mkoeppe
commented
4 years ago comment:1
Most of the time spent seems to come from SR (Maxima). Is it possible to speed things up for this easy special case, avoiding symbolic computation completely?
Closed mkoeppe closed 4 years ago
mkoeppe
commented
4 years ago Most of the time spent seems to come from SR (Maxima). Is it possible to speed things up for this easy special case, avoiding symbolic computation completely?
egourgoulhon
commented
4 years ago EuclideanSpace(n) creates the following objects:
E of dimension nX on E, representing the Cartesian coordinatesXg on E, with its components w.r.t. X initialized to diag(1,1,...,1) g, with its components w.r.t. X initialized to 0Most of the CPU time is spent in step 2, actually in creating the symbolic variables (elements of SR) representing the Cartesian coordinates. The latter operation is equivalent to
sage: def create_coords(n):
....: coords = SR.var(["x{}".format(i) for i in range(n)], domain='real')
....: for x in coords:
....: assume(x, 'real')and we have:
sage: %time EuclideanSpace(5)
CPU times: user 345 ms, sys: 28 ms, total: 373 ms
Wall time: 373 ms
5-dimensional Euclidean space E^5
sage: %time create_coords(5)
CPU times: user 456 ms, sys: 338 µs, total: 456 ms
Wall time: 402 mssage: %time EuclideanSpace(20)
CPU times: user 3.66 s, sys: 58.9 ms, total: 3.72 s
Wall time: 3.48 s
20-dimensional Euclidean space E^20
sage: %time create_coords(20)
CPU times: user 3.38 s, sys: 75.3 ms, total: 3.45 s
Wall time: 3.21 ssage: %time EuclideanSpace(40)
CPU times: user 12.6 s, sys: 253 ms, total: 12.8 s
Wall time: 11.9 s
40-dimensional EuclideanSR.var(["x{}".format(i) for i in range(n)], space E^40
sage: %time create_coords(40)
CPU times: user 12.2 s, sys: 226 ms, total: 12.5 s
Wall time: 11.4 sActually, it turns out that what takes most of CPU time is demanding that the symbolic variables are real. Indeed, if we skip domain='real' and assume(x, 'real') (each uses roughly half of the CPU time and, although they look redundant, both are actually necessary for efficient simplifications), we get very small times:
sage: %time coords = SR.var(["x{}".format(i) for i in range(40)])
CPU times: user 336 µs, sys: 9 µs, total: 345 µs
Wall time: 350 µsI don't know why SR.var('x', domain='real') and assume(x, 'real') are so slow...
NB: in the current setting, chart coordinates are stored as SR variables; but since SymPy can be used as the symbolic backend on manifolds, via
sage: E = EuclideanSpace(5)
sage: E.set_calculus_method('sympy')chart coordinates could be stored as SymPy variables, instead of SR ones.
mkoeppe
commented
4 years ago Thanks very much for the explanation and analysis!
Replying to @egourgoulhon:
I don't know why
SR.var('x', domain='real')andassume(x, 'real')are so slow...
I have created #30065 for this
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
in the current setting, chart coordinates are stored as SR variables; but since SymPy can be used as the symbolic backend on manifolds, via
sage: E = EuclideanSpace(5) sage: E.set_calculus_method('sympy')chart coordinates could be stored as SymPy variables, instead of SR ones.
Thanks, I'll try this.
Actually, would it make sense (at least for simple cases) to have a calculus method that only uses Sage's rational functions?
mkoeppe
commented
4 years ago In a related direction, would it be of interest to have a category of "algebraic" differential manifolds, as a differential geometry view on real varieties and semialgebraic sets (and their boundaries)?
egourgoulhon
commented
4 years ago Replying to @mkoeppe:
Replying to @egourgoulhon:
in the current setting, chart coordinates are stored as SR variables; but since SymPy can be used as the symbolic backend on manifolds, via
sage: E = EuclideanSpace(5) sage: E.set_calculus_method('sympy')chart coordinates could be stored as SymPy variables, instead of SR ones.
Thanks, I'll try this.
Actually, would it make sense (at least for simple cases) to have a calculus method that only uses Sage's rational functions?
Yes, as soon as you don't have to take a square root, as for instance in evaluating the volume n-form on a pseudo-Riemannian manifold (aka Levi-Civita tensor). In particular, this would forbid the computation of the triple scalar product within spherical coordinates in the Euclidean 3-space.
Actually, it should be relatively easy to add a new calculus method, in addition to the two ones already implemented (SR and SymPy), via the classes CalculusMethod and ChartFunction.
egourgoulhon
commented
4 years ago Replying to @mkoeppe:
In a related direction, would it be of interest to have a category of "algebraic" differential manifolds, as a differential geometry view on real varieties and semialgebraic sets (and their boundaries)?
Yes, I think so.
