# Interoperability with Other Languages

As F# can interoperate seamlessly with C# and other CLI languages, DiffSharp can be used with these languages as well. Your project should reference the DiffSharp.dll assembly, its dependencies, and also the FSharp.Core.dll assembly. Please note that your project should target ".NET Framework 4.6" and have "x64" as the platform target. (Also see the installation instructions on the main page.)

For C# and other languages, the DiffSharp.Interop namespace provides a simpler way of using the library. (Without DiffSharp.Interop, you can still use the regular DiffSharp namespaces, but you will need to take care of issues such as converting to and from FSharp.Core.FSharpFunc objects.)

# Using DiffSharp with C#

## Nested Automatic Differentiation

For using the nested forward and reverse AD capability, you need to write the part of your numeric code where you need deriatives (e.g. for optimization) using the D (scalar), DV (vector), and DM (matrix) numeric types under DiffSharp.Interop.Float32 for single precision or DiffSharp.Interop.Float64 for double precision. You can later convert these values to the standard types of float, float[], float[,] or double, double[], double[,]. In other words, for any computation you do with the D, DV, and DM numeric types, you can automatically get exact derivatives. You will also get the benefit of the fast linear algebra computations provided by the BLAS/LAPACK backend (OpenBLAS by default).

The AD class (under DiffSharp.Interop.Float32 or DiffSharp.Interop.Float64 ) provides common mathematical functions (e.g. AD.Exp, AD.Sin, AD.Pow ) for the D, DV, and DM types, similar to the use of the System.Math class with the double type and other types.

C# versions of the differentiation operations are also provided through the AD wrapper class, which internally handles all necessary conversions to and from C# functions. The names of differentiation operations (e.g. diff, grad, hessian ) remain the same, but their first letters are capitalized (e.g. AD.Diff, AD.Grad, AD.Hessian ). Please see the API Overview page for general information about the differentiation API.

Here is a simple example illustrating the creation of values and the computation of derivatives.

  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40: 41: 42: 43: 44: 45: 46: 47: 48: 49: 50: 51: 52: 53:  // Use DiffSharp interop using DiffSharp.Interop.Float64; class Program { // Define a function whose derivative you need // F(x) = Sin(x^2 - Exp(x)) public static D F(D x) { return AD.Sin(x * x - AD.Exp(x)); } public static void Main(string[] args) { // You can compute the value of the derivative of F at a point D da = AD.Diff(F, 2.3); // Or, you can generate a derivative function which you may use for many evaluations // dF is the derivative function of F var dF = AD.Diff(F); // Evaluate the derivative function at different points D db = dF(2.3); D dc = dF(1.4); // Construction and casting of D (scalar) values // Construct new D D a = new D(4.1); // Cast double to D D b = (D)4.1; // Cast D to double double c = (double)b; // Construction and casting of DV (vector) values // Construct new DV DV va = new DV(new double[] { 1, 2, 3 }); // Cast double[] to DV double[] vaa = new double[] { 1, 2, 3 }; DV vb = (DV)vaa; // Cast DV to double[] double[] vc = (double[])vb; // Construction and casting of DM (matrix) values // Construct new DM DM ma = new DM(new double[,] { { 1, 2 }, { 3, 4 } }); // Cast double[,] to DM double[,] maa = new double[,] { { 1, 2 }, { 3, 4 } }; DM mb = (DM)maa; // Cast DM to double[,] double[,] mc = (double[,])mb; } } 

Differentiation operations can be nested, meaning that you can compute higher-order derivatives and differentiate functions that are themselves internally making use of differentiation (also see Nested AD).

  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:  using DiffSharp.Interop.Float64; class Program { // F(x) = Sin(x^2 - Exp(x)) public D F(D x) { return AD.Sin(x * x - AD.Exp(x)); } // G is internally using the derivative of F // G(x) = F'(x) / Exp(x^3) public D G(D x) { return AD.Diff(F, x) / AD.Exp(AD.Pow(x, 3)); } // H is internally using the derivative of G // H(x) = Sin(G'(x) / 2) public D H(D x) { return AD.Sin(AD.Diff(G, x) / 2); } } 

A convenient way of writing functions is to use C# lambda expressions with which you can define local anonymous functions.

