將如何理解 NumPy 的 ndarray 呢？簡短概略之 Quickstart tutorial 恐不足也︰

The Basics

NumPy’s main object is the homogeneous multidimensional array. It is a table of elements (usually numbers), all of the same type, indexed by a tuple of positive integers. In NumPy dimensions are called axes.

For example, the coordinates of a point in 3D space [1, 2, 1] has one axis. That axis has 3 elements in it, so we say it has a length of 3. In the example pictured below, the array has 2 axes. The first axis has a length of 2, the second axis has a length of 3.

[[ 1., 0., 0.],
 [ 0., 1., 2.]]

NumPy’s array class is called ndarray. It is also known by the alias array. Note that numpy.array is not the same as the Standard Python Library class array.array, which only handles one-dimensional arrays and offers less functionality. The more important attributes of anndarray object are:

ndarray.ndim

the number of axes (dimensions) of the array.

ndarray.shape

the dimensions of the array. This is a tuple of integers indicating the size of the array in each dimension. For a matrix with n rows and m columns, shape will be (n,m). The length of the shape tuple is therefore the number of axes, ndim.

ndarray.size

the total number of elements of the array. This is equal to the product of the elements of shape.

ndarray.dtype

an object describing the type of the elements in the array. One can create or specify dtype’s using standard Python types. Additionally NumPy provides types of its own. numpy.int32, numpy.int16, and numpy.float64 are some examples.

ndarray.itemsize

the size in bytes of each element of the array. For example, an array of elements of type float64 has itemsize 8 (=64/8), while one of type complex32 has itemsize 4 (=32/8). It is equivalent to ndarray.dtype.itemsize.

ndarray.data

the buffer containing the actual elements of the array. Normally, we won’t need to use this attribute because we will access the elements in an array using indexing facilities.

 

蓋需要深入認識陣列這個資料結構呦！

Array data structure

In computer science, an array data structure, or simply an array, is a data structure consisting of a collection of elements (values or variables), each identified by at least one array index or key. An array is stored such that the position of each element can be computed from its index tuple by a mathematical formula.^[1][2][3] The simplest type of data structure is a linear array, also called one-dimensional array.

For example, an array of 10 32-bit integer variables, with indices 0 through 9, may be stored as 10 words at memory addresses 2000, 2004, 2008, … 2036, so that the element with index i has the address 2000 + 4 × i.^[4]

The memory address of the first element of an array is called first address or foundation address.

Because the mathematical concept of a matrix can be represented as a two-dimensional grid, two-dimensional arrays are also sometimes called matrices. In some cases the term “vector” is used in computing to refer to an array, although tuples rather than vectors are the more mathematically correct equivalent. Tables are often implemented in the form of arrays, especially lookup tables; the word table is sometimes used as a synonym of array.

Arrays are among the oldest and most important data structures, and are used by almost every program. They are also used to implement many other data structures, such as lists and strings. They effectively exploit the addressing logic of computers. In most modern computers and many external storage devices, the memory is a one-dimensional array of words, whose indices are their addresses. Processors, especially vector processors, are often optimized for array operations.

Arrays are useful mostly because the element indices can be computed at run time. Among other things, this feature allows a single iterative statement to process arbitrarily many elements of an array. For that reason, the elements of an array data structure are required to have the same size and should use the same data representation. The set of valid index tuples and the addresses of the elements (and hence the element addressing formula) are usually,^[3][5] but not always,^[2] fixed while the array is in use.

The term array is often used to mean array data type, a kind of data type provided by most high-level programming languages that consists of a collection of values or variables that can be selected by one or more indices computed at run-time. Array types are often implemented by array structures; however, in some languages they may be implemented by hash tables, linked lists, search trees, or other data structures.

The term is also used, especially in the description of algorithms, to mean associative array or “abstract array”, a theoretical computer science model (an abstract data type or ADT) intended to capture the essential properties of arrays.

History

The first digital computers used machine-language programming to set up and access array structures for data tables, vector and matrix computations, and for many other purposes. John von Neumann wrote the first array-sorting program (merge sort) in 1945, during the building of the first stored-program computer.^[6]p. 159 Array indexing was originally done by self-modifying code, and later using index registers and indirect addressing. Some mainframes designed in the 1960s, such as the Burroughs B5000 and its successors, used memory segmentation to perform index-bounds checking in hardware.^[7]

Assembly languages generally have no special support for arrays, other than what the machine itself provides. The earliest high-level programming languages, including FORTRAN (1957), Lisp (1958), COBOL (1960), and ALGOL 60 (1960), had support for multi-dimensional arrays, and so has C (1972). In C++ (1983), class templates exist for multi-dimensional arrays whose dimension is fixed at runtime^[3][5] as well as for runtime-flexible arrays.^[2]

 

若說小小範例

 

裡的 reshape 費思量？

numpy.reshape

numpy.reshape(a, newshape, order=’C’)

Gives a new shape to an array without changing its data.

Parameters:

a : array_like Array to be reshaped. newshape : int or tuple of ints The new shape should be compatible with the original shape. If an integer, then the result will be a 1-D array of that length. One shape dimension can be -1. In this case, the value is inferred from the length of the array and remaining dimensions. order : {‘C’, ‘F’, ‘A’}, optional Read the elements of a using this index order, and place the elements into the reshaped array using this index order. ‘C’ means to read / write the elements using C-like index order, with the last axis index changing fastest, back to the first axis index changing slowest. ‘F’ means to read / write the elements using Fortran-like index order, with the first index changing fastest, and the last index changing slowest. Note that the ‘C’ and ‘F’ options take no account of the memory layout of the underlying array, and only refer to the order of indexing. ‘A’ means to read / write the elements in Fortran-like index order if a is Fortrancontiguous in memory, C-like order otherwise.

