Re: [Numpy-discussion] matrix power (was: matrix multiply)
On 06/04/2008, Anne Archibald [EMAIL PROTECTED] wrote: On 05/04/2008, Stéfan van der Walt [EMAIL PROTECTED] wrote: Some discussion recently took place around raising a square matrices to integer powers. See ticket #601: http://scipy.org/scipy/numpy/ticket/601 Anne Archibald wrote a patch which factored 'matrix_multiply' out of defmatrix (the matrix power implemented for the Matrix class). After some further discussion on irc, and some careful footwork so that everything imports correctly, I checked in http://projects.scipy.org/scipy/numpy/changeset/4968 The matrix_multiply method is exposed as numpy.linalg.matrix_multiply, and is not in the top-level numpy namespace (it's a bit crowded there already). I'd be glad if you would review the changeset and comment. Just to be clear, the function is called matrix_power, not matrix_multiply. Sorry, 05:00am is maybe not the time to write these posts. Now there is another good reason to review the changes :) Cheers Stéfan ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
[Numpy-discussion] Matrix powers
Hi all, I tried to use the new function matrix_power, but I can't find it. matrix_power(array([[0,1],[-1,0]]),10) Traceback (most recent call last): File stdin, line 1, in ? NameError: name 'matrix_power' is not defined numpy.__version__ '1.0.5.dev4968' Am I missing something ? Nils ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Final push for NumPy 1.0.5 (I need your help!)
OK, here's a patch for:
#718: Bug with numpy.float32.tolist
Can someone commit it (I hope someone has committed the other patches
i've sent)?
James
--- arrayobject.c.old 2008-04-06 13:08:37.0 +0100
+++ arrayobject.c 2008-04-06 13:10:57.0 +0100
@@ -1870,8 +1870,11 @@
if (!PyArray_Check(self)) return (PyObject *)self;
-if (self-nd == 0)
-return self-descr-f-getitem(self-data,self);
+if (self-nd == 0) {
+lp = PyList_New(1);
+PyList_SetItem(lp, 0, self-descr-f-getitem(self-data, self));
+return lp;
+}
sz = self-dimensions[0];
lp = PyList_New(sz);
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Re: [Numpy-discussion] Final push for NumPy 1.0.5 (I need your help!)
James Philbin wrote: OK, here's a patch for: #718: Bug with numpy.float32.tolist Can someone commit it (I hope someone has committed the other patches i've sent)? I don't think this patch should be committed without more discussion. This changes behavior and it is intentional that tolist behaves as it does now. -Travis O. ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Final push for NumPy 1.0.5 (I need your help!)
On Sun, 6 Apr 2008, James Philbin apparently wrote: OK, here's a patch for: #718: Bug with numpy.float32.tolist My impression has always been that to ensure a patch gets appropriate consideration it should be attached to a ticket... fwiw, Alan Isaac ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, 6 Apr 2008, Stéfan van der Walt apparently wrote:
I'd be glad if you would review the changeset and comment.
Just checking:
it's important to me that this won't change
the behavior of boolean matrices, but I don't
see a test for this. E.g., ::
import numpy as N
A = N.mat('1 0;1 1',dtype='bool')
A**2
matrix([[ True, False],
[ True, True]], dtype=bool)
Cheers,
Alan Isaac
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Re: [Numpy-discussion] matrix multiply
On 06/04/2008, Alan G Isaac [EMAIL PROTECTED] wrote:
Just checking:
it's important to me that this won't change
the behavior of boolean matrices, but I don't
see a test for this. E.g., ::
import numpy as N
A = N.mat('1 0;1 1',dtype='bool')
A**2
matrix([[ True, False],
[ True, True]], dtype=bool)
I have no desire to change the behaviour of boolean matrices, and I'll
write a test, but what is it supposed to do with them? Just produce
reduce(dot,[A]*n)? For zero it will give the identity, and for
negative powers some sort of floating-point inverse. Currently for
positive powers it should produce the right answer provided
multiplication is associative (which I think it is).
The inverse actually poses a problem: should it return (A**(-1))**n or
(A**n)**(-1)? (No, they're not the same for boolean matrices.)
Anne
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Re: [Numpy-discussion] matrix multiply
On 06/04/2008, Alan G Isaac [EMAIL PROTECTED] wrote:
Just checking:
it's important to me that this won't change
the behavior of boolean matrices, but I don't
see a test for this. E.g., ::
import numpy as N
A = N.mat('1 0;1 1',dtype='bool')
A**2
matrix([[ True, False],
[ True, True]], dtype=bool)
On Sun, 6 Apr 2008, Anne Archibald apparently wrote:
I have no desire to change the behaviour of boolean matrices, and I'll
write a test, but what is it supposed to do with them? Just produce
reduce(dot,[A]*n)?
That's the part I care about.
For zero it will give the identity,
Yes.
and for negative powers some sort of floating-point
inverse.
That deserves discussion.
Not all invertible boolean matrices have an inverse in the algebra.
Just the orthogonal ones do.
I guess I would special case inverses for Boolean matrices.
Just test if the matrix B is orthogonal (under Boolean
multiplication) and if so return B's transpose.
Currently for positive powers it should produce the right
answer provided multiplication is associative (which
I think it is).
Yes; N×N boolean matrices are I believe a semi-group under multiplication.
The inverse actually poses a problem: should it return
(A**(-1))**n or (A**n)**(-1)? (No, they're not the same
for boolean matrices.)
I think it must be the latter ... ?
By associativity, if B has an inverse A,
then B**n must have inverse A**n.
So you are observing that with boolean matrices
we might find B**n is invertible even though B is not.
Right? So the latter will work in more cases.
So again: form B**n, test for orthogonality,
and return the transpose if the test passes.
Cheers,
Alan Isaac
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Re: [Numpy-discussion] matrix multiply
On Sun, Apr 6, 2008 at 12:59 PM, Alan G Isaac [EMAIL PROTECTED] wrote:
On 06/04/2008, Alan G Isaac [EMAIL PROTECTED] wrote:
Just checking:
it's important to me that this won't change
the behavior of boolean matrices, but I don't
see a test for this. E.g., ::
import numpy as N
A = N.mat('1 0;1 1',dtype='bool')
A**2
matrix([[ True, False],
[ True, True]], dtype=bool)
On Sun, 6 Apr 2008, Anne Archibald apparently wrote:
I have no desire to change the behaviour of boolean matrices, and I'll
write a test, but what is it supposed to do with them? Just produce
reduce(dot,[A]*n)?
That's the part I care about.
For zero it will give the identity,
Yes.
and for negative powers some sort of floating-point
inverse.
That deserves discussion.
Not all invertible boolean matrices have an inverse in the algebra.
Just the orthogonal ones do.
I guess I would special case inverses for Boolean matrices.
Just test if the matrix B is orthogonal (under Boolean
multiplication) and if so return B's transpose.
Currently for positive powers it should produce the right
answer provided multiplication is associative (which
I think it is).
Yes; N×N boolean matrices are I believe a semi-group under multiplication.
The boolean algebra is a field and the correct addition is xor, which is
the same as addition modulo 2. This makes all matrices with determinant 1
invertible. This isn't the current convention, however, as it was when
Caratheodory was writing on measures and rings of sets were actually rings
and the symmetric difference was used instead of union.
Chuck
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Re: [Numpy-discussion] Final push for NumPy 1.0.5 (I need your help!)