  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19:  using DiffSharp.Interop.Float64; class Program { // F(x) = Sin(x^2 - Exp(x)) public static D F(D x) { return AD.Sin(x * x - AD.Exp(x)); } public static void Main(string[] args) { // Derivative of F(x) at x = 3 var a = AD.Diff(F, 3); // This is the same with above, defining the function inline var b = AD.Diff(x => AD.Sin(x * x - AD.Exp(x)), 3); } } 

DiffSharp can handle nested cases such as computing the derivative of a function $$f$$ that takes an argument $$x$$, which, in turn, computes the derivative of another function $$g$$ nested inside $$f$$ that has a free reference to $$x$$, the argument to the surrounding function.

$\frac{d}{dx} \left. \left( x \left( \left. \frac{d}{dy} x y \; \right|_{y=3} \right) \right) \right|_{x=2}$

 1:  var c = AD.Diff(x => x * AD.Diff(y => x * y, 3), 2); 

This allows you to write, for example, nested optimization algorithms of the form

$\mathbf{min} \left( \lambda x \; . \; (f \; x) + \mathbf{min} \left( \lambda y \; . \; g \; x \; y \right) \right)\; ,$

for functions $$f$$ and $$g$$ and a gradient-based minimization procedure $$\mathbf{min}$$.

### Differentiation Operations

Currently the following operations are supported by DiffSharp.Interop:

#### First derivative of a scalar-to-scalar function

Syntax: public static Func<D,D> AD.Diff(Func<D,D> f)

For a function $$f(a): \mathbb{R} \to \mathbb{R}$$, this returns a function that computes the derivative

$\frac{d}{da} f(a) \; .$

 1: 2: 3: 4: 5:  // Derivative of a scalar-to-scalar function var df = AD.Diff(x => AD.Sin(x * x - AD.Exp(x))); // Evaluate df at a point var v = df(3); 

#### First derivative of a scalar-to-scalar function evaluated at a point

Syntax: public static D AD.Diff(Func<D,D> f, D x)

For a function $$f(a): \mathbb{R} \to \mathbb{R}$$, and $$x \in \mathbb{R}$$, this returns the derivative evaluated at $$x$$

$\left. \frac{d}{da} f(a) \right|_{a\; =\; x} \; .$

 1: 2:  // Derivative of a scalar-to-scalar function at a point var v = AD.Diff(x => AD.Sin(x * x - AD.Exp(x)), 3); 

#### Second derivative of a scalar-to-scalar function

Syntax: public static Func<D,D> AD.Diff2(Func<D,D> f)

For a function $$f(a): \mathbb{R} \to \mathbb{R}$$, this returns a function that computes the second derivative

$\frac{d^2}{da^2} f(a) \; .$

 1: 2: 3: 4: 5:  // Second derivative of a scalar-to-scalar function var df = AD.Diff2(x => AD.Sin(x * x - AD.Exp(x))); // Evaluate df at a point var v = df(3); 

#### Second derivative of a scalar-to-scalar function evaluated at a point

Syntax: public static D AD.Diff2(Func<D,D> f, D x)

For a function $$f(a): \mathbb{R} \to \mathbb{R}$$, and $$x \in \mathbb{R}$$, this returns the second derivative evaluated at $$x$$

$\left. \frac{d^2}{da^2} f(a) \right|_{a\; =\; x} \; .$

 1: 2:  // Second derivative of a scalar-to-scalar function at a point var v = AD.Diff2(x => AD.Sin(x * x - AD.Exp(x)), 3); 

#### N-th derivative of a scalar-to-scalar function

Syntax: public static Func<D,D> AD.Diffn(Int32 n, Func<D,D> f)

For $$n \in \mathbb{N}$$ and a function $$f(a): \mathbb{R} \to \mathbb{R}$$, this returns a function that computes the n-th derivative

$\frac{d^n}{da^n} f(a) \; .$

 1: 2: 3: 4: 5:  // Fifth derivative of a scalar-to-scalar function var df = AD.Diffn(5, x => AD.Sin(x * x - AD.Exp(x))); // Evaluate df at a point var v = df(3); 

#### N-th derivative of a scalar-to-scalar function evaluated at a point

Syntax: public static D AD.Diffn(Int32 n, Func<D,D> f, D x)