Returns:

reshaped_array : ndarray This will be a new view object if possible; otherwise, it will be a copy. Note there is no guarantee of thememory layout (C- or Fortran- contiguous) of the returned array.

See also

ndarray.reshape

Equivalent method.

 

怕不知

Element identifier and addressing formulas

When data objects are stored in an array, individual objects are selected by an index that is usually a non-negative scalar integer. Indexes are also called subscripts. An index maps the array value to a stored object.

There are three ways in which the elements of an array can be indexed:

• 0 (zero-based indexing): The first element of the array is indexed by subscript of 0.^[8]
• 1 (one-based indexing): The first element of the array is indexed by subscript of 1.
• n (n-based indexing): The base index of an array can be freely chosen. Usually programming languages allowing n-based indexing also allow negative index values and other scalar data types like enumerations, or characters may be used as an array index.

Using zero based indexing is design choice of many influential programming languages, including C, Java and Lisp. This leads to simpler implementation where the subscript refers to an offset from the starting position of an array, so the first element has an offset of zero.

Arrays can have multiple dimensions, thus it is not uncommon to access an array using multiple indices. For example, a two-dimensional array A with three rows and four columns might provide access to the element at the 2nd row and 4th column by the expressionA[1][3] in the case of a zero-based indexing system. Thus two indices are used for a two-dimensional array, three for a three-dimensional array, and n for an n-dimensional array.

The number of indices needed to specify an element is called the dimension, dimensionality, or rank of the array.

In standard arrays, each index is restricted to a certain range of consecutive integers (or consecutive values of some enumerated type), and the address of an element is computed by a “linear” formula on the indices.

One-dimensional arrays

A one-dimensional array (or single dimension array) is a type of linear array. Accessing its elements involves a single subscript which can either represent a row or column index.

As an example consider the C declaration int anArrayName[10];

Syntax : datatype anArrayname[sizeofArray];

In the given example the array can contain 10 elements of any value available to the int type. In C, the array element indices are 0-9 inclusive in this case. For example, the expressions anArrayName[0] and anArrayName[9] are the first and last elements respectively.

For a vector with linear addressing, the element with index i is located at the address B + c × i, where B is a fixed base address and c a fixed constant, sometimes called the address increment or stride.

If the valid element indices begin at 0, the constant B is simply the address of the first element of the array. For this reason, the C programming language specifies that array indices always begin at 0; and many programmers will call that element “zeroth” rather than “first”.

However, one can choose the index of the first element by an appropriate choice of the base address B. For example, if the array has five elements, indexed 1 through 5, and the base address B is replaced by B + 30c, then the indices of those same elements will be 31 to 35. If the numbering does not start at 0, the constant B may not be the address of any element.

Multidimensional arrays

For a multidimensional array, the element with indices i,j would have address B + c · i + d · j, where the coefficients c and d are the row and column address increments, respectively.

More generally, in a k-dimensional array, the address of an element with indices i₁, i₂, …, i_k is

B + c₁ · i₁ + c₂ · i₂ + … + c_k · i_k.

For example: int a[2][3];

This means that array a has 2 rows and 3 columns, and the array is of integer type. Here we can store 6 elements they will be stored linearly but starting from first row linear then continuing with second row. The above array will be stored as a₁₁, a₁₂, a₁₃, a₂₁, a₂₂, a₂₃.

This formula requires only k multiplications and k additions, for any array that can fit in memory. Moreover, if any coefficient is a fixed power of 2, the multiplication can be replaced by bit shifting.

The coefficients c_k must be chosen so that every valid index tuple maps to the address of a distinct element.

If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero. As in the one-dimensional case, the element indices may be changed by changing the base address B. Thus, if a two-dimensional array has rows and columns indexed from 1 to 10 and 1 to 20, respectively, then replacing B by B + c₁ – − 3 c₁ will cause them to be renumbered from 0 through 9 and 4 through 23, respectively. Taking advantage of this feature, some languages (like FORTRAN 77) specify that array indices begin at 1, as in mathematical tradition while other languages (like Fortran 90, Pascal and Algol) let the user choose the minimum value for each index.

………

Compact layouts

Often the coefficients are chosen so that the elements occupy a contiguous area of memory. However, that is not necessary. Even if arrays are always created with contiguous elements, some array slicing operations may create non-contiguous sub-arrays from them.

Illustration of row- and column-major order

There are two systematic compact layouts for a two-dimensional array. For example, consider the matrix

In the row-major order layout (adopted by C for statically declared arrays), the elements in each row are stored in consecutive positions and all of the elements of a row have a lower address than any of the elements of a consecutive row:

1

2

3

4

5

6

7

8

9

In column-major order (traditionally used by Fortran), the elements in each column are consecutive in memory and all of the elements of a column have a lower address than any of the elements of a consecutive column:

1

4

7

2

5

8

3

6

9

For arrays with three or more indices, “row major order” puts in consecutive positions any two elements whose index tuples differ only by one in the last index. “Column major order” is analogous with respect to the first index.

In systems which use processor cache or virtual memory, scanning an array is much faster if successive elements are stored in consecutive positions in memory, rather than sparsely scattered. Many algorithms that use multidimensional arrays will scan them in a predictable order. A programmer (or a sophisticated compiler) may use this information to choose between row- or column-major layout for each array. For example, when computing the product A·B of two matrices, it would be best to have A stored in row-major order, and B in column-major order.

 

生疑惑哩☂

 

而後才好了陣列運算之基本矣☆

Basic Operations

Arithmetic operators on arrays apply elementwise. A new array is created and filled with the result.

 

Unlike in many matrix languages, the product operator * operates elementwise in NumPy arrays. The matrix product can be performed using the @ operator (in python >=3.5) or the dot function or method: