-- Dr. Paul Barrett Space Telescope Science Institute Phone: 410-338-4475 ESS/Science Software Group FAX: 410-338-4767 Baltimore, MD 21218 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ PEP: 209 Title: Multi-dimensional Arrays Version: Author: barrett@stsci.edu (Paul Barrett), oliphant@ee.byu.edu (Travis Oliphant) Python-Version: 2.2 Status: Draft Type: Standards Track Created: 03-Jan-2001 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 temparory 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 registerd later when the C-extension module for an array-type is imported. The arguments of the regiser 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, resp. 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. Attibutes: 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 deepcopy 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 interpeted 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. Minimall 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 corcion 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 depricated 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 depricated. 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.