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
Actually, would it make sense (at least for simple cases) to have a calculus method that only uses Sage's rational functions?
Yes, as soon as you don't have to take a square root, as for instance in evaluating the volume n-form on a pseudo-Riemannian manifold (aka Levi-Civita tensor). In particular, this would forbid the computation of the triple scalar product within spherical coordinates in the Euclidean 3-space.
Actually, it should be relatively easy to add a new calculus method, in addition to the two ones already implemented (SR and SymPy), via the classes CalculusMethod and ChartFunction.
Thanks! I have created #30070 for this.
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
Replying to @mkoeppe:
In a related direction, would it be of interest to have a category of "algebraic" differential manifolds, as a differential geometry view on real varieties and semialgebraic sets (and their boundaries)?
Yes, I think so.
OK, I have created #30069 for the case of real algebraic manifolds.
Does sage-manifolds currently implement manifolds with boundary, or is it planned to implement them?
egourgoulhon
commented
4 years ago Replying to @egourgoulhon:
Replying to @mkoeppe:
Actually, would it make sense (at least for simple cases) to have a calculus method that only uses Sage's rational functions?
Yes, as soon as you don't have to take a square root, as for instance in evaluating the volume n-form on a pseudo-Riemannian manifold (aka Levi-Civita tensor). In particular, this would forbid the computation of the triple scalar product within spherical coordinates in the Euclidean 3-space.
Actually, for spherical coordinates, there is already some issue with rational functions at the level of the metric tensor itself, since the components of the Euclidean metric involve the sine function.
egourgoulhon
commented
4 years ago Replying to @mkoeppe:
OK, I have created #30069 for the case of real algebraic manifolds.
Very good.
Does sage-manifolds currently implement manifolds with boundary, or is it planned to implement them?
No, manifolds with boundary are not implemented yet; it would be nice to have them.
mkoeppe
commented
4 years ago With #30065 and #30074:
sage: %prun EuclideanSpace(1000)
53513603 function calls (53508842 primitive calls) in 37.037 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
2002006 4.658 0.000 4.786 0.000 tensorfield.py:684(_del_restrictions)
3003010 3.263 0.000 3.766 0.000 tensorfield.py:667(_del_derived)
2002000 3.076 0.000 18.325 0.000 free_module_tensor.py:1191(_set_comp_unsafe)
2002000 3.047 0.000 25.990 0.000 tensorfield_paral.py:820(set_comp)
2002000 2.571 0.000 22.942 0.000 free_module_tensor.py:1263(set_comp)
2002000 2.478 0.000 7.194 0.000 comp.py:864(__setitem__)
2002000 2.147 0.000 2.147 0.000 free_module_tensor.py:1474(del_other_comp)
2003000 2.029 0.000 2.398 0.000 comp.py:616(_check_indices)
2002000 1.825 0.000 20.150 0.000 tensorfield_paral.py:730(_set_comp_unsafe)
2002006 1.697 0.000 9.256 0.000 tensorfield_paral.py:708(_del_derived)
1 1.532 1.532 36.171 36.171 free_module_basis.py:566(__init__)
1002000 1.095 0.000 1.108 0.000 scalarfield.py:1154(is_trivial_zero)
1001003 0.931 0.000 7.942 0.000 vectorfield.py:1612(_del_derived)
8071534 0.813 0.000 0.813 0.000 {built-in method builtins.isinstance}
1 0.722 0.722 15.342 15.342 free_module_basis.py:375(__init__)
1001003 0.675 0.000 0.675 0.000 vectorfield.py:292(_del_dependencies)
2004002 0.656 0.000 0.656 0.000 finite_rank_free_module.py:2037(irange)
1001000 0.603 0.000 5.144 0.000 diff_form.py:1241(_del_derived)
6800 0.532 0.000 0.598 0.000 maxima_lib.py:412(_eval_line)
2045310 0.378 0.000 0.378 0.000 {built-in method builtins.hasattr}
1001003 0.335 0.000 5.137 0.000 multivectorfield.py:966(_del_derived)
5005023 0.334 0.000 0.334 0.000 {method 'clear' of 'dict' objects}Description changed:
---
+++
@@ -1,19 +1,99 @@
+The n-dimensional Euclidean space is available in Sage in many variants. We investigate the speed of the most basic operation: Constructing the space.
