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.