On 06/04/2008, Alan G Isaac [EMAIL PROTECTED] wrote: On Sun, 6 Apr 2008, James Philbin apparently wrote: OK, here's a patch for: #718: Bug with numpy.float32.tolist My impression has always been that to ensure a patch gets appropriate consideration it should be attached to a ticket... And, more importantly, it should be accompanied by a test (I'm guilty on the float to string ticket mentioned above). The test not only verifies the patch's correct behaviour, but also guides the reviewer in understanding possible use cases, as well as illustrating calling convention. Anything that saves the reviewer time improves your chances of having the patch accepted. Finally, it is also considerate of his/her time: you wrote the patch and recently reviewed the code, so writing a test takes you much less time than it would the reviewer. That being said, I am very happy with the quality of patches recently submitted by, amongst others, Pauli, Anne, David and Christoph. Your efforts are very much appreciated. Regards Stéfan ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, 6 Apr 2008, Charles R Harris wrote: The boolean algebra is a field and the correct addition is xor, which is the same as addition modulo 2. This makes all matrices with determinant 1 invertible. This isn't the current convention, however, as it was when Caratheodory was writing on measures and rings of sets were actually rings and the symmetric difference was used instead of union. I am not sure what you are suggesting for matrix behavior, nor what correct means here. Comment: Standard *boolean algebra* axioms include distributivity, but 1 xor (0 and 0) = 1 xor 0 = 1 (1 xor 0) and (1 xor 0) = 1 and 1 = 1 So I guess (?) what you are saying is something like: if we have a boolen algebra with operators 'and' and 'or', we can generate a boolean ring with operations 'xor' and 'and'. When we do so, the '+' is traditionally used for the 'xor' operation. But where in the modern literature on boolean matrices is '+' given this interpretation? IANAM,* Alan Isaac * IANAM = I am not a mathematician. ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, Apr 6, 2008 at 2:34 PM, Alan G Isaac [EMAIL PROTECTED] wrote: On Sun, 6 Apr 2008, Charles R Harris wrote: The boolean algebra is a field and the correct addition is xor, which is the same as addition modulo 2. This makes all matrices with determinant 1 invertible. This isn't the current convention, however, as it was when Caratheodory was writing on measures and rings of sets were actually rings and the symmetric difference was used instead of union. I am not sure what you are suggesting for matrix behavior, nor what correct means here. Comment: Standard *boolean algebra* axioms include distributivity, but 1 xor (0 and 0) = 1 xor 0 = 1 (1 xor 0) and (1 xor 0) = 1 and 1 = 1 So I guess (?) what you are saying is something like: if we have a boolen algebra with operators 'and' and 'or', we can generate a boolean ring with operations 'xor' and 'and'. When we do so, the '+' is traditionally used for the 'xor' operation. But where in the modern literature on boolean matrices is '+' given this interpretation? It's generally not. It used to be that \Sigma and + were used for set union, probably because that was what the printers had on hand and what the mathematicians were used to. Then there was the alternate desire to make boolean algebra conform to the standad ring structure which led to using the symmetric difference, \Delta. For instance, if + is 'or', then 1 has no additive inverse, whereas 1 xor 1 = 0. I'm just pointing to some of the history here that I've noticed in old papers. I prefer the modern usage myself as it is closer to the accepted logic operations, but applying algebraic manipulations like powers and matrix inverses in that context leads to strange results. Chuck ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
[Numpy-discussion] New project : Spyke python-to-C compiler
Note this message has been posted to numpy-discussion and python-dev. Sorry for the multiple posting but I thought both python devs and numpy users will be interested. If you believe your list should not receive this email, let me know. Also I just wanted to introduce myself since I may ask doubts about Python and Numpy internals from time to time :) Hi. I am a student at Univ of Alberta doing my masters in computing science. I am writing a Python-to-C compiler as one part of my thesis. The compiler, named Spyke, will be made available in a couple of weeks and is geared towards scientific applications and will therefore focus mostly on needs of scientific app developers. What is Spyke? In many performance critical projects, it is often necessary to rewrite parts of the application in C. However writing C wrappers can be time consuming. Spyke offers an alternative approach. You add annotations to your Python code as strings. These strings are discarded by the Python interpreter but these are interpreted as types by Spyke compiler to convert to C. Example : int - int def f(x): return 2*x In this case