+sage: VectorSpace(QQ, 5).category()
+Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)
+sage: %time VectorSpace(QQ, 5)
+CPU times: user 28 µs, sys: 9 µs, total: 37 µs
+Wall time: 40.1 µs
+Vector space of dimension 5 over Rational Field
+sage: %time VectorSpace(QQ, 80)
+CPU times: user 218 µs, sys: 1 µs, total: 219 µs
+Wall time: 223 µs
+Vector space of dimension 80 over Rational Field
+sage: %time VectorSpace(QQ, 1000)
+CPU times: user 208 µs, sys: 1 µs, total: 209 µs
+Wall time: 213 µs
+Vector space of dimension 1000 over Rational Field
+ + +
+sage: CombinatorialFreeModule(QQ, range(5)).category()
+Category of finite dimensional vector spaces with basis over Rational Field
+sage: %time CombinatorialFreeModule(QQ, range(5))
+CPU times: user 80 µs, sys: 0 ns, total: 80 µs
+Wall time: 86.1 µs
+Free module generated by {0, 1, 2, 3, 4} over Rational Field
+sage: %time CombinatorialFreeModule(QQ, range(80))
+CPU times: user 244 µs, sys: 10 µs, total: 254 µs
+Wall time: 259 µs
+Free module generated by {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79} over Rational Field
+sage: %time CombinatorialFreeModule(QQ, range(1000))
+CPU times: user 322 µs, sys: 26 µs, total: 348 µs
+Wall time: 354 µs
+ + + +
+sage: AffineSpace(RR, 5).category()
+Category of schemes over Real Field with 53 bits of precision
+
+sage: %time AffineSpace(RR, 5)
+CPU times: user 47 µs, sys: 1 µs, total: 48 µs
+Wall time: 51 µs
+Affine Space of dimension 5 over Real Field with 53 bits of precision
+sage: %time AffineSpace(RR, 80)
+CPU times: user 2 ms, sys: 39 µs, total: 2.04 ms
+Wall time: 2.04 ms
+Affine Space of dimension 80 over Real Field with 53 bits of precision
+sage: %time AffineSpace(RR, 1000)
+CPU times: user 62.8 ms, sys: 1.45 ms, total: 64.3 ms
+Wall time: 63.5 ms
+Affine Space of dimension 1000 over Real Field with 53 bits of precision
+ + + +
+sage: VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5)).category()
+Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)
+sage: %time VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5))
+CPU times: user 6.55 ms, sys: 666 µs, total: 7.22 ms
+Wall time: 6.88 ms
+Ambient quadratic space of dimension 5 over Rational Field
+Inner product matrix:
+[1 0 0 0 0]
+[0 1 0 0 0]
+[0 0 1 0 0]
+[0 0 0 1 0]
+[0 0 0 0 1]
+sage: %time VectorSpace(QQ, 80, inner_product_matrix=matrix.identity(80))
+CPU times: user 14.1 ms, sys: 439 µs, total: 14.6 ms
+Wall time: 14.4 ms
+Ambient quadratic space of dimension 80 over Rational Field
+Inner product matrix:
+sage: %time VectorSpace(QQ, 1000, inner_product_matrix=matrix.identity(1000))
+CPU times: user 1.44 s, sys: 34.4 ms, total: 1.47 s
+Wall time: 1.48 s
+Ambient quadratic space of dimension 1000 over Rational Field
+Inner product matrix: ...
+ + + +
+sage: EuclideanSpace(5).category()
+Category of smooth manifolds over Real Field with 53 bits of precision
+
sage: %time EuclideanSpace(5)
CPU times: user 177 ms, sys: 10.4 ms, total: 187 ms
Wall time: 196 ms
5-dimensional Euclidean space E^5
-sage: %time EuclideanSpace(20)
-CPU times: user 2.16 s, sys: 56 ms, total: 2.21 s
-Wall time: 2.08 s
-20-dimensional Euclidean space E^20
-sage: %time EuclideanSpace(40)
-CPU times: user 8.06 s, sys: 188 ms, total: 8.25 s
-Wall time: 7.84 s
-40-dimensional Euclidean space E^40
sage: %time EuclideanSpace(80)
CPU times: user 32.8 s, sys: 758 ms, total: 33.6 s
Wall time: 31.5 s
80-dimensional Euclidean space E^80
+
+
+
+
+Description changed:
---
+++
@@ -91,6 +91,8 @@
CPU times: user 32.8 s, sys: 758 ms, total: 33.6 s
Wall time: 31.5 s
80-dimensional Euclidean space E^80
+sage: %time EuclideanSpace(80)
+(timeout)Description changed:
---
+++
@@ -91,7 +91,7 @@
CPU times: user 32.8 s, sys: 758 ms, total: 33.6 s
Wall time: 31.5 s
80-dimensional Euclidean space E^80
-sage: %time EuclideanSpace(80)
+sage: %time EuclideanSpace(1000)
(timeout)Replying to @mkoeppe:
Regarding the new ticket description, I would not say that the first three examples, namely VectorSpace(QQ, 5), CombinatorialFreeModule(QQ, range(5)) and AffineSpace(RR, 5) do construct an Euclidean space, because their ouputs are not endowed with a scalar product, are they?
mkoeppe
commented
4 years ago Description changed:
---
+++
@@ -1,4 +1,6 @@
The n-dimensional Euclidean space is available in Sage in many variants. We investigate the speed of the most basic operation: Constructing the space.
