樹莓派、樹莓派之學習、樹莓派之教育

數字派 NumPy ︰陣列運算《三》

仔細閱讀清晰思考『普及函數』

Universal functions (ufunc)

A universal function (or ufunc for short) is a function that operates on ndarrays in an element-by-element fashion, supporting array broadcasting, type casting, and several other standard features. That is, a ufunc is a “vectorized” wrapper for a function that takes a fixed number of specific inputs and produces a fixed number of specific outputs.

In NumPy, universal functions are instances of the numpy.ufunc class. Many of the built-in functions are implemented in compiled C code. The basic ufuncs operate on scalars, but there is also a generalized kind for which the basic elements are sub-arrays (vectors, matrices, etc.), and broadcasting is done over other dimensions. One can also produce custom ufunc instances using the frompyfunc factory function.

Broadcasting

Each universal function takes array inputs and produces array outputs by performing the core function element-wise on the inputs (where an element is generally a scalar, but can be a vector or higher-order sub-array for generalized ufuncs). Standard broadcasting rules are applied so that inputs not sharing exactly the same shapes can still be usefully operated on. Broadcasting can be understood by four rules:

  1. All input arrays with ndim smaller than the input array of largest ndim, have 1’s prepended to their shapes.
  2. The size in each dimension of the output shape is the maximum of all the input sizes in that dimension.
  3. An input can be used in the calculation if its size in a particular dimension either matches the output size in that dimension, or has value exactly 1.
  4. If an input has a dimension size of 1 in its shape, the first data entry in that dimension will be used for all calculations along that dimension. In other words, the stepping machinery of the ufunc will simply not step along that dimension (the stride will be 0 for that dimension).

Broadcasting is used throughout NumPy to decide how to handle disparately shaped arrays; for example, all arithmetic operations (+,-, *, …) between ndarrays broadcast the arrays before operation.

A set of arrays is called “broadcastable” to the same shape if the above rules produce a valid result, i.e., one of the following is true:

  1. The arrays all have exactly the same shape.
  2. The arrays all have the same number of dimensions and the length of each dimensions is either a common length or 1.
  3. The arrays that have too few dimensions can have their shapes prepended with a dimension of length 1 to satisfy property 2.

Example

If a.shape is (5,1), b.shape is (1,6), c.shape is (6,) and d.shape is () so that d is a scalar, then a, b, c, and d are all broadcastable to dimension (5,6); and

  • a acts like a (5,6) array where a[:,0] is broadcast to the other columns,
  • b acts like a (5,6) array where b[0,:] is broadcast to the other rows,
  • c acts like a (1,6) array and therefore like a (5,6) array where c[:] is broadcast to every row, and finally,
  • d acts like a (5,6) array where the single value is repeated.

………

 

文本內容，然後藉著

numpy.vectorize

class numpy.vectorize(pyfunc, otypes=None, doc=None, excluded=None, cache=False, signature=None)
 

Generalized function class.

Define a vectorized function which takes a nested sequence of objects or numpy arrays as inputs and returns a single numpy array or a tuple of numpy arrays. The vectorized function evaluates pyfunc over successive tuples of the input arrays like the python map function, except it uses the broadcasting rules of numpy.

The data type of the output of vectorized is determined by calling the function with the first element of the input. This can be avoided by specifying the otypes argument.

Parameters:
pyfunc : callable

A python function or method.

otypes : str or list of dtypes, optional

The output data type. It must be specified as either a string of typecode characters or a list of data type specifiers. There should be one data type specifier for each output.

doc : str, optional

The docstring for the function. If None, the docstring will be the pyfunc.__doc__.

excluded : set, optional

Set of strings or integers representing the positional or keyword arguments for which the function will not be vectorized. These will be passed directly to pyfunc unmodified.

New in version 1.7.0.

cache : bool, optional

If True, then cache the first function call that determines the number of outputs if otypes is not provided.

New in version 1.7.0.

signature : string, optional

Generalized universal function signature, e.g., (m,n),(n)->(m) for vectorized matrix-vector multiplication. If provided, pyfunc will be called with (and expected to return) arrays with shapes given by the size of corresponding core dimensions. By default, pyfunc is assumed to take scalars as input and output.

New in version 1.12.0.
Returns:
vectorized : callable

Vectorized function.

See also

frompyfunc
Takes an arbitrary Python function and returns a ufunc

Notes

The vectorize function is provided primarily for convenience, not for performance. The implementation is essentially a for loop.

If otypes is not specified, then a call to the function with the first argument will be used to determine the number of outputs. The results of this call will be cached if cache is True to prevent calling the function twice. However, to implement the cache, the original function must be wrapped which will slow down subsequent calls, so only do this if your function is expensive.

The new keyword argument interface and excluded argument support further degrades performance.

References

[1] NumPy Reference, section Generalized Universal Function API.

 

玩味理蘊深化了解『向量化』，自可知

Nicolas P. Rougier 《From Python to Numpy》一書

第二章論旨也。

Introduction

Simple example

Note

You can execute any code below from the code folder using the regular python shell or from inside an IPython session or Jupyter notebook. In such a case, you might want to use the magic command %timeit instead of thecustom one I wrote.

Numpy is all about vectorization. If you are familiar with Python, this is the main difficulty you’ll face because you’ll need to change your way of thinking and your new friends (among others) are named “vectors”, “arrays”, “views” or “ufuncs”.

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