I will not answer your question since
That said, Sage has numerical capapilities. In particular, it is interfaced with low-level numeric libraries that are as fast as any C code you could write yourself. Somehow, the issue (which is also a strength) is that it also has other capabilities (symbolic, algebraic, ...) so that numerics is not the default, and if you are not careful, then you might work in some other framework and not benefit from numerical software, hence be slower. Here are two advices for numerical computation in Sage.
First advice, the real (or complex) numbers you are working with should explicitely belong to RDF, the Real Double Field, that is, the floating-point numbers that the CPU can handle directly (i mean, not emulated). The default floatin-poing numbers in Sage are emulated with MPFR.
A simpe example:
If you write:
sage: a = 1/10
You define a rational number:
sage: a.parent()
Rational Field
If you write:
sage: a = 0.1
You define a floating-point number:
sage: a.parent()
Real Field with 53 bits of precision
BUT the computations with those numbers (e.g. multiplication) are emulated and do not rely on the CPU floating-point arithmetics. To use such hardware implementation, you need to specify:
sage: a = RDF(0.1)
sage: a.parent()
Real Double Field
Here is a quick benchmark to be explicit, what is going on if you want to elevate a 100x100 matrix to the power 100:
With rational numbers:
sage: d = 100
sage: M = matrix(QQ,d,d)
sage: for i in range(d):
....: for j in range(d):
....: M[i,j] = QQ.random_element(distribution='1/n')
sage: %time m=M^100
CPU times: user 15.9 s, sys: 36 ms, total: 15.9 s
Wall time: 15.9 s
This is pretty slow, but actually not ridiculous. Note that the computation uses the blazing fast FLINT library for dealing with rational matrices:
sage: from sage.misc.citation import get_systems
sage: get_systems('M^100')
['FLINT']
However, as it is an exact algebraic computation, the rational numbers that are involved along the computation become very huge, which explains the slowness:
sage: m[42,42]
-143403877757962034908229648979166715246023070908121800149357850500509382879155469212403509993324936983836034231601408025604512188042155054510185506983671055836518268283296097981910299096285130974077147512218469448356315775614176654302917462135918450328466624682433535244279651136503200572437791427649637495142716599330424756533649845742064881459812701278305804645425047594913762093446441972957817257126514503143050407460705950011710084565531827542704863095434609967927914423579742343869924125225487620283312414413274849013393390983736909801421258757308472686338302240842651126642009736715824530821318978456872448162954996631679420804303636154872969963348989833116134054387170749229727283037539004381749906799086536804833060409474136172555366941728618585406059541252034119743871324250678700384872982451171781187582425415036174859973807308791740126700638971051970130984253220733018069676465821436211189459166314816274312856343743847662935629297385039058995867638657628145507773864526108055960240555022481606199202785925058024830432579575255979350088206582498334200765250127346197462183445273244313791481019358399101974261045866228112643969381260915000633692085584764674590962348778981913376088884428003028884525380391147846960296170242506610362212453097147899059246752039470838651568756936447860255647839420934555768524254936307869325709458298713942099571699038352532605173724401046409835891068947995918563622939580020975403667793972445824314924660071421910280960543001126509465348171578165274136712030616591124370278163264756418240035656344485977356981516571170184503490367247329839029918905331047244192646388781483694722070933208276834926440877061903462400310310912714971691777215944382466558195875724235464126574871359191505402431685753780224309872229197968954568292419800681149500774736283850034371556868937848642903100728570736544781094527730083798193064273329641035102107294849482857941069225426202700986705864486343792581539778720880082472939094132176790819879063083321549033366758411669542076025022153052874530561753566578544319391200352505804404263160393897255267328141341573519260317206587567124721064115544033521885878060454490629232621801703261192543213789940147570897598576539236613542839596065958971620169496607381685702579378743276056299101042330609992226050038406371749624774551692227688179116742092296970031968515816849488829812308838973713812548816822625768321587146870870902983652872039699799443046347590284856879915348783897704689183559695047654245372544600457831804001868574295269358336129152534108239647857265940835850541084976890878718168564019735733404413868473483925209291074662970899003410894678407600370270218659911158024027068908727792092608139478540593070795368029820411059560063770818806300201468089597682281464503558441794599205826355228832689993091165952402266711860196750750016774904076511126182712628448362415567227933921041383599916774741258107269767540823396637585176722118472859896814523237395488865515100105081169354315143940698655994219990316698833562289676307780180403165199454157784522590797094478527374783501574320605401121026459046776407888405131418936829305804667907886324195276589175624033831537090441965631102401115345029530651865015732082027908555810312251396237141029959527165808305219771879666805822719914981669182927300885157938700980642128297901433519612881625146340498799586514929278189798256153105789049971719443957612232561467762663752293854016397327300219477211339390143610214020779541658705009870234823049706317349536402840156086539233268937871985432517515463172604709770339182871369732928056025299517733716713747314091327675123334926745606275249259431691897722774610868184641879355032567152931374739472265649186964811897385922336343519610063911873060758568647677747715523847663529768249278160478089091016419362489297291835670381126599721981391398104800068362337473997065285669862259560466340900225586401805671925988517977227517932779620515459839830061885380471763711923114804873916975264575368654385873055716459504494520723831558083395652262993116300740553064319612842833420246247557544382720728988451119281433524116699786263755596727000062127722598041113800958261882375836196955289660885700197622000468013546008085927065493797593246260797703800289284981/306823391035634599061493161762358005202536348382121570056404590599970887213057943563563839984148126283053162488537350819710001959097870145886191485891947545393333583162633701886922108379673838290293728779922019772260286341159624924201799995359186939002255135893158884841322713171385346840699100216487574892799749644501696450006584445413321628227799590291273525694741543490636127358826327851416072721246655168813967838330980985840803440229923820236397591830018847625222802935778235706526624534976167723324300939035684355163783572494193569783445045039164106717134983335826929969144281049696600091977987036934310039124663146037754977464304436539004705165781758594665823452429464977613818712053435153482184077357982961075131083244748660072506052650955006459512298836819525612486405842371728260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Now, with the default floating-point emulated "real" numbers:
sage: M = M.change_ring(RR)
sage: %time m = M^100
CPU times: user 3.63 s, sys: 8 ms, total: 3.64 s
Wall time: 3.64 s
The timing is about 4 times better, but you lose exactness of precision, since the space of representation of numbers stays bounded:
sage: m[42,42]
-4.67382481087635e217
Note that the only external library that is used is MPFR, which is a library for emulated floating-point numbers, not matrices. Hence, the matrix computations use a slow naive Python implementation:
sage: get_systems('M^100')
['MPFR']
Now, let us use the CPU floating-point numbers:
sage: M = M.change_ring(RDF)
sage: %time m = M^100
CPU times: user 4 ms, sys: 0 ns, total: 4 ms
Wall time: 1.08 ms
This is more that 1000 times faster for a very similar result !
sage: m[42,42]
-4.673824810876391e+217
Unfortunately, we do not get any information from the get_systems function:
sage: get_systems('M^100')
[]
This is because there are only Python calls. So, we have to use another tool:
sage: %prun -r m = M^100
134 function calls in 0.001 seconds
Ordered by: internal time
ncalls tottime percall cumtime percall filename:lineno(function)
8 0.001 0.000 0.001 0.000 matrix_double_dense.pyx:358(_matrix_times_matrix_)
8 0.000 0.000 0.000 0.000 matrix_real_double_dense.pyx:90(__cinit__)
8 0.000 0.000 0.000 0.000 matrix_double_dense.pyx:253(_new)
1 0.000 0.000 0.001 0.001 power.pyx:105(__pyx_fuse_0generic_power_pos)
1 0.000 0.000 0.001 0.001 matrix0.pyx:5365(__pow__)
8 0.000 0.000 0.000 0.000 matrix0.pyx:97(__cinit__)
8 0.000 0.000 0.000 0.000 matrix1.pyx:2273(matrix_space)
1 0.000 0.000 0.001 0.001 <string>:1(<module>)
8 0.000 0.000 0.000 0.000 matrix_double_dense.pyx:116(__create_matrix__)
8 0.000 0.000 0.001 0.000 element.pyx:3435(__mul__)
8 0.000 0.000 0.000 0.000 matrix_space.py:1870(nrows)
16 0.000 0.000 0.000 0.000 matrix0.pyx:4122(is_sparse)
1 0.000 0.000 0.001 0.001 power.pyx:23(generic_power)
8 0.000 0.000 0.000 0.000 element.pxd:65(classify_elements)
16 0.000 0.000 0.000 0.000 matrix_dense.pyx:22(is_sparse_c)
1 0.000 0.000 0.001 0.001 power.pyx:91(generic_power_long)
1 0.000 0.000 0.000 0.000 long.pxd:82(integer_check_long)
1 0.000 0.000 0.000 0.000 long.pxd:195(integer_check_long_py)
1 0.000 0.000 0.000 0.000 integer.pyx:493(__init__)
8 0.000 0.000 0.000 0.000 matrix_space.py:1858(ncols)
8 0.000 0.000 0.000 0.000 element.pxd:108(HAVE_SAME_PARENT)
1 0.000 0.000 0.000 0.000 integer.pyx:7187(fast_tp_new)
1 0.000 0.000 0.000 0.000 integer_fake.pxd:48(is_Integer)
1 0.000 0.000 0.000 0.000 integer.pyx:7265(fast_tp_dealloc)
1 0.000 0.000 0.000 0.000 integer.pyx:4554(__nonzero__)
1 0.000 0.000 0.000 0.000 matrix0.pyx:4138(is_square)
1 0.000 0.000 0.000 0.000 {method 'disable' of '_lsprof.Profiler' objects}
<pstats.Stats instance at 0x7f4b893ffb00>
Now, we see that the computation relies on _matrix_times_matrix_ metod of matrix_double_dense.pyx, which you can see read online at https://git.sagemath.org/sage.git/tre...
So, we can see that the computation relies on numpy, which is a Python package for numerical matrix operations. This package relies on BLAS, as you can see in the dependencies of numpy: https://git.sagemath.org/sage.git/tre... BLAS is a highly optimized low-level library for dealing with huge floating-point matrices developped by physicists, which is basically what other dedicated numerical software will use.
Second advice, Sage can run any Python package, and there are lots of packages dedicated to numerics. If you want to install a particular Python package to be run within Sage, you can do (from a terminal):
sage --pip install <package>
A pretty exhaustive of Python packages for scientific computation can be found at http://scipy.org/topical-software.html and in particular https://scipy.org/topical-software.ht... for PDEs.
Last, i would be glad if you report any feedback from your experience, so that we could get a better idea and what could be improved in Sage.