For $$n \in \mathbb{N}$$, a function $$f(a): \mathbb{R} \to \mathbb{R}$$, and $$x \in \mathbb{R}$$, this returns the n-th derivative evaluated at $$x$$

$\left. \frac{d^n}{da^n} f(a) \right|_{a\; =\; x} \; .$

 1: 2:  // Fifth derivative of a scalar-to-scalar function at a point var v = AD.Diffn(5, x => AD.Sin(x * x - AD.Exp(x)), 3); 

#### Gradient of a vector-to-scalar function

Syntax: public static Func<DV,DV> AD.Grad(Func<DV,D> f)

For a function $$f(a_1, \dots, a_n): \mathbb{R}^n \to \mathbb{R}$$, this returns a function that computes the gradient

$\nabla f = \left[ \frac{\partial f}{{\partial a}_1}, \dots, \frac{\partial f}{{\partial a}_n} \right] \; .$

 1: 2: 3: 4: 5:  // Gradient of a vector-to-scalar function var gf = AD.Grad(x => AD.Sin(x[0] * x[1])); // Evaluate gf at a point var v = gf(new DV(new double[] { 3, 2 })); 

#### Gradient of a vector-to-scalar function evaluated at a point

Syntax: public static DV AD.Grad(Func<DV,D> f, DV x)

For a function $$f(a_1, \dots, a_n): \mathbb{R}^n \to \mathbb{R}$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns the gradient evaluated at $$\mathbf{x}$$

$\left( \nabla f \right)_\mathbf{x} = \left. \left[ \frac{\partial f}{{\partial a}_1}, \dots, \frac{\partial f}{{\partial a}_n} \right] \right|_{\mathbf{a}\; = \; \mathbf{x}} \; .$

 1: 2:  // Gradient of a vector-to-scalar function at a point var v = AD.Grad(x => AD.Sin(x[0] * x[1]), new DV(new double[] { 3, 2 })); 

Syntax: public static D AD.Gradv(Func<DV,D> f, DV x, DV v)

For a function $$f: \mathbb{R}^n \to \mathbb{R}$$, and $$\mathbf{x}, \mathbf{v} \in \mathbb{R}^n$$, this returns the gradient-vector product (directional derivative), that is, the dot product of the gradient of $$f$$ at $$\mathbf{x}$$ with $$\mathbf{v}$$

$\left( \nabla f \right)_\mathbf{x} \cdot \mathbf{v} \; .$

With AD, this value is computed efficiently in one forward evaluation of the function, without computing the full gradient.

 1: 2:  // Gradient-vector product of a vector-to-scalar function var v = AD.Gradv(x => AD.Sin(x[0] * x[1]), new DV(new double[] { 3, 2 }), new DV(new double[] { 5, 3 })); 

#### Hessian of a vector-to-scalar function

Syntax: public static Func<DV,DM> AD.Hessian(Func<DV,D> f)

For a function $$f(a_1, \dots, a_n): \mathbb{R}^n \to \mathbb{R}$$, this returns a function that computes the Hessian matrix

$\mathbf{H}_f = \begin{bmatrix} \frac{\partial ^2 f}{\partial a_1^2} & \frac{\partial ^2 f}{\partial a_1 \partial a_2} & \cdots & \frac{\partial ^2 f}{\partial a_1 \partial a_n} \\ \frac{\partial ^2 f}{\partial a_2 \partial a_1} & \frac{\partial ^2 f}{\partial a_2^2} & \cdots & \frac{\partial ^2 f}{\partial a_2 \partial a_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial ^2 f}{\partial a_n \partial a_1} & \frac{\partial ^2 f}{\partial a_n \partial a_2} & \cdots & \frac{\partial ^2 f}{\partial a_n^2} \end{bmatrix} \; .$

 1: 2: 3: 4: 5:  // Hessian of a vector-to-scalar function var hf = AD.Hessian(x => AD.Sin(x[0] * x[1])); // Evaluate hf at a point var v = hf(new DV(new double[] { 3, 2 })); 

#### Hessian of a vector-to-scalar function evaluated at a point

Syntax: public static DM AD.Hessian(Func<DV,D> f, DV x)

