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A ay p og amming wi h NumPy
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Published e sion
Ha is, Cha les R.; Millman, K. Ja od; an de Wal , S é an J.; Gomme s, Ral ;
Vi anen, Pauli; Cou napeau, Da id; Wiese , E ic; Taylo , Julian; Be g, Sebas ian;
Smi h, Na haniel J.; Ke n, Robe ; Picus, Ma i; Hoye , S ephan; an Ke kwijk,
Ma en H.; B e , Ma hew; Haldane, Allan; del Río, Jaime Fe nández; Wiebe,
Ma k; Pe e son, Pea u; Gé a d-Ma chan , Pie e; Sheppa d, Ke in; Reddy, Tyle ;
Weckesse , Wa en; Abbasi, Hamee ; Gohlke, Ch is oph; Oliphan , T a is E.
Ha is, C. R., Millman, K. J., an de Wal , S. J., Gomme s, R., Vi anen, P., Cou napeau, D.,
Wiese , E., Taylo , J., Be g, S., Smi h, N. J., Ke n, R., Picus, M., Hoye , S., an Ke kwijk, M. H.,
B e , M., Haldane, A., del Río, J. F., Wiebe, M., Pe e son, P., . . . Oliphan , T. E. (2020). A ay
p og amming wi h NumPy. Na u e, 585(7825), 357-362. h ps://doi.o g/10.1038/s41586-020-
2649-2
2020
Na u e | Vol 585 | 17 Sep embe 2020 | 357
Re iew
A ay p og amming wi h NumPy
Cha les R. Ha is1, K. Ja od Millman2,3,4 ✉, S é an J. an de Wal 2,4,5 ✉, Ral Gomme s6 ✉,
Pauli Vi anen7,8, Da id Cou napeau9, E ic Wiese 10, Julian Taylo 11, Sebas ian Be g4,
Na haniel J. Smi h12, Robe Ke n13, Ma i Picus4, S ephan Hoye 14, Ma en H. an Ke kwijk15,
Ma hew B e 2,16, Allan Haldane17, Jaime Fe nández del Río18, Ma k Wiebe19,20,
Pea u Pe e son6,21,22, Pie e Gé a d-Ma chan 23,24, Ke in Sheppa d25, Tyle Reddy26,
Wa en Weckesse 4, Hamee Abbasi6, Ch is oph Gohlke27 & T a is E. Oliphan 6
A ay p og amming p o ides a powe ul, compac and exp essi e syn ax o
accessing, manipula ing and ope a ing on da a in ec o s, ma ices and
highe -dimensional a ays. NumPy is he p ima y a ay p og amming lib a y o he
Py hon language. I has an essen ial ole in esea ch analysis pipelines in ields as
di e se as physics, chemis y, as onomy, geoscience, biology, psychology, ma e ials
science, enginee ing, inance and economics. Fo example, in as onomy, NumPy was
an impo an pa o he so wa e s ack used in he disco e y o g a i a ional wa es1
and in he i s imaging o a black hole2. He e we e iew how a ew undamen al a ay
concep s lead o a simple and powe ul p og amming pa adigm o o ganizing,
explo ing and analysing scien i ic da a. NumPy is he ounda ion upon which he
scien i ic Py hon ecosys em is cons uc ed. I is so pe asi e ha se e al p ojec s,
a ge ing audiences wi h specialized needs, ha e de eloped hei own NumPy-like
in e aces and a ay objec s. Owing o i s cen al posi ion in he ecosys em, NumPy
inc easingly ac s as an in e ope abili y laye be ween such a ay compu a ion
lib a ies and, oge he wi h i s applica ion p og amming in e ace (API), p o ides a
lexible amewo k o suppo he nex decade o scien i ic and indus ial analysis.
Two Py hon a ay packages exis ed be o e NumPy. The Nume ic pack-
age was de eloped in he mid-1990s and p o ided a ay objec s and
a ay-awa e unc ions in Py hon. I was w i en in C and linked o s and-
a d as implemen a ions o linea algeb a3,4. One o i s ea lies uses was
o s ee C++ applica ions o ine ial con inemen usion esea ch a
Law ence Li e mo e Na ional Labo a o y
5
. To handle la ge as onomi-
cal images coming om he Hubble Space Telescope, a eimplemen a-
ion o Nume ic, called Numa ay, added suppo o s uc u ed a ays,
lexible indexing, memo y mapping, by e-o de a ian s, mo e e icien
memo y use, lexible IEEE 754-s anda d e o -handling capabili ies, and
be e ype-cas ing ules6. Al hough Numa ay was highly compa ible
wi h Nume ic, he wo packages had enough di e ences ha i di ided
he communi y; howe e , in 2005 NumPy eme ged as a ‘bes o bo h
wo lds’ uni ica ion
7
—combining he ea u es o Numa ay wi h he
small-a ay pe o mance o Nume ic and i s ich C API.