the Spyke compiler will consider the string int - int as a decalration that the function accepts int as parameter and returns int. Spyke will then generate a C function and a wrapper function. This idea is directly copied from PLW (Python Language Wrapper) project. Once Python3k arrives, much of these declarations will be moved to function annotations and class decorators. This way you can do all your development and debugging interactively using the standard Python interpreter. When you need to compile to C, you just add type annotations to places that you want to convert and invoke spyke on the annotated module. This is different from Pyrex because Pyrex does not accept Python code. With Spyke, your code is 100% pure python. Spyke has basic support for functions and classes. Spyke can do very basic type inference for local variables in function bodies. Spyke also has partial support for homogenous lists and dictionaries and fixed length tuples. One big advantage of Spyke is that it understands at least part of numpy. Numpy arrays are treated as fundamental types and Spyke knows what C code to generate for slicing/indexing of numpy arrays etc. This should help a lot in scientific applications. Note that Spyke can handle only a subset of Python. Exceptions, iterators, generators, runtime code generation of any kind etc is not handled. Nested functions will be added soon. I will definitely add some of these missing features based on what is actually required for real world Python codes. Currently if Spyke does not understand a function, it just leaves it as Python code. Classes can be handled but special methods are not currently supported. The support of classes is a little brittle because I am trying to resolve some issues b/w old and new style of classes. Where is Spyke? Spyke will be available as a binary only release in a couple of weeks. I intend to make it open source after a few months. Spyke is written in Python and Java and should be platform independant. I do intend to make the source open in a few months. Right now its undergoing very rapid development and has negligible amounts of documentation so the source code right now is pretty useless to anyone else anyway. I need help: However I need a bit of help. I am having a couple of problems : a) I am finding it hard to get pure Python+NumPy testing codes. I need more codes to test the compiler. Developing a compiler without a test-suite is kind of useless. If you have some pure Python codes which need better performance, please contact me. I guarantee that your codes will not be released to public without your permission but might be referenced in academic publications. I can also make the compiler available to you hopefully after 10th of April. Its kind of unstable currently. I will also need your help in annotating the provided testing codes since I probably wont know what your application is doing. b) Libraries which interface with C/C++ : Many codes in SciPy for instance have mixed language codes. Part of the code is written in C/C++. Spyke only knows how to annotated Python codes. For C/C++ libraries wrapped into Python modules, Spyke will therefore need to know at least 2 things : i) The mapping of a C function name/struct etc to Python ii) The type information of the said C function. There are many many ways that people interact with C code. People either write wrappers manually, or use autogenerated wrappers using SWIG or SIP Boost.Python etc., use Pyrex or Cython while some people use ctypes. I dont have the time or resources to support these multitude of methods. I considered trying to parse the C code implementing wrappers but its non-trivial to put it mildly. Parsing only SWIG generated code is another possibility but its still hard. Another approach that I am
Re: [Numpy-discussion] matrix multiply
and for negative powers some sort of floating-point inverse. That deserves discussion. Not all invertible boolean matrices have an inverse in the algebra. Just the orthogonal ones do. I guess I would special case inverses for Boolean matrices. Just test if the matrix B is orthogonal (under Boolean multiplication) and if so return B's transpose. This is actually a limitation of the whole linear algebra subsystem: we only support floating-point linear algebra (apart from dot()). There are algorithms for doing integer linear algebra, which might be interesting to implement. But that's a big job. Boolean matrices as you use them are another step again: because they are not a group under +, almost all of linear algebra has to be reinterpreted. For example it's not obvious that matrix multiplication is associative; it turns out to be, because you can replace the matrices by integer matrices containing ones for True and zeros for False, then do the math, then interpret any nonzero integer as True, and zero as False. As an aside, if you use + to mean xor (which I am not suggesting!) all of linear algebra continues more or less unchanged; you're just working over the field of integers modulo 2. If you want eigenvalues