For a function $$f(a_1, \dots, a_n): \mathbb{R}^n \to \mathbb{R}$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns the Hessian matrix evaluated at $$\mathbf{x}$$

$\left( \mathbf{H}_f \right)_\mathbf{x} = \left. \begin{bmatrix} \frac{\partial ^2 f}{\partial a_1^2} & \frac{\partial ^2 f}{\partial a_1 \partial a_2} & \cdots & \frac{\partial ^2 f}{\partial a_1 \partial a_n} \\ \frac{\partial ^2 f}{\partial a_2 \partial a_1} & \frac{\partial ^2 f}{\partial a_2^2} & \cdots & \frac{\partial ^2 f}{\partial a_2 \partial a_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial ^2 f}{\partial a_n \partial a_1} & \frac{\partial ^2 f}{\partial a_n \partial a_2} & \cdots & \frac{\partial ^2 f}{\partial a_n^2} \end{bmatrix} \right|_{\mathbf{a}\; = \; \mathbf{x}} \; .$

 1: 2:  // Hessian of a vector-to-scalar function at a point var v = AD.Hessian(x => AD.Sin(x[0] * x[1]), new DV(new double[] { 3, 2 })); 

#### Hessian-vector product

Syntax: public static DV AD.Hessianv(Func<DV,D> f, DV x, DV v)

For a function $$f: \mathbb{R}^n \to \mathbb{R}$$, and $$\mathbf{x}, \mathbf{v} \in \mathbb{R}^n$$, this returns the Hessian-vector product, that is, the multiplication of the Hessian matrix of $$f$$ at $$\mathbf{x}$$ with $$\mathbf{v}$$

$\left( \mathbf{H}_f \right)_\mathbf{x} \; \mathbf{v} \; .$

With AD, this value is computed efficiently using one forward and one reverse evaluation of the function, in a matrix-free way (without computing the full Hessian matrix).

 1: 2:  // Hessian-vector product of a vector-to-scalar function var hv = AD.Hessianv(x => AD.Sin(x[0] * x[1]), new DV(new double[] { 3, 2 }), new DV(new double[] { 5, 3 })); 

#### Laplacian of a vector-to-scalar function

Syntax: public static Func<DV,D> AD.Laplacian(Func<DV,D> f)

For a function $$f(a_1, \dots, a_n): \mathbb{R}^n \to \mathbb{R}$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns a function that computes the sum of second derivatives evaluated at $$\mathbf{x}$$

$\mathrm{tr}\left(\mathbf{H}_f \right) = \left(\frac{\partial ^2 f}{\partial a_1^2} + \dots + \frac{\partial ^2 f}{\partial a_n^2}\right) \; ,$

which is the trace of the Hessian matrix.

With AD, this value is computed efficiently in a Matrix-free way, without computing the full Hessian matrix.

 1: 2: 3: 4: 5:  // Laplacian of a vector-to-scalar function var lf = AD.Laplacian(x => AD.Sin(x[0] * x[1])); // Evaluate lf at a point var v = lf(new DV(new double[] { 3, 2 })); 

#### Laplacian of a vector-to-scalar function evaluated at a point

Syntax: public static D AD.Laplacian(Func<DV,D> f, DV x)

For a function $$f(a_1, \dots, a_n): \mathbb{R}^n \to \mathbb{R}$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns the sum of second derivatives evaluated at $$\mathbf{x}$$

$\mathrm{tr}\left(\mathbf{H}_f \right)_\mathbf{x} = \left. \left(\frac{\partial ^2 f}{\partial a_1^2} + \dots + \frac{\partial ^2 f}{\partial a_n^2}\right) \right|_{\mathbf{a} \; = \; \mathbf{x}} \; .$

 1: 2:  // Laplacian of a vector-to-scalar function at a point var v = AD.Laplacian(x => AD.Sin(x[0] * x[1]), new DV(new double[] { 3, 2 })); 

#### Jacobian of a vector-to-vector function

Syntax: public static Func<DV,DM> AD.Jacobian(Func<DV,DV> f)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m$$ with components $$F_1 (a_1, \dots, a_n), \dots, F_m (a_1, \dots, a_n)$$, this returns a function that computes the $$m$$-by-$$n$$ Jacobian matrix

