < 3e-08
    integrate_1d_inf_test(0.5, Inf)
    # $approximate
    # [1] 0.4845702
    #
    # $error_estimate
    # [1] 1.633995e-08
    #
    # $error_code
    # [1] 0

Similarly, for multi-dimensional integration, infinite limits are supported in each dimension by specifying `std::numeric_limits<double>::infinity()` or `-std::numeric_limits<double>::infinity()`:

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]
    #include <RcppNumerical.h>
    using namespace Numer;

    // Test 1: Semi-infinite [0, +Inf) x [0, 1]
    // Integrate exp(-x) over x in [0, +Inf) and y in [0, 1]
    // True value: 1.0
    class SemiInfiniteTest: public MFunc
    {
    public:
        double operator()(Constvec& x)
        {
            return std::exp(-x[0]);
        }
    };

    // Test 2: Doubly-infinite (-Inf, +Inf) x [0, 1]
    // Integrate exp(-x^2) over x in (-Inf, +Inf) and y in [0, 1]
    // True value: sqrt(pi)
    class DoublyInfiniteTest: public MFunc
    {
    public:
        double operator()(Constvec& x)
        {
            return std::exp(-x[0] * x[0]);
        }
    };

    // Test 3: All infinite bounds
    // Integrate exp(-x^2 - y^2) over (-Inf, +Inf) x (-Inf, +Inf)
    // Expected: pi
    class Gaussian2D: public MFunc
    {
    public:
        double operator()(Constvec& x)
        {
            return std::exp(-x[0] * x[0] - x[1] * x[1]);
        }
    };

    // [[Rcpp::export]]
    Rcpp::List integrate_md_inf_test()
    {
        constexpr double Inf = std::numeric_limits<double>::infinity();
        double err_est;
        int err_code;

        Eigen::VectorXd lower(2), upper(2);

        // Test 1: Semi-infinite
        lower[0] = 0.0; upper[0] = Inf;
        lower[1] = 0.0; upper[1] = 1.0;
        SemiInfiniteTest f1;
        double res1 = integrate(f1, lower, upper, err_est, err_code);

        // Test 2: Doubly-infinite
        lower[0] = -Inf; upper[0] = Inf;
        lower[1] = 0.0; upper[1] = 1.0;
        DoublyInfiniteTest f2;
        double res2 = integrate(f2, lower, upper, err_est, err_code);

        // Test 3: All infinite
        lower[0] = -Inf; upper[0] = Inf;
        lower[1] = -Inf; upper[1] = Inf;
        Gaussian2D f3;
        double res3 = integrate(f3, lower, upper, err_est, err_code);

        return Rcpp::List::create(
            Rcpp::Named("semi_infinite") = res1,
            Rcpp::Named("doubly_infinite") = res2,
            Rcpp::Named("all_infinite") = res3
        );
    }

Calling the generated R function `integrate_md_inf_test()` gives

    integrate_md_inf_test()
    # $semi_infinite
    # [1] 1
    #
    # $doubly_infinite
    # [1] 1.772454
    #
    # $all_infinite
    # [1] 3.141542

## Numerical Optimization

### Unconstrained Minimization

Currently RcppNumerical uses the L-BFGS algorithm to solve unconstrained minimization problems based on the LBFGS++ library.

Again, one needs to first define a functor to represent the multivariate function to be minimized.

    class MFuncGrad
    {
    public:
        virtual double f_grad(Constvec& x, Refvec grad) = 0;

        virtual ~MFuncGrad() {}
    };

Same as the case in multi-dimensional integration, `Constvec` represents a read-only vector and `Refvec` a writable vector. Their definitions are

    // Reference to a vector
    using RefVec = Eigen::Ref<Eigen::VectorXd>;
    using Constvec = const Eigen::Ref<const Eigen::VectorXd>;

The `f_grad()` member function returns the function value on vector `x`, and overwrites `grad` by the gradient.