+
+Spaces without scalar product:
sage: VectorSpace(QQ, 5).category() @@ -53,6 +55,8 @@
+Spaces with scalar product:
+sage: VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5)).category() Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)
Of course, I agree. The summary already showed the categories of the spaces, but I've added two headers to emphasize this point.
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
Replying to @mkoeppe:
Does sage-manifolds currently implement manifolds with boundary, or is it planned to implement them?
No, manifolds with boundary are not implemented yet; it would be nice to have them.
I have created #30080 for this
mkoeppe
commented
4 years ago Description changed:
---
+++
@@ -100,6 +100,26 @@+Scalar product without a space:
+
+ +sage: %time DiagonalQuadraticForm(QQ, [1]*5) +CPU times: user 60 µs, sys: 1e+03 ns, total: 61 µs +Wall time: 62.9 µs +Quadratic form in 5 variables over Rational Field with coefficients: +[ 1 0 0 0 0 ] +[ * 1 0 0 0 ] +[ * * 1 0 0 ] +[ * * * 1 0 ] +[ * * * * 1 ] +sage: %time DiagonalQuadraticForm(QQ, [1]*80) +CPU times: user 1.35 ms, sys: 31 µs, total: 1.39 ms +Wall time: 1.44 ms +Quadratic form in 80 variables over Rational Field with coefficients: +sage: %time DiagonalQuadraticForm(QQ, [1]*1000) +CPU times: user 144 ms, sys: 2.85 ms, total: 147 ms +Wall time: 147 ms +Quadratic form in 1000 variables over Rational Field with coefficients: +
-
Dependencies: #30065, #30074
mkoeppe
commented
4 years ago For further speedups of EuclideanSpace(1000), it seems one would need to work on
sage.tensor.modules.free_module_basis.FreeModuleCoBasis.__init__ and FreeModuleBasis.__init__. They dominate the computation now, using non-sparse operations, calling sage.manifolds.differentiable.tensorfield_paral.TensorFieldParal.set_comp 10002 times. Does it make sense to make this sparse in some way? Would major speedups be expected from cythonizing this module?
egourgoulhon
commented
4 years ago Replying to @mkoeppe:
For further speedups of
EuclideanSpace(1000), it seems one would need to work onsage.tensor.modules.free_module_basis.FreeModuleCoBasis.__init__andFreeModuleBasis.__init__. They dominate the computation now, using non-sparse operations, callingsage.manifolds.differentiable.tensorfield_paral.TensorFieldParal.set_comp10002 times. Does it make sense to make this sparse in some way?
Yes, actually in the current framework, any uninitialized component of a vector/tensor is considered to be zero. So in FreeModuleBasis.__init__, we could skip the loop initializing the components of the basis vector to zero:
for i in fmodule.irange():
v = fmodule.element_class(fmodule)
- for j in fmodule.irange():
- v.set_comp(self)[j] = fmodule._ring.zero()
v.set_comp(self)[i] = fmodule._ring.one()
vl.append(v)This would make the number of calls to TensorFieldParal.set_comp to be 1000, instead in of 10002 !
Similarly in FreeModuleCoBasis.__init_:
for i in self._fmodule.irange():
v = self._fmodule.linear_form()
- for j in self._fmodule.irange():
- v.set_comp(basis)[j] = 0
- v.set_comp(basis)[i] = 1
+ v.set_comp(basis)[i] = self._fmodule._ring.one()
vl.append(v)The above unnecessary initializations to zero have been implemented at the early stage of the manifolds project, when we were not certain to keep the storage convention of having uninitialized components to be zero. Given the heavy use of that convention now, it's pretty safe to remove these lines.
egourgoulhon
commented
4 years ago egourgoulhon
commented
4 years ago The changes suggested in comment:22 are implemented in the above branch.
New commits:
fccbf43 | Improve initialization of elements of a free module basis |
egourgoulhon
commented
4 years ago Commit: fccbf43
7ed8c4ca-6d56-4ae9-953a-41e42b4ed313
commented
4 years ago 7ed8c4ca-6d56-4ae9-953a-41e42b4ed313
commented
4 years ago Branch pushed to git repo; I updated commit sha1. New commits:
134da39 | sage.symbolic.assumptions.GenericDeclaration.assume: Validate against cached valid features first |
1fbfb68 | Update instead of overwriting |
a99aa3f | Merge branch 't/30065/faster_maxima_assumptions' into t/30061/public/manifolds/init_module_basis-30061 |
8b31261 | sage.symbolic.assumptions.GenericDeclaration: Make it a UniqueRepresentation |
51ea23d | sage.symbolic.assumptions.GenericDeclaration.assume: Check first if already in _assumptions |
9119e82 | sage.symbolic.assumptions._assumptions: Change from list to OrderDict |
cda81ff | sage.symbolic.assumptions.GenericDeclaration: Make it hashable by inheriting comparisons from UniqueRepresentation |
642836d | sage.symbolic.assumptions: Use dict instead of OrderedDict for _assumptions |
3ce0c15 | Merge branch 't/30074/even_faster_maxima_assumptions' into t/30061/public/manifolds/init_module_basis-30061 |
mkoeppe
commented
4 years ago Merged in the tickets listed as dependencies
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
The changes suggested in comment:22 are implemented in the above branch.