$\mathbf{J}_\mathbf{F} = \begin{bmatrix} \frac{\partial F_1}{\partial a_1} & \cdots & \frac{\partial F_1}{\partial a_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial F_m}{\partial a_1} & \cdots & \frac{\partial F_m}{\partial a_n} \end{bmatrix} \; .$

 1: 2: 3: 4: 5:  // Jacobian of a vector-to-vector function var jf = AD.Jacobian(x => new DV(new D[]{ AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] })); // Evaluate jf at a point var v = jf(new DV(new double[] { 3, 2, 4 })); 

#### Jacobian of a vector-to-vector function evaluated at a point

Syntax: public static DM AD.Jacobian(Func<DV,DV> f, DV x)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m$$ with components $$F_1 (a_1, \dots, a_n), \dots, F_m (a_1, \dots, a_n)$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns the $$m$$-by-$$n$$ Jacobian matrix evaluated at $$\mathbf{x}$$

$\left( \mathbf{J}_\mathbf{F} \right)_\mathbf{x} = \left. \begin{bmatrix} \frac{\partial F_1}{\partial a_1} & \cdots & \frac{\partial F_1}{\partial a_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial F_m}{\partial a_1} & \cdots & \frac{\partial F_m}{\partial a_n} \end{bmatrix} \right|_{\mathbf{a}\; = \; \mathbf{x}} \; .$

 1: 2:  // Jacobian of a vector-to-vector function at a point var v = AD.Jacobian(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] }), new DV(new double[] { 3, 2, 4 })); 

#### Jacobian-vector product

Syntax: public static DV AD.Jacobianv(Func<DV,DV> f, DV x, DV v)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m$$, and $$\mathbf{x}, \mathbf{v} \in \mathbb{R}^n$$, this returns the Jacobian-vector product, that is, the matrix product of the Jacobian of $$\mathbf{F}$$ at $$\mathbf{x}$$ with $$\mathbf{v}$$

$\left( \mathbf{J}_\mathbf{F} \right)_\mathbf{x} \mathbf{v} \; .$

With AD, this value is computed efficiently in one forward evaluation of the function, in a matrix-free way (without computing the full Jacobian matrix).

 1: 2:  // Jacobian-vector product of a vector-to-vector function var v = AD.Jacobianv(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] }), new DV(new double[] { 3, 2, 4 }), new DV(new double[] { 1, 2, 3 })); 

#### Transposed Jacobian of a vector-to-vector function

Syntax: public static Func<DV,DM> AD.JacobianT(Func<DV,DV> f)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m$$ with components $$F_1 (a_1, \dots, a_n), \dots, F_m (a_1, \dots, a_n)$$, this returns a function that computes the $$n$$-by-$$m$$ transposed Jacobian matrix

$\mathbf{J}_\mathbf{F}^\textrm{T} = \begin{bmatrix} \frac{\partial F_1}{\partial a_1} & \cdots & \frac{\partial F_m}{\partial a_1} \\ \vdots & \ddots & \vdots \\ \frac{\partial F_1}{\partial a_n} & \cdots & \frac{\partial F_m}{\partial a_n} \end{bmatrix} \; .$

 1: 2: 3: 4: 5:  // Transposed Jacobian of a vector-to-vector function var jf = AD.JacobianT(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] })); // Evaluate jf at a point var v = jf(new DV(new double[] { 3, 2, 4 })); 

#### Transposed Jacobian of a vector-to-vector function evaluated at a point

Syntax: public static DM AD.JacobianT(Func<DV,DV> f, DV x)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m$$ with components $$F_1 (a_1, \dots, a_n), \dots, F_m (a_1, \dots, a_n)$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns the $$n$$-by-$$m$$ transposed Jacobian matrix evaluated at $$\mathbf{x}$$

$\left( \mathbf{J}_\mathbf{F}^\textrm{T} \right)_\mathbf{x} = \left. \begin{bmatrix} \frac{\partial F_1}{\partial a_1} & \cdots & \frac{\partial F_m}{\partial a_1} \\ \vdots & \ddots & \vdots \\ \frac{\partial F_1}{\partial a_n} & \cdots & \frac{\partial F_m}{\partial a_n} \end{bmatrix} \right|_{\mathbf{a}\; = \; \mathbf{x}} \; .$