you can pass to an algebraic closure. The inverse actually poses a problem: should it return (A**(-1))**n or (A**n)**(-1)? (No, they're not the same for boolean matrices.) I think it must be the latter ... ? By associativity, if B has an inverse A, then B**n must have inverse A**n. So you are observing that with boolean matrices we might find B**n is invertible even though B is not. Right? So the latter will work in more cases. So again: form B**n, test for orthogonality, and return the transpose if the test passes. Well, as it stands, since we have no integer linear algebra, inverses are always floating-point. When you do that, you find that, for example, ([[True,False],[True,True]]**(-1))**2 = [[1.,0.],[-2.,1.]] but ([[True,False],[True,True]]**2)**(-1) = [[1.,0.],[-1.,1.]] I am not aware of any algorithm for finding inverses, or even determining which matrices are invertible, in the peculiar Boolean arithmetic we use. Anne ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] Ticket #605 Incorrect behavior of numpy.histogram
On Apr 5, 2008, at 2:01 PM, Bruce Southey wrote: Hi, I have been investigating Ticket #605 'Incorrect behavior of numpy.histogram' (http://scipy.org/scipy/numpy/ticket/605 ). I think that my preference depends on the definition of what the bin number means. If the bin numbers are the lower bounds of the bins (numpy default behavior) then it would make little sense to exclude anything above the largest bin. I don't have access to numpy on my laptop at the moment, so I can't remember wether numpy has a keyword for what the bins array is defining? Having this as a keyword of (lower,middle,upper) of the bin would be very helpful. Cheers Tommy ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, 6 Apr 2008, Anne Archibald apparently wrote: I am not aware of any algorithm for finding inverses, or even determining which matrices are invertible, in the peculiar Boolean arithmetic we use. Again, it is *not* peculiar, it is very standard for boolean matrices. And with this behavior, a nonnegative integer power has an obvious graph theoretic interpretation. Such boolean matrices are obviously invertible if they are orthogonal. It turn out this is a necessary condition as well. [1]_ Orthogonality is obviously easy to test. Cheers, Alan Isaac .. [1] Luce, D., 1952, A Note on Boolean Matrix Theory, Proceeding of the Am. Math. Soc 3(3), p.382-8. ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, 6 Apr 2008, Charles R Harris apparently wrote: I prefer the modern usage myself as it is closer to the accepted logic operations, but applying algebraic manipulations like powers and matrix inverses in that context leads to strange results. I have not really thought much about inverses, but nonnegative integer powers have a natural interpretation in graph theory (with the boolean algebra operations, not the boolean ring operations). This is exactly what I was requesting be preserved. Cheers, Alan Isaac ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
[Numpy-discussion] ma's stdu and varu
Hi, I was just going through tidying up the documentation for all the many functions in numpy that compute standard deviations or variances (the functions, the methods, the methods on matrices, the methods on maskedarrays, all needed their docstrings updated in approximately the same way). I noticed that maskedarrays seem to have the functions stdu and varu, for computing the unbiased standard deviation and variance (where one divides by N-1 rather than N). These are now available, I think, via std and var with ddof=1. More seriously, they still provide the peculiar older definition of var, where varu([1,1.j,-1,-1.j])==0. I don't use maskedarrays, and in fact I haven't been keeping track of what's going on with the two different implementations. So: what should be done about stdu and varu? Do the two implementations both need their definitions of std and var examined? Thanks, Anne ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, Apr 6, 2008 at 8:51 PM, Alan G Isaac [EMAIL PROTECTED] wrote: On Sun, 6 Apr 2008, Charles R Harris apparently wrote: I prefer the modern usage myself as it is closer to the accepted logic operations, but applying algebraic manipulations like powers and matrix inverses in that context leads to strange results. I have not really thought much about inverses, but nonnegative integer powers have a natural interpretation in graph theory (with the boolean algebra operations, not the boolean ring operations). This is exactly what I was requesting be preserved. You mean as edges in a directed graph? I suppose so, although graph algorithms tend to use different structures, lists of lists, trees, and such. I would think plain old integer matrices might actually carry more information; let me count the ways. And positive matrices have their own oddities. Hmm... I wonder what matrices over Z_2 mean in that context? Chuck ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] New project : Spyke python-to-C compiler