Great! Together with the merged tickets, this now gives:
sage: %time EuclideanSpace(5)
CPU times: user 4.87 ms, sys: 372 µs, total: 5.24 ms
Wall time: 4.93 ms
5-dimensional Euclidean space E^5
sage: %time EuclideanSpace(80)
CPU times: user 106 ms, sys: 4.52 ms, total: 110 ms
Wall time: 92 ms
80-dimensional Euclidean space E^80
sage: %time EuclideanSpace(1000)
CPU times: user 1.04 s, sys: 33.5 ms, total: 1.07 s
Wall time: 986 msAnd some caching is happening too. The second time:
sage: %time EuclideanSpace(1000)
CPU times: user 207 ms, sys: 5.63 ms, total: 213 ms
Wall time: 212 ms
1000-dimensional Euclidean space E^1000That's faster than VectorSpace(QQ, 1000, inner_product_matrix=matrix.identity(1000))
mkoeppe
commented
4 years ago Most of the caching is coming from #30074. Making the real-declared symbolic variables known to Maxima is costly the first time; after that, it's free.
mkoeppe
commented
4 years ago First time:
sage: %prun EuclideanSpace(10000)
4614917 function calls (4585885 primitive calls) in 42.685 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
80096 35.217 0.000 35.779 0.000 maxima_lib.py:412(_eval_line)
25 0.529 0.021 0.529 0.021 {built-in method sage.libs.ecl.ecl_eval}
10000 0.425 0.000 39.373 0.004 {method 'var' of 'sage.symbolic.ring.SymbolicRing' objects}
20000 0.377 0.000 38.550 0.002 assumptions.py:213(assume)
20048 0.323 0.000 0.382 0.000 maxima_lib.py:274(max_to_string)
20006 0.275 0.000 0.288 0.000 free_module_tensor.py:258(__init__)
20000 0.208 0.000 0.460 0.000 assumptions.py:396(preprocess_assumptions)
56 0.207 0.004 0.207 0.004 {built-in method _imp.create_dynamic}
50000 0.190 0.000 0.190 0.000 latex.py:463(__add__)
209119 0.165 0.000 0.165 0.000 {built-in method builtins.hasattr}
20007 0.164 0.000 0.180 0.000 tensorfield_paral.py:689(_init_derived)
10032/1 0.139 0.000 42.050 42.050 {sage.misc.classcall_metaclass.typecall}
10139 0.129 0.000 0.130 0.000 {built-in method builtins.repr}
20001 0.125 0.000 0.143 0.000 interface.py:620(__getattr__)
714 0.119 0.000 0.119 0.000 {built-in method marshal.loads}
546190 0.114 0.000 0.114 0.000 {built-in method builtins.isinstance}
1 0.102 0.102 39.733 39.733 chart.py:1610(_init_coordinates)
10003 0.091 0.000 0.107 0.000 chart_func.py:318(__init__)
10000 0.091 0.000 0.460 0.000 {method '_maxima_init_' of 'sage.structure.sage_object.SageObject' objects}
40024 0.090 0.000 0.099 0.000 interface.py:491(_next_var_name)
20000 0.087 0.000 13.550 0.001 interface.py:587(function_call)
40024 0.086 0.000 15.460 0.000 maxima_lib.py:492(set)
30024 0.086 0.000 13.245 0.000 interface.py:251(__call__)
30024 0.082 0.000 13.139 0.000 interface.py:706(__init__)
30000 0.080 0.000 0.237 0.000 scalarfield.py:1154(is_trivial_zero)
20000 0.079 0.000 3.032 0.000 interface.py:534(_convert_args_kwds)
20024 0.078 0.000 16.639 0.001 interface.py:981(__del__)
20000 0.077 0.000 0.416 0.000 free_module_tensor.py:1191(_set_comp_unsafe)
160195 0.074 0.000 0.074 0.000 {method 'find' of 'str' objects}
30000 0.069 0.000 0.126 0.000 chart_func.py:810(is_trivial_zero)
10000 0.065 0.000 0.118 0.000 diff_form.py:1241(_del_derived)
10001 0.061 0.000 0.069 0.000 interface.py:419(_relation_symbols)
20048 0.059 0.000 0.059 0.000 {method 'python' of 'sage.libs.ecl.EclObject' objects}
90434 0.057 0.000 0.057 0.000 {method 'split' of 'str' objects}
20006 0.051 0.000 0.053 0.000 tensorfield.py:684(_del_restrictions)
30024 0.051 0.000 13.056 0.000 maxima_lib.py:561(_create)
56/50 0.049 0.001 0.054 0.001 {built-in method _imp.exec_dynamic}
20000 0.045 0.000 0.045 0.000 infinity.py:1020(gen)
30010 0.044 0.000 0.051 0.000 tensorfield.py:667(_del_derived)
20000 0.044 0.000 21.876 0.001 interface.py:653(__call__)
20001 0.044 0.000 0.070 0.000 vectorfield_module.py:1765(dual_exterior_power)
