[Numpy-discussion] Cython numerical syntax revisited
Sturla Molden
sturla@molden...
Thu Mar 5 09:04:44 CST 2009
> A Thursday 05 March 2009, Dag Sverre Seljebotn escrigué:
> Good point. I was not aware of this subtlity. In fact, numexpr does
> not get well with transposed views of NumPy arrays. Filed the bug in:
>
> http://code.google.com/p/numexpr/issues/detail?id=18
>
>> Not sure whether it is possible with other kinds of views
>> (where e.g. a middle dimension varies fastest), but the NumPy model
>> doesn't preclude it and I suppose it would be possible with
>> stride_tricks.
>
> Middle dimensions varying first? Oh my! :)
I cannot see any obvious justification for letting the middle dimension
varying first.
C ordering is natural if we work with a "pointer to an array of pointers"
or an "array of arrays", which in both cases will be indexed as
array[i][j] in C:
array[i][j] = (array[i])[j]
= *(array[i]+j) = *(*(array+i)+j)
While C has arrays and pointers, the difference is almost never visible to
the programmer. This has lead some to erroneously believe that "C has no
arrays, only pointers". However:
double array[512];
double *p = array;
Now sizeof(array) will be sizeof(double)*512, whereas sizeof(p) will be
sizeof(long). This is one of very few cases where C arrays and pointers
behave differently, but demonstrates the existence of arrays in C.
The justification for Fortran ordering is in the mathematics. Say we have
a set of linear equations
A * X = B
and are going to solve for X, using some magical subroutine 'solve'. The
most efficient way to store these arrays becomes the Fortran ordering.
That is,
call solve(A, X, B)
will be mathematically equivalent to the loop
do i = i, n
call solve(A, X(:,i), B)
end do
All the arrays in the call to solve are still kept contigous! This would
not be the case with C ordering, which is an important reason that C sucks
so much for numerical computing. To write efficient linear algebra in C,
we have to store matrices in memory transposed to how they appear in
mathematical equations. In fact, Matlab uses Fortran ordering because of
this.
While C ordering feels natural to computer scientists, who loves the
beauty of pointer and array symmetries, it is a major obstacle for
scientists and engineers from other fields. It is perhaps the most
important reason why numerical code written in Fortran tend to be the
faster: If a matrix is rank n x m in the equation, it should be rank n x m
in the program as well, right? Not so with C. The better performance of
Fortran for numerical code is often blamed on pointer aliasing in C. I
believe wrong memory layout by the hands of the programmer is actually the
more important reason. In fact, whenever I have done comparisons, the C
compiler has always produced the faster machine code (gcc vs. gfortran or
g77, icc vs. ifort). But to avoid the pitfall, one must be aware of it.
And when a programmer's specialization is in another field, this is
usually not the case. Most scientists doing some casual C, Java or Python
programming fall straight into the trap. That is also why I personally
feel it was a bad choice to let C ordering be the default in NumPy.
S.M.
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