Rahul Garg wrote: Note this message has been posted to numpy-discussion and python-dev. Sorry for the multiple posting but I thought both python devs and numpy users will be interested. If you believe your list should not receive this email, let me know. Also I just wanted to introduce myself since I may ask doubts about Python and Numpy internals from time to time :) Hey Rahul, This is a very interesting project. I've been interested in something like this for a while. I'd love to see more about what you are doing. Best regards, -Travis Oliphant ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] New project : Spyke python-to-C compiler
What will be the licensing of this project? Do you know yet? I have a couple of comments because I've been thinking along these lines. What is Spyke? In many performance critical projects, it is often necessary to rewrite parts of the application in C. However writing C wrappers can be time consuming. Spyke offers an alternative approach. You add annotations to your Python code as strings. These strings are discarded by the Python interpreter but these are interpreted as types by Spyke compiler to convert to C. Example : int - int def f(x): return 2*x In this case the Spyke compiler will consider the string int - int as a decalration that the function accepts int as parameter and returns int. Spyke will then generate a C function and a wrapper function. What about the use of decorators in this case? Also, it would be great to be able to create ufuncs (and general numpy funcs) using this approach. A decorator would work well here as well. Where is Spyke? Spyke will be available as a binary only release in a couple of weeks. I intend to make it open source after a few months. I'd like to encourage you to make it available as open source as early as possible.I think you are likely to get help in ways you didn't expect. People are used to reading code, so even an alpha project can get help early. In fact given that you are looking for help. I think this may be the best way to get it. If you need help getting set up in terms of hosting it somewhere, I can help you do that. Spyke is written in Python and Java and should be platform independant. I do intend to make the source open in a few months. Right now its undergoing very rapid development and has negligible amounts of documentation so the source code right now is pretty useless to anyone else anyway. c) Strings as type declarations : Do you think I should use decorators instead at least for function type declarations? I think you should use decorators. That way you can work towards having the compiler embedded in the decorator and happen seamlessly without invoking a separte program (it just happens when the module is loaded -- a.l.a weave). Best regards, -Travis Oliphant ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] ma's stdu and varu
Anne Archibald wrote: Hi, I was just going through tidying up the documentation for all the many functions in numpy that compute standard deviations or variances (the functions, the methods, the methods on matrices, the methods on maskedarrays, all needed their docstrings updated in approximately the same way). I noticed that maskedarrays seem to have the functions stdu and varu, for computing the unbiased standard deviation and variance (where one divides by N-1 rather than N). These are now available, I think, via std and var with ddof=1. More seriously, they still provide the peculiar older definition of var, where varu([1,1.j,-1,-1.j])==0. I don't use maskedarrays, and in fact I haven't been keeping track of what's going on with the two different implementations. So: what should be done about stdu and varu? Do the two implementations both need their definitions of std and var examined? Yes, the masked arrays should be changed and stdu and varu eliminated. -Travis ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, 6 Apr 2008, Charles R Harris apparently wrote: You mean as edges in a directed graph? Yes. Naturally a boolean matrix is not the most compact representation of a directed graph, especially a sparse one. However it can be convenient. If B is a boolean matrix such that Bij=1 if there is and edge from i to j, then B**2 has unit entries where there is a path of length 2 from i to j. The transitive closure is similarly easy to represent (as a matrix power). Cheers, Alan Isaac ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
Re: [Numpy-discussion] matrix multiply
On Sun, Apr 6, 2008 at 10:38 PM, Alan G Isaac [EMAIL PROTECTED] wrote: On Sun, 6 Apr 2008, Charles R Harris apparently wrote: You mean as edges in a directed graph? Yes. Naturally a boolean matrix is not the most compact representation of a directed graph, especially a sparse one. However it can be convenient. I've had occasional thoughts of adding a computer science kit to scipy with equivalence relations, trees, tries, graphs, and other such things that come in handy for some sorts of problems. Chuck ___ Numpy-discussion mailing list Numpy-discussion@scipy.org http://projects.scipy.org/mailman/listinfo/numpy-discussion
