PEP: 209

Title: Multi-dimensional Arrays

Version: $Revision$

Last-Modified: $Date$

Author: [email protected] (Paul Barrett), [email protected] (Travis Oliphant)

Status: Withdrawn

Type: Standards Track

Created: 03-Jan-2001

Python-Version: 2.2

Post-History:

Abstract

This PEP proposes a redesign and re-implementation of the multi-

dimensional array module, Numeric, to make it easier to add new

features and functionality to the module. Aspects of Numeric 2

that will receive special attention are efficient access to arrays

exceeding a gigabyte in size and composed of inhomogeneous data

structures or records. The proposed design uses four Python

classes: ArrayType, UFunc, Array, and ArrayView; and a low-level

C-extension module, _ufunc, to handle the array operations

efficiently. In addition, each array type has its own C-extension

module which defines the coercion rules, operations, and methods

for that type. This design enables new types, features, and

functionality to be added in a modular fashion. The new version

will introduce some incompatibilities with the current Numeric.

Motivation

Multi-dimensional arrays are commonly used to store and manipulate

data in science, engineering, and computing. Python currently has

an extension module, named Numeric (henceforth called Numeric 1),

which provides a satisfactory set of functionality for users

manipulating homogeneous arrays of data of moderate size (of order

10 MB). For access to larger arrays (of order 100 MB or more) of

possibly inhomogeneous data, the implementation of Numeric 1 is

inefficient and cumbersome. In the future, requests by the

Numerical Python community for additional functionality is also

likely as PEPs 211: Adding New Linear Operators to Python, and

225: Elementwise/Objectwise Operators illustrate.

Proposal

This proposal recommends a re-design and re-implementation of

Numeric 1, henceforth called Numeric 2, which will enable new

types, features, and functionality to be added in an easy and

modular manner. The initial design of Numeric 2 should focus on

providing a generic framework for manipulating arrays of various

types and should enable a straightforward mechanism for adding new

array types and UFuncs. Functional methods that are more specific

to various disciplines can then be layered on top of this core.

This new module will still be called Numeric and most of the

behavior found in Numeric 1 will be preserved.

The proposed design uses four Python classes: ArrayType, UFunc,

Array, and ArrayView; and a low-level C-extension module to handle

the array operations efficiently. In addition, each array type

has its own C-extension module which defines the coercion rules,

operations, and methods for that type. At a later date, when core

functionality is stable, some Python classes can be converted to

C-extension types.

Some planned features are:

1. Improved memory usage

This feature is particularly important when handling large arrays

and can produce significant improvements in performance as well as

memory usage. We have identified several areas where memory usage

can be improved:

a. Use a local coercion model

Instead of using Python's global coercion model which creates

temporary arrays, Numeric 2, like Numeric 1, will implement a

local coercion model as described in PEP 208 which defers the

responsibility of coercion to the operator. By using internal

buffers, a coercion operation can be done for each array

(including output arrays), if necessary, at the time of the

operation. Benchmarks [1] have shown that performance is at

most degraded only slightly and is improved in cases where the

internal buffers are less than the L2 cache size and the

processor is under load. To avoid array coercion altogether,

C functions having arguments of mixed type are allowed in

Numeric 2.

b. Avoid creation of temporary arrays

In complex array expressions (i.e. having more than one

operation), each operation will create a temporary array which

will be used and then deleted by the succeeding operation. A

better approach would be to identify these temporary arrays

and reuse their data buffers when possible, namely when the

array shape and type are the same as the temporary array being

created. This can be done by checking the temporary array's

reference count. If it is 1, then it will be deleted once the

operation is done and is a candidate for reuse.

c. Optional use of memory-mapped files

Numeric users sometimes need to access data from very large

files or to handle data that is greater than the available

memory. Memory-mapped arrays provide a mechanism to do this

by storing the data on disk while making it appear to be in

memory. Memory- mapped arrays should improve access to all

files by eliminating one of two copy steps during a file

access. Numeric should be able to access in-memory and

memory-mapped arrays transparently.

d. Record access

In some fields of science, data is stored in files as binary

records. For example, in astronomy, photon data is stored as a

1 dimensional list of photons in order of arrival time. These

records or C-like structures contain information about the

detected photon, such as its arrival time, its position on the

detector, and its energy. Each field may be of a different

type, such as char, int, or float. Such arrays introduce new

issues that must be dealt with, in particular byte alignment

or byte swapping may need to be performed for the numeric

values to be properly accessed (though byte swapping is also

an issue for memory mapped data). Numeric 2 is designed to

automatically handle alignment and representational issues

when data is accessed or operated on. There are two

approaches to implementing records; as either a derived array

class or a special array type, depending on your point-of-

view. We defer this discussion to the Open Issues section.