The wrapper function for L-BFGS is

    inline int optim_lbfgs(
        MFuncGrad& f, Refvec x, double& fx_opt,
        const int maxit = 300, const double& eps_f = 1e-6, const double& eps_g = 1e-5
    )

- `f`: The function to be minimized.
- `x`: In: The initial guess. Out: Best value of variables found.
- `fx_opt`: Out: Function value on the output `x`.
- `maxit`: Maximum number of iterations.
- `eps_f`: Algorithm stops if `|f_{k+1} - f_k| < eps_f * |f_k|`.
- `eps_g`: Algorithm stops if `||g|| < eps_g * max(1, ||x||)`.
- Return value: Error code. Negative values indicate errors.

Below is an example that illustrates the optimization of the Rosenbrock function `f(x1, x2) = 100 * (x2 - x1^2)^2 + (1 - x1)^2`:

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]

    #include <RcppNumerical.h>

    using namespace Numer;

    // f = 100 * (x2 - x1^2)^2 + (1 - x1)^2
    // True minimum: x1 = x2 = 1
    class Rosenbrock: public MFuncGrad
    {
    public:
        double f_grad(Constvec& x, Refvec grad)
        {
            double t1 = x[1] - x[0] * x[0];
            double t2 = 1 - x[0];
            grad[0] = -400 * x[0] * t1 - 2 * t2;
            grad[1] = 200 * t1;
            return 100 * t1 * t1 + t2 * t2;
        }
    };

    // [[Rcpp::export]]
    Rcpp::List optim_test()
    {
        Eigen::VectorXd x(2);
        x[0] = -1.2;
        x[1] = 1;
        double fopt;
        Rosenbrock f;
        int res = optim_lbfgs(f, x, fopt);
        return Rcpp::List::create(
            Rcpp::Named("xopt") = x,
            Rcpp::Named("fopt") = fopt,
            Rcpp::Named("status") = res
        );
    }

Calling the generated R function `optim_test()` gives

    optim_test()
    # $xopt
    # [1] 0.9999683 0.9999354
    #
    # $fopt
    # [1] 1.150395e-09
    #
    # $status
    # [1] 0

### Box-constrained Minimization

For optimization problems with box constraints (i.e., each variable has a lower and upper bound), RcppNumerical provides the L-BFGS-B algorithm, also based on the LBFGS++ library.

The functor definition is the same as that in the unconstrained minimization problems, i.e., inheriting from `MFuncGrad`.

The wrapper function for box-constrained optimization is

    inline int optim_lbfgsb(
        MFuncGrad& f, Refvec x, double& fx_opt,
        Constvec& lb, Constvec& ub,
        const int maxit = 300, const double& eps_f = 1e-6, const double& eps_g = 1e-5
    )

- `f`: The function to be minimized.
- `x`: In: The initial guess. Out: Best value of variables found.
- `fx_opt`: Out: Function value on the output `x`.
- `lb`: In: Lower bounds for each variable.
- `ub`: In: Upper bounds for each variable.
- `maxit`: Maximum number of iterations.
- `eps_f`: Algorithm stops if `|f_{k+1} - f_k| < eps_f * |f_k|`.
- `eps_g`: Algorithm stops if the projected gradient norm satisfies the tolerance.
- Return value: Error code. Negative values indicate errors.

Below is an example that minimizes the same Rosenbrock function, but this time with box constraints that force the first variable in [-2, 0.5] and second variable in [0, +Inf).

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]

    #include <RcppNumerical.h>

    using namespace Numer;

    // f = 100 * (x2 - x1^2)^2 + (1 - x1)^2
    class Rosenbrock: public MFuncGrad
    {
    public:
        double f_grad(Constvec& x, Refvec grad)
        {
            double t1 = x[1] - x[0] * x[0];
            double t2 = 1 - x[0];
            grad[0] = -400 * x[0] * t1 - 2 * t2;
            grad[1] = 200 * t1;
            return 100 * t1 * t1 + t2 * t2;
        }
    };