20000 0.042 0.000 0.251 0.000 comp.py:864(__setitem__)
30000 0.042 0.000 0.050 0.000 comp.py:616(_check_indices)
20000 0.041 0.000 0.043 0.000 interface.py:603(_function_call_string)
10001 0.040 0.000 0.332 0.000 vectorfield.py:1533(__init__)
10000 0.040 0.000 0.224 0.000 expression_conversions.py:155(__call__)
20024 0.039 0.000 16.518 0.001 maxima_lib.py:516(clear)Second time:
2221257 function calls (2221199 primitive calls) in 2.982 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
50000 0.297 0.000 0.297 0.000 latex.py:463(__add__)
30010 0.243 0.000 0.251 0.000 tensorfield.py:667(_del_derived)
20006 0.214 0.000 0.226 0.000 free_module_tensor.py:258(__init__)
10000 0.178 0.000 0.266 0.000 {method 'var' of 'sage.symbolic.ring.SymbolicRing' objects}
10003 0.091 0.000 0.107 0.000 chart_func.py:318(__init__)
30000 0.086 0.000 0.254 0.000 scalarfield.py:1154(is_trivial_zero)
20000 0.085 0.000 0.656 0.000 free_module_tensor.py:1191(_set_comp_unsafe)
10000 0.077 0.000 0.136 0.000 diff_form.py:1241(_del_derived)
130046 0.075 0.000 0.075 0.000 {built-in method builtins.hasattr}
30000 0.074 0.000 0.134 0.000 chart_func.py:810(is_trivial_zero)
20007 0.073 0.000 0.090 0.000 tensorfield_paral.py:689(_init_derived)
20000 0.068 0.000 0.085 0.000 assumptions.py:396(preprocess_assumptions)
20006 0.055 0.000 0.057 0.000 tensorfield.py:684(_del_restrictions)
20001 0.049 0.000 0.076 0.000 vectorfield_module.py:1765(dual_exterior_power)
270213 0.048 0.000 0.048 0.000 {built-in method builtins.isinstance}
30000 0.045 0.000 0.054 0.000 comp.py:616(_check_indices)
1 0.044 0.044 0.412 0.412 chart.py:1610(_init_coordinates)
20000 0.044 0.000 0.271 0.000 comp.py:864(__setitem__)
10001 0.042 0.000 0.276 0.000 vectorfield.py:1533(__init__)
30001 0.040 0.000 0.040 0.000 chart_func.py:460(expr)
20000 0.034 0.000 0.714 0.000 free_module_tensor.py:1263(set_comp)
10001 0.034 0.000 0.243 0.000 diff_form.py:1137(__init__)
20000 0.033 0.000 0.746 0.000 tensorfield_paral.py:820(set_comp)
30000 0.033 0.000 0.033 0.000 {method '_latex_' of 'sage.symbolic.expression.Expression' objects}
10001 0.033 0.000 0.059 0.000 free_module_alt_form.py:298(_new_comp)
30000 0.033 0.000 0.033 0.000 free_module_tensor.py:923(set_name)
10001 0.028 0.000 0.054 0.000 free_module_element.py:230(_new_comp)
10000 0.027 0.000 0.065 0.000 comp.py:3061(_ordered_indices)
20000 0.026 0.000 0.026 0.000 free_module_tensor.py:1474(del_other_comp)
10001 0.026 0.000 0.026 0.000 chart.py:311(<genexpr>)
10000 0.025 0.000 0.178 0.000 comp.py:4784(__setitem__)
120055 0.025 0.000 0.042 0.000 {built-in method builtins.len}
10001 0.024 0.000 0.024 0.000 euclidean.py:858(<genexpr>)
30010 0.024 0.000 0.024 0.000 tensorfield.py:654(_init_derived)
10001 0.023 0.000 0.023 0.000 vectorframe.py:1764(<genexpr>)
10001 0.023 0.000 0.038 0.000 vectorfield_module.py:1709(exterior_power)
3 0.022 0.007 0.339 0.113 free_module_basis.py:208(set_name)
20005 0.022 0.000 0.052 0.000 comp.py:496(__init__)
1 0.022 0.022 1.688 1.688 free_module_basis.py:566(__init__)
20006 0.022 0.000 0.318 0.000 tensorfield_paral.py:708(_del_derived)First time profile, filtered to methods in maxima_lib and sage.libs.ecl:
sage: %prun EuclideanSpace(5000)
40096 7.970 0.000 8.203 0.000 maxima_lib.py:412(_eval_line)
25 0.511 0.020 0.511 0.020 {built-in method sage.libs.ecl.ecl_eval}
10048 0.133 0.000 0.158 0.000 maxima_lib.py:274(max_to_string)
20024 0.037 0.000 3.505 0.000 maxima_lib.py:492(set)
15024 0.021 0.000 2.934 0.000 maxima_lib.py:561(_create)
10024 0.017 0.000 3.597 0.000 maxima_lib.py:516(clear)
40096 0.015 0.000 0.017 0.000 maxima_lib.py:463(<listcomp>)
5023 0.007 0.000 0.275 0.000 maxima_lib.py:541(get)The time spent on maxima_lib can probably still be reduced a little by using ECL objects directly for "context", but I will not work on this at the moment.