2. Additional array types

Numeric 1 has 11 defined types: char, ubyte, sbyte, short, int,

long, float, double, cfloat, cdouble, and object. There are no

ushort, uint, or ulong types, nor are there more complex types

such as a bit type which is of use to some fields of science and

possibly for implementing masked-arrays. The design of Numeric 1

makes the addition of these and other types a difficult and

error-prone process. To enable the easy addition (and deletion)

of new array types such as a bit type described below, a re-design

of Numeric is necessary.

a. Bit type

The result of a rich comparison between arrays is an array of

boolean values. The result can be stored in an array of type

char, but this is an unnecessary waste of memory. A better

implementation would use a bit or boolean type, compressing

the array size by a factor of eight. This is currently being

implemented for Numeric 1 (by Travis Oliphant) and should be

included in Numeric 2.

3. Enhanced array indexing syntax

The extended slicing syntax was added to Python to provide greater

flexibility when manipulating Numeric arrays by allowing

step-sizes greater than 1. This syntax works well as a shorthand

for a list of regularly spaced indices. For those situations

where a list of irregularly spaced indices are needed, an enhanced

array indexing syntax would allow 1-D arrays to be arguments.

4. Rich comparisons

The implementation of PEP 207: Rich Comparisons in Python 2.1

provides additional flexibility when manipulating arrays. We

intend to implement this feature in Numeric 2.

5. Array broadcasting rules

When an operation between a scalar and an array is done, the

implied behavior is to create a new array having the same shape as

the array operand containing the scalar value. This is called

array broadcasting. It also works with arrays of lesser rank,

such as vectors. This implicit behavior is implemented in Numeric

1 and will also be implemented in Numeric 2.

Design and Implementation

The design of Numeric 2 has four primary classes:

1. ArrayType:

This is a simple class that describes the fundamental properties

of an array-type, e.g. its name, its size in bytes, its coercion

relations with respect to other types, etc., e.g.

> Int32 = ArrayType('Int32', 4, 'doc-string')

Its relation to the other types is defined when the C-extension

module for that type is imported. The corresponding Python code

is:

> Int32.astype[Real64] = Real64

This says that the Real64 array-type has higher priority than the

Int32 array-type.

The following attributes and methods are proposed for the core

implementation. Additional attributes can be added on an

individual basis, e.g. .bitsize or .bitstrides for the bit type.

Attributes:

.name: e.g. "Int32", "Float64", etc.

.typecode: e.g. 'i', 'f', etc.

(for backward compatibility)

.size (in bytes): e.g. 4, 8, etc.

.array_rules (mapping): rules between array types

.pyobj_rules (mapping): rules between array and python types

.doc: documentation string

Methods:

__init__(): initialization

__del__(): destruction

__repr__(): representation

C-API:

This still needs to be fleshed-out.

2. UFunc:

This class is the heart of Numeric 2. Its design is similar to

that of ArrayType in that the UFunc creates a singleton callable

object whose attributes are name, total and input number of

arguments, a document string, and an empty CFunc dictionary; e.g.

> add = UFunc('add', 3, 2, 'doc-string')

When defined the add instance has no C functions associated with

it and therefore can do no work. The CFunc dictionary is

populated or registered later when the C-extension module for an

array-type is imported. The arguments of the register method are:

function name, function descriptor, and the CUFunc object. The

corresponding Python code is

> add.register('add', (Int32, Int32, Int32), cfunc-add)

In the initialization function of an array type module, e.g.

Int32, there are two C API functions: one to initialize the

coercion rules and the other to register the CFunc objects.

When an operation is applied to some arrays, the __call__ method

is invoked. It gets the type of each array (if the output array

is not given, it is created from the coercion rules) and checks

the CFunc dictionary for a key that matches the argument types.

