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scipy.differentiate.

# hessian

scipy.differentiate.hessian(f, x, *, tolerances=None, maxiter=10, order=8, initial_step=0.5, step_factor=2.0)

  Evaluate the Hessian of a function numerically.

  Parameters:
    f callable

      The function whose Hessian is desired. The signature must be:

          f(xi: ndarray) -> ndarray

      where each element of `xi` is a finite real. If the function to be differentiated accepts additional arguments, wrap it (e.g. using `functools.partial` or `lambda`) and pass the wrapped callable into `hessian`. f must not mutate the array `xi`. See Notes regarding vectorization and the dimensionality of the input and output.

    x float array_like

      Points at which to evaluate the Hessian. Must have at least one dimension. See Notes regarding the dimensionality and vectorization.

    tolerances dictionary of floats, optional

      Absolute and relative tolerances. Valid keys of the dictionary are:

      - `atol` - absolute tolerance on the derivative

      - `rtol` - relative tolerance on the derivative

      Iteration will stop when `res.error < atol + rtol * abs(res.df)`. The default atol is the smallest normal number of the appropriate dtype, and the default rtol is the square root of the precision of the appropriate dtype.

    order int, default: 8

      The (positive integer) order of the finite difference formula to be used. Odd integers will be rounded up to the next even integer.

    initial_step float, default: 0.5

      The (absolute) initial step size for the finite difference derivative approximation.

    step_factor float, default: 2.0

      The factor by which the step size is reduced in each iteration; i.e. the step size in iteration 1 is `initial_step/step_factor`. If `step_factor < 1`, subsequent steps will be greater than the initial step; this may be useful if steps smaller than some threshold are undesirable (e.g. due to subtractive cancellation error).

    maxiter int, default: 10

      The maximum number of iterations of the algorithm to perform. See Notes.

  Returns:
    res_RichResult

      An object similar to an instance of `scipy.optimize.OptimizeResult` with the following attributes. The descriptions are written as though the values will be scalars; however, if f returns an array, the outputs will be arrays of the same shape.

      successbool array

        `True` where the algorithm terminated successfully (status `0`); `False` otherwise.

      statusint array

        An integer representing the exit status of the algorithm.

        - `0` : The algorithm converged to the specified tolerances.

        - `-1` : The error estimate increased, so iteration was terminated.

        - `-2` : The maximum number of iterations was reached.

        - `-3` : A non-finite value was encountered.

      ddffloat array

        The Hessian of f at x, if the algorithm terminated successfully.

      errorfloat array

        An estimate of the error: the magnitude of the difference between the current estimate of the Hessian and the estimate in the previous iteration.

      nfevint array

        The number of points at which f was evaluated.

      Each element of an attribute is associated with the corresponding element of ddf. For instance, element `[i, j]` of nfev is the number of points at which f was evaluated for the sake of computing element `[i, j]` of ddf.

  See also

  `derivative`, `jacobian`

  Notes

  Suppose we wish to evaluate the Hessian of a function \(f: \mathbf{R}^m \rightarrow \mathbf{R}\), and we assign to variable `m` the positive integer value of \(m\). If we wish to evaluate the Hessian at a single point, then:

  - argument x must be an array of shape `(m,)`

  - argument f must be vectorized to accept an array of shape `(m, ...)`. The first axis represents the \(m\) inputs of \(f\); the remaining axes indicated by ellipses are for evaluating the function at several abscissae in a single call.

  - argument f must return an array of shape `(...)`.

  - attribute `dff` of the result object will be an array of shape `(m, m)`, the Hessian.

  This function is also vectorized in the sense that the Hessian can be evaluated at `k` points in a single call. In this case, x would be an array of shape `(m, k)`, f would accept an array of shape `(m, ...)` and return an array of shape `(...)`, and the `ddf` attribute of the result would have shape `(m, m, k)`. Note that the axis associated with the `k` points is included within the axes denoted by `(...)`.

  Currently, `hessian` is implemented by nesting calls to `jacobian`. All options passed to `hessian` are used for both the inner and outer calls with one exception: the rtol used in the inner `jacobian` call is tightened by a factor of 100 with the expectation that the inner error can be ignored. A consequence is that rtol should not be set less than 100 times the precision of the dtype of x; a warning is emitted otherwise.

  Array API Standard Support

  `hessian` has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable `SCIPY_ARRAY_API=1` and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

  Library | CPU | GPU
  --- | --- | ---
  NumPy | ✅ | n/a
  CuPy | n/a | ✅
  PyTorch | ✅ | ✅
  JAX | ⚠️ no JIT | ⚠️ no JIT
  Dask | ⛔ | n/a

  > See Support for the array API standard for more information.

  References

  [1]

  Hessian matrix, Wikipedia, https://en.wikipedia.org/wiki/Hessian_matrix

  Examples

  The Rosenbrock function maps from \(\mathbf{R}^m \rightarrow \mathbf{R}\); the SciPy implementation `scipy.optimize.rosen` is vectorized to accept an array of shape `(m, ...)` and return an array of shape `...`. Suppose we wish to evaluate the Hessian at `[0.5, 0.5, 0.5]`.

      >>> import numpy as np
      >>> from scipy.differentiate import hessian
      >>> from scipy.optimize import rosen, rosen_hess
      >>> m = 3
      >>> x = np.full(m, 0.5)
      >>> res = hessian(rosen, x)
      >>> ref = rosen_hess(x)  # reference value of the Hessian
      >>> np.allclose(res.ddf, ref)
      True

  `hessian` is vectorized to evaluate the Hessian at multiple points in a single call.

      >>> rng = np.random.default_rng()
      >>> x = rng.random((m, 10))
      >>> res = hessian(rosen, x)
      >>> ref = [rosen_hess(xi) for xi in x.T]
      >>> ref = np.moveaxis(ref, 0, -1)
      >>> np.allclose(res.ddf, ref)
      True