mkoeppe
commented
4 years ago Replying to @mkoeppe:
The time spent on
maxima_libcan probably still be reduced a little by using ECL objects directly for "context", but I will not work on this at the moment.
This is #30086 (just for saving the thought and some preliminary work on it).
mkoeppe
commented
4 years ago Author: Eric Gourgoulhon
mkoeppe
commented
4 years ago Next I think we can benchmark some other basic operations in the spaces
mkoeppe
commented
4 years ago Description changed:
---
+++
@@ -17,6 +17,23 @@
CPU times: user 208 µs, sys: 1 µs, total: 209 µs
Wall time: 213 µs
Vector space of dimension 1000 over Rational Field
+
+sage: for n in 5, 80, 1000, 10000, 100000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {})".format(n))); u = [ x for x in range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n); print("Distance: ", timeit("norm(V(u) - V(v))"))
+n = 5
+Construction: 625 loops, best of 3: 1.88 μs per loop
+Distance: 625 loops, best of 3: 75.6 μs per loop
+n = 80
+Construction: 625 loops, best of 3: 1.27 μs per loop
+Distance: 625 loops, best of 3: 280 μs per loop
+n = 1000
+Construction: 625 loops, best of 3: 1.26 μs per loop
+Distance: 125 loops, best of 3: 2.82 ms per loop
+n = 10000
+Construction: 625 loops, best of 3: 1.29 μs per loop
+Distance: 25 loops, best of 3: 27.9 ms per loop
+n = 100000
+Construction: 625 loops, best of 3: 1.25 μs per loop
+Distance: 5 loops, best of 3: 290 ms per loop@@ -80,6 +97,9 @@
Wall time: 1.48 s
Ambient quadratic space of dimension 1000 over Rational Field
Inner product matrix: ...
+
+sage: for n in 5, 80, 1000, 10000, 100000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {}, inner_product_matrix=matrix.identity({}))".format(n, n))); u = [ x for x in range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n, inner_product_matrix=matrix.identity(n)); print("Distance: ", timeit("norm(V(u) - V(v))"))
+Description changed:
---
+++
@@ -1,6 +1,6 @@
The n-dimensional Euclidean space is available in Sage in many variants. We investigate the speed of the most basic operation: Constructing the space.
-Spaces without scalar product:
+**Spaces without scalar product:**
sage: VectorSpace(QQ, 5).category() @@ -18,22 +18,26 @@ Wall time: 213 µs Vector space of dimension 1000 over Rational Field
-sage: for n in 5, 80, 1000, 10000, 100000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {})".format(n))); u = [ x for x in range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n); print("Distance: ", timeit("norm(V(u) - V(v))")) +sage: for n in 5, 80, 1000, 4000, 10000, 100000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {})".format(n),number=1,repeat=1)); u = [ x for x i +....: n range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n); print("Distance: ", timeit("norm(V(u) - V(v))")) n = 5 -Construction: 625 loops, best of 3: 1.88 μs per loop -Distance: 625 loops, best of 3: 75.6 μs per loop +Construction: 1 loop, best of 1: 56.1 ms per loop +Distance: 625 loops, best of 3: 81.7 μs per loop n = 80 -Construction: 625 loops, best of 3: 1.27 μs per loop -Distance: 625 loops, best of 3: 280 μs per loop +Construction: 1 loop, best of 1: 159 μs per loop +Distance: 625 loops, best of 3: 299 μs per loop n = 1000 -Construction: 625 loops, best of 3: 1.26 μs per loop -Distance: 125 loops, best of 3: 2.82 ms per loop +Construction: 1 loop, best of 1: 236 μs per loop +Distance: 125 loops, best of 3: 3.18 ms per loop +n = 4000 +Construction: 1 loop, best of 1: 147 μs per loop +Distance: 25 loops, best of 3: 11.3 ms per loop n = 10000 -Construction: 625 loops, best of 3: 1.29 μs per loop -Distance: 25 loops, best of 3: 27.9 ms per loop +Construction: 1 loop, best of 1: 149 μs per loop +Distance: 25 loops, best of 3: 28.7 ms per loop n = 100000 -Construction: 625 loops, best of 3: 1.25 μs per loop -Distance: 5 loops, best of 3: 290 ms per loop +Construction: 1 loop, best of 1: 162 μs per loop +Distance: 5 loops, best of 3: 296 ms per loop
@@ -72,7 +76,7 @@
-Spaces with scalar product:
+**Spaces with scalar product:**
sage: VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5)).category() @@ -98,10 +102,21 @@ Ambient quadratic space of dimension 1000 over Rational Field Inner product matrix: ...