If it exists the operation is performed immediately, otherwise the

coercion rules are used to search for a related operation and set

of conversion functions. The __call__ method then invokes a

compute method written in C to iterate over slices of each array,

namely:

> _ufunc.compute(slice, data, func, swap, conv)

The 'func' argument is a CFuncObject, while the 'swap' and 'conv'

arguments are lists of CFuncObjects for those arrays needing pre-

or post-processing, otherwise None is used. The data argument is

a list of buffer objects, and the slice argument gives the number

of iterations for each dimension along with the buffer offset and

step size for each array and each dimension.

We have predefined several UFuncs for use by the __call__ method:

cast, swap, getobj, and setobj. The cast and swap functions do

coercion and byte-swapping, respectively and the getobj and setobj

functions do coercion between Numeric arrays and Python sequences.

The following attributes and methods are proposed for the core

implementation.

Attributes:

.name: e.g. "add", "subtract", etc.

.nargs: number of total arguments

.iargs: number of input arguments

.cfuncs (mapping): the set C functions

.doc: documentation string

Methods:

__init__(): initialization

__del__(): destruction

__repr__(): representation

__call__(): look-up and dispatch method

initrule(): initialize coercion rule

uninitrule(): uninitialize coercion rule

register(): register a CUFunc

unregister(): unregister a CUFunc

C-API:

This still needs to be fleshed-out.

3. Array:

This class contains information about the array, such as shape,

type, endian-ness of the data, etc.. Its operators, ' ', '-',

etc. just invoke the corresponding UFunc function, e.g.

> def __add__(self, other):

> return ufunc.add(self, other)

The following attributes, methods, and functions are proposed for

the core implementation.

Attributes:

.shape: shape of the array

.format: type of the array

.real (only complex): real part of a complex array

.imag (only complex): imaginary part of a complex array

Methods:

__init__(): initialization

__del__(): destruction

__repr_(): representation

__str__(): pretty representation

__cmp__(): rich comparison

__len__():

__getitem__():

__setitem__():

__getslice__():

__setslice__():

numeric methods:

copy(): copy of array

aslist(): create list from array

asstring(): create string from array

Functions:

fromlist(): create array from sequence

fromstring(): create array from string

array(): create array with shape and value

concat(): concatenate two arrays

resize(): resize array

C-API:

This still needs to be fleshed-out.

4. ArrayView

This class is similar to the Array class except that the reshape

and flat methods will raise exceptions, since non-contiguous

arrays cannot be reshaped or flattened using just pointer and

step-size information.

C-API:

This still needs to be fleshed-out.

5. C-extension modules:

Numeric2 will have several C-extension modules.

a. _ufunc:

The primary module of this set is the _ufuncmodule.c. The

intention of this module is to do the bare minimum,

i.e. iterate over arrays using a specified C function. The

interface of these functions is the same as Numeric 1, i.e.

int (*CFunc)(char *data, int *steps, int repeat, void *func);

and their functionality is expected to be the same, i.e. they

iterate over the inner-most dimension.

The following attributes and methods are proposed for the core

implementation.

Attributes:

Methods:

compute():

C-API:

This still needs to be fleshed-out.

b. _int32, _real64, etc.:

There will also be C-extension modules for each array type,

e.g. _int32module.c, _real64module.c, etc. As mentioned

previously, when these modules are imported by the UFunc

module, they will automatically register their functions and

coercion rules. New or improved versions of these modules can

be easily implemented and used without affecting the rest of

Numeric 2.

Open Issues

1. Does slicing syntax default to copy or view behavior?

The default behavior of Python is to return a copy of a sub-list

or tuple when slicing syntax is used, whereas Numeric 1 returns a

view into the array. The choice made for Numeric 1 is apparently

for reasons of performance: the developers wish to avoid the

penalty of allocating and copying the data buffer during each

array operation and feel that the need for a deep copy of an array

to be rare. Yet, some have argued that Numeric's slice notation

should also have copy behavior to be consistent with Python lists.

In this case the performance penalty associated with copy behavior

can be minimized by implementing copy-on-write. This scheme has

both arrays sharing one data buffer (as in view behavior) until

either array is assigned new data at which point a copy of the

data buffer is made. View behavior would then be implemented by

an ArrayView class, whose behavior be similar to Numeric 1 arrays,

i.e. .shape is not settable for non-contiguous arrays. The use of

an ArrayView class also makes explicit what type of data the array

contains.