Now, 15 yea s la e , NumPy unde pins almos e e y Py hon lib a y
ha does scien i ic o nume ical compu a ion8–11, including SciPy12,
Ma plo lib13, pandas14, sciki -lea n15 and sciki -image16. NumPy is a
communi y-de eloped, open-sou ce lib a y, which p o ides a mul-
idimensional Py hon a ay objec along wi h a ay-awa e unc ions
ha ope a e on i . Because o i s inhe en simplici y, he NumPy a ay
is he de ac o exchange o ma o a ay da a in Py hon.
NumPy ope a es on in-memo y a ays using he cen al p ocessing
uni (CPU). To u ilize mode n, specialized s o age and ha dwa e, he e
has been a ecen p oli e a ion o Py hon a ay packages. Unlike wi h
he Numa ay–Nume ic di ide, i is now much ha de o hese new
lib a ies o ac u e he use communi y—gi en how much wo k is
al eady buil on op o NumPy. Howe e , o p o ide he communi y wi h
access o new and explo a o y echnologies, NumPy is ansi ioning
in o a cen al coo dina ing mechanism ha speci ies a well de ined
a ay p og amming API and dispa ches i , as app op ia e, o special-
ized a ay implemen a ions.
NumPy a ays
The NumPy a ay is a da a s uc u e ha e icien ly s o es and accesses
mul idimensional a ays
17
(also known as enso s), and enables a wide
a ie y o scien i ic compu a ion. I consis s o a poin e o memo y,
along wi h me ada a used o in e p e he da a s o ed he e, no ably
‘da a ype’, ‘shape’ and ‘s ides’ (Fig.1a).
h ps://doi.o g/10.1038/s41586-020-2649-2
Recei ed: 21 Feb ua y 2020
Accep ed: 17 June 2020
Published online: 16 Sep embe 2020
Open access
Check o upda es
1Independen esea che , Logan, UT, USA. 2B ain Imaging Cen e , Uni e si y o Cali o nia, Be keley, Be keley, CA, USA. 3Di ision o Bios a is ics, Uni e si y o Cali o nia, Be keley, Be keley, CA,
USA. 4Be keley Ins i u e o Da a Science, Uni e si y o Cali o nia, Be keley, Be keley, CA, USA. 5Applied Ma hema ics, S ellenbosch Uni e si y, S ellenbosch, Sou h A ica. 6Quansigh , Aus in,
TX, USA. 7Depa men o Physics, Uni e si y o Jy äskylä, Jy äskylä, Finland. 8Nanoscience Cen e , Uni e si y o Jy äskylä, Jy äskylä, Finland. 9Me ca i JP, Tokyo, Japan. 10Depa men o
Enginee ing, Uni e si y o Camb idge, Camb idge, UK. 11Independen esea che , Ka ls uhe, Ge many. 12Independen esea che , Be keley, CA, USA. 13En hough , Aus in, TX, USA. 14Google
Resea ch, Moun ain View, CA, USA. 15Depa men o As onomy and As ophysics, Uni e si y o To on o, To on o, On a io, Canada. 16School o Psychology, Uni e si y o Bi mingham,
Edgbas on, Bi mingham, UK. 17Depa men o Physics, Temple Uni e si y, Philadelphia, PA, USA. 18Google, Zu ich, Swi ze land. 19Depa men o Physics and As onomy, The Uni e si y o
B i ish Columbia, Vancou e , B i ish Columbia, Canada. 20Amazon, Sea le, WA, USA. 21Independen esea che , Saue, Es onia. 22Depa men o Mechanics and Applied Ma hema ics, Ins i u e
o Cybe ne ics a Tallinn Technical Uni e si y, Tallinn, Es onia. 23Depa men o Biological and Ag icul u al Enginee ing, Uni e si y o Geo gia, A hens, GA, USA. 24F ance-IX Se ices, Pa is,
F ance. 25Depa men o Economics, Uni e si y o Ox o d, Ox o d, UK. 26CCS-7, Los Alamos Na ional Labo a o y, Los Alamos, NM, USA. 27Labo a o y o Fluo escence Dynamics, Biomedical
Enginee ing Depa men , Uni e si y o Cali o nia, I ine, I ine, CA, USA. ✉e-mail: millman@be keley.edu; s e an @be keley.edu; al .gomme [email protected]
358 | Na u e | Vol 585 | 17 Sep embe 2020
Re iew
The da a ype desc ibes he na u e o elemen s s o ed in an a ay.
An a ay has a single da a ype, and each elemen o an a ay occupies
he same numbe o by es in memo y. Examples o da a ypes include
eal and complex numbe s (o lowe and highe p ecision), s ings,
imes amps and poin e s o Py hon objec s.