    // [[Rcpp::export]]
    Rcpp::List optim_box_test()
    {
        Eigen::VectorXd x(2), lb(2), ub(2);
        // Initial guess
        x[0] = -1.2;
        x[1] = 1;
        // Bounds [-2, 0.5] for firsts variable
        lb[0] = -2;
        ub[0] = 0.5;
        // Bounds [0, +Inf) for second variable -- infinite values are supported
        lb[1] = 0;
        ub[1] = std::numeric_limits<double>::infinity();

        double fopt;
        Rosenbrock f;
        int res = optim_lbfgsb(f, x, fopt, lb, ub);
        return Rcpp::List::create(
            Rcpp::Named("xopt") = x,
            Rcpp::Named("fopt") = fopt,
            Rcpp::Named("status") = res
        );
    }

Calling the generated R function `optim_box_test()` gives

    optim_box_test()
    # $xopt
    # [1] 0.50 0.25
    #
    # $fopt
    # [1] 0.25
    #
    # $status
    # [1] 0

## A Practical Example

It may be more meaningful to look at a real application of the RcppNumerical package. Below is an example to fit logistic regression using the L-BFGS algorithm. It also demonstrates the performance of the library.

    // [[Rcpp::depends(RcppEigen)]]
    // [[Rcpp::depends(RcppNumerical)]]

    #include <RcppNumerical.h>

    using namespace Numer;

    using MapMat = Eigen::Map<Eigen::MatrixXd>;
    using MapVec = Eigen::Map<Eigen::VectorXd>;

    class LogisticReg: public MFuncGrad
    {
    private:
        const MapMat X;
        const MapVec Y;
    public:
        LogisticReg(const MapMat x_, const MapVec y_) : X(x_), Y(y_) {}

        double f_grad(Constvec& beta, Refvec grad)
        {
            // Negative log likelihood
            //   sum(log(1 + exp(X * beta))) - y' * X * beta

            Eigen::VectorXd xbeta = X * beta;
            const double yxbeta = Y.dot(xbeta);
            // X * beta => exp(X * beta)
            xbeta = xbeta.array().exp();
            const double f = (xbeta.array() + 1.0).log().sum() - yxbeta;

            // Gradient
            //   X' * (p - y), p = exp(X * beta) / (1 + exp(X * beta))

            // exp(X * beta) => p
            xbeta.array() /= (xbeta.array() + 1.0);
            grad.noalias() = X.transpose() * (xbeta - Y);

            return f;
        }
    };

    // [[Rcpp::export]]
    Rcpp::NumericVector logistic_reg(Rcpp::NumericMatrix x, Rcpp::NumericVector y)
    {
        const MapMat xx = Rcpp::as<MapMat>(x);
        const MapVec yy = Rcpp::as<MapVec>(y);
        // Negative log likelihood
        LogisticReg nll(xx, yy);
        // Initial guess
        Eigen::VectorXd beta(xx.cols());
        beta.setZero();

        double fopt;
        int status = optim_lbfgs(nll, beta, fopt);
        if(status < 0)
            Rcpp::stop("fail to converge");

        return Rcpp::wrap(beta);
    }

Here is the R code to test the function:

    set.seed(123)
    n = 5000
    p = 100
    x = matrix(rnorm(n * p), n)
    beta = runif(p)
    xb = c(x %*% beta)
    p = exp(xb) / (1 + exp(xb))
    y = rbinom(n, 1, p)

    system.time(res1 <- glm.fit(x, y, family = binomial())$coefficients)
    #    user  system elapsed
    #   0.391   0.018   0.410
    system.time(res2 <- logistic_reg(x, y))
    #    user  system elapsed
    #   0.008   0.000   0.009
    max(abs(res1 - res2))
    # [1] 0.0001873564

It is much faster than the standard `glm.fit()` function in R! (Although `glm.fit()` calculates some other quantities besides beta.)

RcppNumerical also provides the `fastLR()` function to run fast logistic regression, which is a modified and more stable version of the code above.

    system.time(res3 <- fastLR(x, y)$coefficients)
    #    user  system elapsed
    #   0.011   0.000   0.012
    max(abs(res1 - res3))
    # [1] 7.066969e-06