2. Does item syntax default to copy or view behavior?

A similar question arises with the item syntax. For example, if a

= [[0,1,2], [3,4,5]] and b = a[0], then changing b[0] also changes

a[0][0], because a[0] is a reference or view of the first row of

a. Therefore, if c is a 2-d array, it would appear that c[i]

should return a 1-d array which is a view into, instead of a copy

of, c for consistency. Yet, c[i] can be considered just a

shorthand for c[i,:] which would imply copy behavior assuming

slicing syntax returns a copy. Should Numeric 2 behave the same

way as lists and return a view or should it return a copy.

3. How is scalar coercion implemented?

Python has fewer numeric types than Numeric which can cause

coercion problems. For example, when multiplying a Python scalar

of type float and a Numeric array of type float, the Numeric array

is converted to a double, since the Python float type is actually

a double. This is often not the desired behavior, since the

Numeric array will be doubled in size which is likely to be

annoying, particularly for very large arrays. We prefer that the

array type trumps the python type for the same type class, namely

integer, float, and complex. Therefore, an operation between a

Python integer and an Int16 (short) array will return an Int16

array. Whereas an operation between a Python float and an Int16

array would return a Float64 (double) array. Operations between

two arrays use normal coercion rules.

4. How is integer division handled?

In a future version of Python, the behavior of integer division

will change. The operands will be converted to floats, so the

result will be a float. If we implement the proposed scalar

coercion rules where arrays have precedence over Python scalars,

then dividing an array by an integer will return an integer array

and will not be consistent with a future version of Python which

would return an array of type double. Scientific programmers are

familiar with the distinction between integer and float-point

division, so should Numeric 2 continue with this behavior?

5. How should records be implemented?

There are two approaches to implementing records depending on your

point-of-view. The first is two divide arrays into separate

classes depending on the behavior of their types. For example,

numeric arrays are one class, strings a second, and records a

third, because the range and type of operations of each class

differ. As such, a record array is not a new type, but a

mechanism for a more flexible form of array. To easily access and

manipulate such complex data, the class is comprised of numeric

arrays having different byte offsets into the data buffer. For

example, one might have a table consisting of an array of Int16,

Real32 values. Two numeric arrays, one with an offset of 0 bytes

and a stride of 6 bytes to be interpreted as Int16, and one with an

offset of 2 bytes and a stride of 6 bytes to be interpreted as

Real32 would represent the record array. Both numeric arrays

would refer to the same data buffer, but have different offset and

stride attributes, and a different numeric type.

The second approach is to consider a record as one of many array

types, albeit with fewer, and possibly different, array operations

than for numeric arrays. This approach considers an array type to

be a mapping of a fixed-length string. The mapping can either be

simple, like integer and floating-point numbers, or complex, like

a complex number, a byte string, and a C-structure. The record

type effectively merges the struct and Numeric modules into a

multi-dimensional struct array. This approach implies certain

changes to the array interface. For example, the 'typecode'

keyword argument should probably be changed to the more

descriptive 'format' keyword.

a. How are record semantics defined and implemented?

Which ever implementation approach is taken for records, the

syntax and semantics of how they are to be accessed and

manipulated must be decided, if one wishes to have access to

sub-fields of records. In this case, the record type can

essentially be considered an inhomogeneous list, like a tuple

returned by the unpack method of the struct module; and a 1-d

array of records may be interpreted as a 2-d array with the

second dimension being the index into the list of fields.

This enhanced array semantics makes access to an array of one

or more of the fields easy and straightforward. It also

allows a user to do array operations on a field in a natural

and intuitive way. If we assume that records are implemented

as an array type, then last dimension defaults to 0 and can

therefore be neglected for arrays comprised of simple types,

like numeric.

6. How are masked-arrays implemented?

Masked-arrays in Numeric 1 are implemented as a separate array

class. With the ability to add new array types to Numeric 2, it

is possible that masked-arrays in Numeric 2 could be implemented

as a new array type instead of an array class.

7. How are numerical errors handled (IEEE floating-point errors in

particular)?

It is not clear to the proposers (Paul Barrett and Travis

Oliphant) what is the best or preferred way of handling errors.

Since most of the C functions that do the operation, iterate over

the inner-most (last) dimension of the array. This dimension

could contain a thousand or more items having one or more errors

of differing type, such as divide-by-zero, underflow, and

overflow. Additionally, keeping track of these errors may come at

the expense of performance. Therefore, we suggest several

options:

a. Print a message of the most severe error, leaving it to

the user to locate the errors.

b. Print a message of all errors that occurred and the number

of occurrences, leaving it to the user to locate the errors.

c. Print a message of all errors that occurred and a list of

where they occurred.

d. Or use a hybrid approach, printing only the most severe

error, yet keeping track of what and where the errors

occurred. This would allow the user to locate the errors

while keeping the error message brief.