 1: 2:  // Transposed Jacobian of a vector-to-vector function at a point var v = AD.JacobianT(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] }), new DV(new D[] { 3, 2, 4 })); 

#### Transposed Jacobian-vector product

Syntax: public static DV AD.JacobianTv(Func<DV,DV> f, DV x, DV v)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^m$$, $$\mathbf{x} \in \mathbb{R}^n$$, and $$\mathbf{v} \in \mathbb{R}^m$$, this returns the matrix product of the transposed Jacobian of $$\mathbf{F}$$ at $$\mathbf{x}$$ with $$\mathbf{v}$$

$\left( \mathbf{J}_\mathbf{F}^\textrm{T} \right)_\mathbf{x} \mathbf{v} \; .$

With AD, this value is computed efficiently in one forward and one reverse evaluation of the function, in a matrix-free way (without computing the full Jacobian matrix).

 1: 2:  // Transposed Jacobian-vector product of a vector-to-vector function var v = AD.JacobianTv(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] }), new DV(new double[] { 3, 2, 4 }), new DV(new double[] { 1, 2, 3 })); 

#### Curl of a vector-to-vector function

Syntax: public static Func<DV,DV> AD.Curl(Func<DV,DV> f)

For a function $$\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$$ with components $$F_1(a_1, a_2, a_3),\; F_2(a_1, a_2, a_3),\; F_3(a_1, a_2, a_3)$$ this returns a function that computes the curl, that is,

$\textrm{curl} \, \mathbf{F} = \nabla \times \mathbf{F} = \left[ \frac{\partial F_3}{\partial a_2} - \frac{\partial F_2}{\partial a_3}, \; \frac{\partial F_1}{\partial a_3} - \frac{\partial F_3}{\partial a_1}, \; \frac{\partial F_2}{\partial a_1} - \frac{\partial F_1}{\partial a_2} \right] \; .$

 1: 2: 3: 4: 5:  // Curl of a vector-to-vector function var cf = AD.Curl(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] })); // Evaluate cf at a point var v = cf(new DV (new double[] { 3, 2, 4 })); 

#### Curl of a vector-to-vector function evaluated at a point

Syntax: public static DV AD.Curl(Func<DV,DV> f, DV x)

For a function $$\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$$ with components $$F_1(a_1, a_2, a_3),\; F_2(a_1, a_2, a_3),\; F_3(a_1, a_2, a_3)$$, and $$\mathbf{x} \in \mathbb{R}^3$$, this returns the curl evaluated at $$\mathbf{x}$$

$\left( \textrm{curl} \, \mathbf{F} \right)_{\mathbf{x}} = \left( \nabla \times \mathbf{F} \right)_{\mathbf{x}}= \left. \left[ \frac{\partial F_3}{\partial a_2} - \frac{\partial F_2}{\partial a_3}, \; \frac{\partial F_1}{\partial a_3} - \frac{\partial F_3}{\partial a_1}, \; \frac{\partial F_2}{\partial a_1} - \frac{\partial F_1}{\partial a_2} \right] \right|_{\mathbf{a}\; = \; \mathbf{x}} \; .$

 1: 2:  // Curl of a vector-to-vector function at a point var v = AD.Curl(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] }), new DV(new double[] { 3, 2, 4 })); 

#### Divergence of a vector-to-vector function

Syntax: public static Func<DV,D> AD.Div(Func<DV,D[]> f)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^n$$ with components $$F_1(a_1, \dots, a_n),\; \dots, \; F_n(a_1, \dots, a_n)$$, this returns a function that computes the divergence, that is, the trace of the Jacobian matrix

$\textrm{div} \, \mathbf{F} = \nabla \cdot \mathbf{F} = \textrm{tr}\left( \mathbf{J}_{\mathbf{F}} \right) = \left( \frac{\partial F_1}{\partial a_1} + \dots + \frac{\partial F_n}{\partial a_n}\right) \; .$

 1: 2: 3: 4: 5:  // Divergence of a vector-to-vector function var df = AD.Curl(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] })); // Evaluate df at a point var v = df(new DV(new double[] { 3, 2, 4 })); 

#### Divergence of a vector-to-vector function evaluated at a point

Syntax: public static D AD.Div(Func<DV,DV> f, DV x)

