The Calculus package provides tools for working with the basic calculus operations of differentiation and integration. You can use the Calculus package to produce approximate derivatives by several forms of finite differencing or to produce exact derivative using symbolic differentiation. You can also compute definite integrals by different numerical methods.

Most users will want to work with a limited set of basic functions:

• derivative(): Use this for functions from R to R
• second_derivative(): Use this for functions from R to R
• gradient(): Use this for functions from R^n to R
• hessian(): Use this for functions from R^n to R
• integrate(): Use this to integrate functions from R to R
• differentiate(): Use this to perform symbolic differentiation
• simplify(): Use this to perform symbolic simplification
• deparse(): Use this to get usual infix representation of expressions

There are a few basic approaches to using the Calculus package:

• Use finite-differencing to evaluate the derivative at a specific point
• Use higher-order functions to create new functions that evaluate derivatives
• Use integration by Simpson's rule or Monte Carlo method to evaluate definite integrals
• Use symbolic differentiation to produce exact derivatives for simple functions

using Calculus

# Compare with cos(0.0)
derivative(sin, 0.0)
# Compare with cos(1.0)
derivative(sin, 1.0)
# Compare with cos(pi)
derivative(sin, float64(pi))

# Compare with [cos(0.0), -sin(0.0)]
gradient(x -> sin(x[1]) + cos(x[2]), [0.0, 0.0])
# Compare with [cos(1.0), -sin(1.0)]
gradient(x -> sin(x[1]) + cos(x[2]), [1.0, 1.0])
# Compare with [cos(pi), -sin(pi)]
gradient(x -> sin(x[1]) + cos(x[2]), [float64(pi), float64(pi)])

# Compare with -sin(0.0)
second_derivative(sin, 0.0)
# Compare with -sin(1.0)
second_derivative(sin, 1.0)
# Compare with -sin(pi)
second_derivative(sin, float64(pi))

# Compare with [-sin(0.0) 0.0; 0.0 -cos(0.0)]
hessian(x -> sin(x[1]) + cos(x[2]), [0.0, 0.0])
# Compare with [-sin(1.0) 0.0; 0.0 -cos(1.0)]
hessian(x -> sin(x[1]) + cos(x[2]), [1.0, 1.0])
# Compare with [-sin(pi) 0.0; 0.0 -cos(pi)]
hessian(x -> sin(x[1]) + cos(x[2]), [float64(pi), float64(pi)])

using Calculus

g1 = derivative(sin)
g1(0.0)
g1(1.0)
g1(pi)

g2 = gradient(x -> sin(x[1]) + cos(x[2]))
g2([0.0, 0.0])
g2([1.0, 1.0])
g2([pi, pi])

h1 = second_derivative(sin)
h1(0.0)
h1(1.0)
h1(pi)

h2 = hessian(x -> sin(x[1]) + cos(x[2]))
h2([0.0, 0.0])
h2([1.0, 1.0])
h2([pi, pi])

For scalar functions that map R to R, you can use the ' operator to calculate derivatives as well. This operator can be used abritratily many times, but you should keep in mind that the approximation degrades with each approximate derivative you calculate:

using Calculus

f(x) = sin(x)
f'(1.0) - cos(1.0)
f''(1.0) - (-sin(1.0))
f'''(1.0) - (-cos(1.0))

using Calculus

# Compare with log(2)
integrate(x -> 1 / x, 1.0, 2.0)

# Compare with cos(pi) - cos(0)
integrate(x -> -sin(x), 0.0, float64(pi))

using Calculus

# Compare with cos(pi) - cos(0)
integrate(x -> -sin(x), 0.0, float64(pi), :monte_carlo)

using Calculus

differentiate("cos(x) + sin(x) + exp(-x) * cos(x)", :x)
differentiate("cos(x) + sin(y) + exp(-x) * cos(y)", [:x, :y])

• Finite differencing based on complex numbers

Calculus.jl is built on contributions from:

• John Myles White
• Tim Holy
• Andreas Noack Jensen
• Nathaniel Daw
• Blake Johnson
• Avik Sengupta

And draws inspiration and ideas from:

• Mark Schmidt
• Jonas Rauch