8. What features are needed to ease the integration of FORTRAN

libraries and code?

It would be a good idea at this stage to consider how to ease the

integration of FORTRAN libraries and user code in Numeric 2.

Implementation Steps

1. Implement basic UFunc capability

a. Minimal Array class:

Necessary class attributes and methods, e.g. .shape, .data,

.type, etc.

b. Minimal ArrayType class:

Int32, Real64, Complex64, Char, Object

c. Minimal UFunc class:

UFunc instantiation, CFunction registration, UFunc call for

1-D arrays including the rules for doing alignment,

byte-swapping, and coercion.

d. Minimal C-extension module:

_UFunc, which does the innermost array loop in C.

This step implements whatever is needed to do: 'c = add(a, b)'

where a, b, and c are 1-D arrays. It teaches us how to add

new UFuncs, to coerce the arrays, to pass the necessary

information to a C iterator method and to do the actually

computation.

2. Continue enhancing the UFunc iterator and Array class

a. Implement some access methods for the Array class:

print, repr, getitem, setitem, etc.

b. Implement multidimensional arrays

c. Implement some of basic Array methods using UFuncs:

, -, *, /, etc.

d. Enable UFuncs to use Python sequences.

3. Complete the standard UFunc and Array class behavior

a. Implement getslice and setslice behavior

b. Work on Array broadcasting rules

c. Implement Record type

4. Add additional functionality

a. Add more UFuncs

b. Implement buffer or mmap access

Incompatibilities

The following is a list of incompatibilities in behavior between

Numeric 1 and Numeric 2.

1. Scalar coercion rules

Numeric 1 has single set of coercion rules for array and Python

numeric types. This can cause unexpected and annoying problems

during the calculation of an array expression. Numeric 2 intends

to overcome these problems by having two sets of coercion rules:

one for arrays and Python numeric types, and another just for

arrays.

2. No savespace attribute

The savespace attribute in Numeric 1 makes arrays with this

attribute set take precedence over those that do not have it set.

Numeric 2 will not have such an attribute and therefore normal

array coercion rules will be in effect.

3. Slicing syntax returns a copy

The slicing syntax in Numeric 1 returns a view into the original

array. The slicing behavior for Numeric 2 will be a copy. You

should use the ArrayView class to get a view into an array.

4. Boolean comparisons return a boolean array

A comparison between arrays in Numeric 1 results in a Boolean

scalar, because of current limitations in Python. The advent of

Rich Comparisons in Python 2.1 will allow an array of Booleans to

be returned.

5. Type characters are deprecated

Numeric 2 will have an ArrayType class composed of Type instances,

for example Int8, Int16, Int32, and Int for signed integers. The

typecode scheme in Numeric 1 will be available for backward

compatibility, but will be deprecated.

Appendices

A. Implicit sub-arrays iteration

A computer animation is composed of a number of 2-D images or

frames of identical shape. By stacking these images into a single

block of memory, a 3-D array is created. Yet the operations to be

performed are not meant for the entire 3-D array, but on the set

of 2-D sub-arrays. In most array languages, each frame has to be

extracted, operated on, and then reinserted into the output array

using a for-like loop. The J language allows the programmer to

perform such operations implicitly by having a rank for the frame

and array. By default these ranks will be the same during the

creation of the array. It was the intention of the Numeric 1

developers to implement this feature, since it is based on the

language J. The Numeric 1 code has the required variables for

implementing this behavior, but was never implemented. We intend

to implement implicit sub-array iteration in Numeric 2, if the

array broadcasting rules found in Numeric 1 do not fully support

this behavior.

Copyright

This document is placed in the public domain.

Related PEPs

PEP 207: Rich Comparisons

by Guido van Rossum and David Ascher

PEP 208: Reworking the Coercion Model

by Neil Schemenauer and Marc-Andre' Lemburg

PEP 211: Adding New Linear Algebra Operators to Python

by Greg Wilson

PEP 225: Elementwise/Objectwise Operators

by Huaiyu Zhu

PEP 228: Reworking Python's Numeric Model

by Moshe Zadka

References

[1] P. Greenfield 2000. private communication.

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