For a function $$\mathbf{F}: \mathbb{R}^n \to \mathbb{R}^n$$ with components $$F_1(a_1, \dots, a_n),\; \dots, \; F_n(a_1, \dots, a_n)$$, and $$\mathbf{x} \in \mathbb{R}^n$$, this returns the trace of the Jacobian matrix evaluated at $$\mathbf{x}$$

$\left( \textrm{div} \, \mathbf{F} \right)_{\mathbf{x}} = \left( \nabla \cdot \mathbf{F} \right)_{\mathbf{x}} = \textrm{tr}\left( \mathbf{J}_{\mathbf{F}} \right)_{\mathbf{x}} = \left. \left( \frac{\partial F_1}{\partial a_1} + \dots + \frac{\partial F_n}{\partial a_n}\right) \right|_{\mathbf{a}\; = \; \mathbf{x}} \; .$

 1: 2:  // Divergence of a vector-to-vector function at a point var v = AD.Curl(x => new DV(new D[] { AD.Sin(x[0] * x[1]), x[0] - x[1], x[2] }), new DV(new double[] { 3, 2, 4 })); 

## Numerical Differentiation

DiffSharp.Interop also provides numerical differentiation, through the DiffSharp.Interop.Float32.Numerical (for single precision) and DiffSharp.Interop.Float64.Numerical (for double precision) classes.

Numerical differentiation operations are used with the float or double numeric type, and the common mathematical functions can be accessed using the System.Math class as usual (e.g. Math.Exp, Math.Sin, Math.Pow ).

Currently, the following operations are supported:

• AD.Diff: First derivative of a scalar-to-scalar function
• AD.Diff2: Second derivative of a scalar-to-scalar function
  1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40: 41: 42: 43: 44: 45: 46: 47: 48: 49: 50: 51: 52: 53: 54: 55: 56: 57: 58: 59: 60: 61: 62: 63: 64: 65:  using System; using DiffSharp.Interop.Float64; class Program { // A scalar-to-scalar function // F(x) = Sin(x^2 - Exp(x)) public static double F(double x) { return Math.Sin(x * x - Math.Exp(x)); } // A vector-to-scalar function // G(x1, x2) = Sin(x1 * x2) public static double G(double[] x) { return Math.Sin(x[0] * x[1]); } // A vector-to-vector function // H(x1, x2, x3) = (Sin(x1 * x2), Exp(x1 - x2), x3) public static double[] H(double[] x) { return new double[] { Math.Sin(x[0] * x[1]), Math.Exp(x[0] - x[1]), x[2] }; } public static void Main(string[] args) { // Derivative of F(x) at x = 3 var a = Numerical.Diff(F, 3); // Second derivative of F(x) at x = 3 var b = Numerical.Diff2(F, 3); // Gradient of G(x) at x = (4, 3) var c = Numerical.Grad(G, new double[] { 4, 3 }); // Directional derivative of G(x) at x = (4, 3) along v = (2, 5) var d = Numerical.Gradv(G, new double[] { 4, 3 }, new double[] { 2, 5 }); // Hessian of G(x) at x = (4, 3) var e = Numerical.Hessian(G, new double[] { 4, 3 }); // Hessian-vector product of G(x), with x = (4, 3) and v = (2, 5) var f = Numerical.Hessianv(G, new double[] { 4, 3 }, new double[] { 2, 5 }); // Laplacian of G(x) at x = (4, 3) var g = Numerical.Laplacian(G, new double[] { 4, 3 }); // Jacobian of H(x) at x = (5, 2, 1) var h = Numerical.Jacobian(H, new double[] { 5, 2, 1 }); // Transposed Jacobian of H(x) at x = (5, 2, 1) var i = Numerical.JacobianT(H, new double[] { 5, 2, 1 }); // Jacobian-vector product of H(x), with x = (5, 2, 1) and v = (2, 5, 3) var j = Numerical.Jacobianv(H, new double[] { 5, 2, 1 }, new double[] { 2, 5, 3 }); // Curl of H(x) at x = (5, 2, 1) var k = Numerical.Curl(H, new double[] { 5, 2, 1 }); // Divergence of H(x) at x = (5, 2, 1) var l = Numerical.Div(H, new double[] { 5, 2, 1 }); } }