Compute derivatives with finite-difference methods

## Project description

# FDM: Finite Difference Methods

FDM estimates derivatives with finite differences. See also FiniteDifferences.jl.

- Installation
- Multivariate Derivatives
- Scalar Derivatives
- Testing Sensitivities in a Reverse-Mode Automatic Differentation Framework

## Installation

FDM requires Python 3.6 or higher.

```
pip install fdm
```

## Multivariate Derivatives

```
from fdm import gradient, jacobian, jvp, hvp
```

For the purpose of illustration, let us consider a quadratic function:

```
>>> a = np.random.randn(3, 3); a = a @ a.T
>>> a
array([[ 3.57224794, 0.22646662, -1.80432262],
[ 0.22646662, 4.72596213, 3.46435663],
[-1.80432262, 3.46435663, 3.70938152]])
>>> def f(x):
... return 0.5 * x @ a @ x
```

Consider the following input value:

```
>>> x = np.array([1.0, 2.0, 3.0])
```

### Gradients

```
>>> grad = gradient(f)
>>> grad(x)
array([-1.38778668, 20.07146076, 16.25253519])
>>> a @ x
array([-1.38778668, 20.07146076, 16.25253519])
```

### Jacobians

```
>>> jac = jacobian(f)
>>> jac(x)
array([[-1.38778668, 20.07146076, 16.25253519]])
>>> a @ x
array([-1.38778668, 20.07146076, 16.25253519])
```

But `jacobian`

also works for multi-valued functions.

```
>>> def f2(x):
... return a @ x
>>> jac2 = jacobian(f2)
>>> jac2(x)
array([[ 3.57224794, 0.22646662, -1.80432262],
[ 0.22646662, 4.72596213, 3.46435663],
[-1.80432262, 3.46435663, 3.70938152]])
>>> a
array([[ 3.57224794, 0.22646662, -1.80432262],
[ 0.22646662, 4.72596213, 3.46435663],
[-1.80432262, 3.46435663, 3.70938152]])
```

### Jacobian-Vector Products (Directional Derivatives)

In the scalar case, `jvp`

computes directional derivatives:

```
>>> v = np.array([0.5, 0.6, 0.7]) # A direction
>>> dir_deriv = jvp(f, v)
>>> dir_deriv(x)
22.725757753354657
>>> np.sum(grad(x) * v)
22.72575775335481
```

In the multivariate case, `jvp`

generalises to Jacobian-vector products:

```
>>> prod = jvp(f2, v)
>>> prod(x)
array([0.65897811, 5.37386023, 3.77301973])
>>> a @ v
array([0.65897811, 5.37386023, 3.77301973])
```

### Hessian-Vector Products

```
>>> prod = hvp(f, v)
>>> prod(x)
array([[0.6589781 , 5.37386023, 3.77301973]])
>>> 0.5 * (a + a.T) @ v
array([0.65897811, 5.37386023, 3.77301973])
```

## Scalar Derivatives

```
>>> from fdm import central_fdm
```

Let's try to estimate the first derivative of `np.sin`

at `1`

with a
second-order method.

```
>>> central_fdm(order=2, deriv=1)(np.sin, 1) - np.cos(1)
-1.2914319613699377e-09
```

And let's try to estimate the second derivative of `np.sin`

at `1`

with a
third-order method.

```
>>> central_fdm(order=3, deriv=2)(np.sin, 1) + np.sin(1)
1.6342919018086377e-08
```

Hm.
Let's check the accuracy of this third-order method.
The step size and accuracy of the method are computed upon calling
`FDM.estimate`

.

```
>>> central_fdm(order=3, deriv=2).estimate(np.sin, 1).acc
5.476137293912896e-06
```

We might want a little more accuracy. Let's check the accuracy of a fifth-order method.

```
>>> central_fdm(order=5, deriv=2).estimate(np.sin, 1).acc
7.343652562575157e-10
```

And let's estimate the second derivative of `np.sin`

at `1`

with a
fifth-order method.

```
>>> central_fdm(order=5, deriv=2)(np.sin, 1) + np.sin(1)
-1.7121615236703747e-10
```

Hooray!

Finally, let us verify that increasing the order generally increases the accuracy.

```
>>> for i in range(3, 10):
... print(central_fdm(order=i, deriv=2)(np.sin, 1) + np.sin(1))
1.6342919018086377e-08
8.604865264771888e-09
-1.7121615236703747e-10
8.558931341440257e-12
-2.147615418834903e-12
6.80566714095221e-13
-1.2434497875801753e-14
```

## Testing Sensitivities in a Reverse-Mode Automatic Differentation Framework

Consider the function

```
def mul(a, b):
return a * b
```

and its sensitivity

```
def s_mul(s_y, y, a, b):
return s_y * b, a * s_y
```

The sensitivity `s_mul`

takes in the sensitivity `s_y`

of the output `y`

,
the output `y`

, and the arguments of the function `mul`

; and returns a tuple
containing the sensitivities with respect to `a`

and `b`

.
Then function `check_sensitivity`

can be used to assert that the
implementation of `s_mul`

is correct:

```
>>> from fdm import check_sensitivity
>>> check_sensitivity(mul, s_mul, (2, 3)) # Test at arguments `2` and `3`.
```

Suppose that the implementation were wrong, for example

```
def s_mul_wrong(s_y, y, a, b):
return s_y * b, b * s_y # Used `b` instead of `a` for the second sensitivity!
```

Then `check_sensitivity`

should throw an `AssertionError`

:

```
>>> check_sensitivity(mul, s_mul, (2, 3))
AssertionError: Sensitivity of argument 2 of function "mul" did not match numerical estimate.
```

## Project details

## Download files

Download the file for your platform. If you're not sure which to choose, learn more about installing packages.