torch.trapezoid

torch.trapezoid(y, x=None, *, dx=None, dim=- 1) → Tensor

Computes the trapezoidal rule along dim. By default the spacing between elements is assumed to be 1, but dx can be used to specify a different constant spacing, and x can be used to specify arbitrary spacing along dim.

Assuming y is a one-dimensional tensor with elements ${y_0, y_1, ..., y_n}$, the default computation is

$$\begin{aligned} \sum_{i = 1}^{n-1} \frac{1}{2} (y_i + y_{i-1}) \end{aligned}$$

When dx is specified the computation becomes

$$\begin{aligned} \sum_{i = 1}^{n-1} \frac{\Delta x}{2} (y_i + y_{i-1}) \end{aligned}$$

effectively multiplying the result by dx. When x is specified, assuming x is also a one-dimensional tensor with elements ${x_0, x_1, ..., x_n}$, the computation becomes

$$\begin{aligned} \sum_{i = 1}^{n-1} \frac{(x_i - x_{i-1})}{2} (y_i + y_{i-1}) \end{aligned}$$

When y is two or more dimensions, this computation is performed independently along dimension dim. If x is also specified and is one-dimensional, then that dimension defines the spacing for each computation. If x is also specified and is not one-dimensional, then it is broadcast to the shape of y and the corresponding sizes are used for each computation. See the examples below for details.

Note

The trapezoidal rule is a technique for approximating the definite integral of a function by averaging its left and right Riemann sums. The approximation becomes more accurate as the resolution of the partition increases.

Parameters

• y (Tensor) – Values to use when computing the trapezoidal rule.

• x (Tensor) – If specified, defines spacing between values as specified above.

Keyword Arguments

• dx (float) – constant spacing between values. If neither x or dx are specified then this defaults to 1. Effectively multiplies the result by its value.

• dim (int) – The dimension along which to compute the trapezoidal rule. The last (inner-most) dimension by default.

Examples:

>>> # Computes the trapezoidal rule in 1D, spacing is implicitly 1
>>> y = torch.tensor([1, 5, 10])
>>> torch.trapezoid(y)
tensor(10.5)

>>> # Computes the same trapezoidal rule directly to verify
>>> (1 + 10 + 10) / 2
10.5

>>> # Computes the trapezoidal rule in 1D with constant spacing of 2
>>> # NOTE: the result is the same as before, but multiplied by 2
>>> torch.trapezoid(y, dx=2)
21.0

>>> # Computes the trapezoidal rule in 1D with arbitrary spacing
>>> x = torch.tensor([1, 3, 6])
>>> torch.trapezoid(y, x)
28.5

>>> # Computes the same trapezoidal rule directly to verify
>>> ((3 - 1) * (1 + 5) + (6 - 3) * (5 + 10)) / 2
28.5

>>> # Computes the trapezoidal rule for each row of a 3x3 matrix
>>> y = torch.arange(9).reshape(3, 3)
tensor([[0, 1, 2],
        [3, 4, 5],
        [6, 7, 8]])
>>> torch.trapezoid(y)
tensor([ 2., 8., 14.])

>>> # Computes the trapezoidal rule for each column of the matrix
>>> torch.trapezoid(y, dim=0)
tensor([ 6., 8., 10.])

>>> # Computes the trapezoidal rule for each row of a 3x3 ones matrix
>>> #   with the same arbitrary spacing
>>> y = torch.ones(3, 3)
>>> x = torch.tensor([1, 3, 6])
>>> torch.trapezoid(y, x)
array([5., 5., 5.])

>>> # Computes the trapezoidal rule for each row of a 3x3 ones matrix
>>> #   with different arbitrary spacing per row
>>> y = torch.ones(3, 3)
>>> x = torch.tensor([[1, 2, 3], [1, 3, 5], [1, 4, 7]])
>>> torch.trapezoid(y, x)
array([2., 4., 6.])