+sage: for n in 5, 80, 1000, 4000: print("n = {}".format(n)); print("Construction: ", timeit("V = VectorSpace(QQ, {}, inner_product_matrix=matrix.identity({}))".format(n, n),number=1,repeat=1)); u = [ x for x in range(n) ]; v = [ x + 1 for x in range(n) ]; V = VectorSpace(QQ, n, inner_product_matrix=matrix.identity(n)); print("Distance: ", timeit("t = V(u)-V(v); sqrt(t.inner_product(t))")) +n = 5 +Construction: 1 loop, best of 1: 61.9 ms per loop +Distance: 625 loops, best of 3: 49.4 μs per loop +n = 80 +Construction: 1 loop, best of 1: 9.75 ms per loop +Distance: 625 loops, best of 3: 243 μs per loop +n = 1000 +Construction: 1 loop, best of 1: 1.51 s per loop +Distance: 25 loops, best of 3: 14.4 ms per loop +n = 4000 +Construction: 1 loop, best of 1: 23.8 s per loop +Distance: 5 loops, best of 3: 225 ms per loop
-
+NB: It is not in the category `MetricSpaces`, and thus the element methods `dist` and `abs` (?!) are missing... The methods `__abs__`, `norm`, and `dot_product` are unrelated to the inner product matrix; only `inner_product` uses `inner_product_matrix`.
sage: EuclideanSpace(5).category() @@ -120,7 +135,7 @@
-Scalar product without a space:
+**Scalar product without a space:**
sage: %time DiagonalQuadraticForm(QQ, [1]*5)
Does EuclideanSpace know how to compute geodesics, and thus distances?
egourgoulhon
commented
4 years ago Replying to @mkoeppe:
Does
EuclideanSpaceknow how to compute geodesics, and thus distances?
Yes it does, via IntegratedGeodesic. Note however that for Euclidean spaces, geodesics are trivial (straight lines).
egourgoulhon
commented
4 years ago By the way, an overview of EuclideanSpace capabilities is in this thematic tutorial; see also the associated notebooks and this section of the reference manual.
mkoeppe
commented
4 years ago Thanks for the pointers. It's on my list of things to learn more systematically about sage-manifolds.
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
Replying to @mkoeppe:
Does
EuclideanSpaceknow how to compute geodesics, and thus distances?Yes it does, via IntegratedGeodesic.
Should it (and presumably other manifold classes) be added to the category MetricSpaces then (#30062)?
egourgoulhon
commented
4 years ago Replying to @mkoeppe:
Should it (and presumably other manifold classes) be added to the category
MetricSpacesthen (#30062)?
Yes. It will be easy to implement the distance function by means of the Cartesian chart.
mkoeppe
commented
4 years ago Replying to @egourgoulhon:
Replying to @mkoeppe:
Should it (and presumably other manifold classes) be added to the category
MetricSpacesthen (#30062)?Yes. It will be easy to implement the distance function by means of the Cartesian chart.
Thanks. I have added this task to #30062.
mkoeppe
commented
4 years ago Should we set this ticket to "needs_review"?
The n-dimensional Euclidean space is available in Sage in many variants.
This ticket brings the speed of the most basic operation (constructing the space) of
EuclideanSpace(from sage-manifolds) closer to that of the other variants.Spaces without scalar product:
Spaces with scalar product:
NB: It is not in the category
MetricSpaces, and thus the element methodsdistandabs(?!) are missing... The methods__abs__,norm, anddot_productare unrelated to the inner product matrix; onlyinner_productusesinner_product_matrix.With this ticket (and its dependencies #30065, #30074):
Some caching is happening too. The second time:
Scalar product without a space:
Depends on #30065 Depends on #30074
CC: @egourgoulhon @tscrim @nbruin @mwageringel @mjungmath
Component: geometry
Author: Eric Gourgoulhon
Branch/Commit:
3ce0c15Reviewer: Matthias Koeppe
Issue created by migration from https://trac.sagemath.org/ticket/30061