The shape o an a ay de e mines he numbe o elemen s along
each axis, and he numbe o axes is he dimensionali y o he a ay.
Fo example, a ec o o numbe s can be s o ed as a one-dimensional
a ay o shape N, whe eas colou ideos a e ou -dimensional a ays
o shape (T,M,N,3).
S ides a e necessa y o in e p e compu e memo y, which s o es
elemen s linea ly, as mul idimensional a ays. They desc ibe he num-
be o by es o mo e o wa d in memo y o jump om ow o ow, col-
umn o column, and so o h. Conside , o example, a wo-dimensional
a ay o loa ing-poin numbe s wi h shape (4,3), whe e each elemen
occupies 8by es in memo y. To mo e be ween consecu i e columns,
we need o jump o wa d 8by es in memo y, and o access he nex ow,
3×8=24by es. The s ides o ha a ay a e he e o e (24,8). NumPy
can s o e a ays in ei he C o Fo an memo y o de , i e a ing i s o e
ei he ows o columns. This allows ex e nal lib a ies w i en in hose
languages o access NumPy a ay da a in memo y di ec ly.
Use s in e ac wi h NumPy a ays using ‘indexing’ ( o access sub-
a ays o indi idual elemen s), ‘ope a o s’ ( o example, +, − and ×
o ec o ized ope a ions and @ o ma ix mul iplica ion), as well
as ‘a ay-awa e unc ions’; oge he , hese p o ide an easily eadable,
exp essi e, high-le el API o a ay p og amming while NumPy deals
wi h he unde lying mechanics o making ope a ions as .
Indexing an a ay e u ns single elemen s, suba ays o elemen s
ha sa is y a speci ic condi ion (Fig.1b). A ays can e en be indexed
using o he a ays (Fig.1c). Whe e e possible, indexing ha e ie es a
suba ay e u ns a ‘ iew’ on he o iginal a ay such ha da a a e sha ed
be ween he wo a ays. This p o ides a powe ul way o ope a e on
subse s o a ay da a while limi ing memo y usage.
To complemen he a ay syn ax, NumPy includes unc ions ha
pe o m ec o ized calcula ions on a ays, including a i hme ic,
s a is ics and igonome y (Fig.1d). Vec o iza ion—ope a ing on
en i e a ays a he han hei indi idual elemen s—is essen ial o a ay
p og amming. This means ha ope a ions ha would ake many ens
o lines o exp ess in languages such as C can o en be implemen ed as
a single, clea Py hon exp ession. This esul s in concise code and ees
use s o ocus on he de ails o hei analysis, while NumPy handles
looping o e a ay elemen s nea -op imally— o example, aking
s ides in o conside a ion o bes u ilize he compu e ’s as cache
memo y.
When pe o ming a ec o ized ope a ion (such as addi ion) on wo
a ays wi h he same shape, i is clea wha should happen. Th ough
‘b oadcas ing’ NumPy allows he dimensions o di e , and p oduces
esul s ha appeal o in ui ion. A i ial example is he addi ion o a
scala alue o an a ay, bu b oadcas ing also gene alizes o mo e com-
plex examples such as scaling each column o an a ay o gene a ing
a g id o coo dina es. In b oadcas ing, one o bo h a ays a e i ually
duplica ed ( ha is, wi hou copying any da a in memo y), so ha he
shapes o he ope ands ma ch (Fig.1d). B oadcas ing is also applied
when an a ay is indexed using a ays o indices (Fig.1c).
O he a ay-awa e unc ions, such as sum, mean and maximum,
pe o m elemen -by-elemen ‘ educ ions’, agg ega ing esul s ac oss
one, mul iple o all axes o a single a ay. Fo example, summing an
n-dimensional a ay o e d axes esul s in an a ay o dimension n−d
(Fig.1 ).
NumPy also includes a ay-awa e unc ions o c ea ing, eshaping,
conca ena ing and padding a ays; sea ching, so ing and coun ing
da a; and eading and w i ing iles. I p o ides ex ensi e suppo o
gene a ing pseudo andom numbe s, includes an asso men o p ob-
abili y dis ibu ions, and pe o ms accele a ed linea algeb a, using
one o se e al backends such as OpenBLAS18,19 o In el MKL op imized
o he CPUs a hand (see Supplemen a y Me hods o mo e de ails).
Al oge he , he combina ion o a simple in-memo y a ay ep e-
sen a ion, a syn ax ha closely mimics ma hema ics, and a a ie y
o a ay-awa e u ili y unc ions o ms a p oduc i e and powe ully
exp essi e a ay p og amming language.
In [1]: impo numpy as np
In [2]: x = np.a ange(12)
In [3]: x = x. eshape(4, 3)
In [4]: x
Ou [4]:
a ay([[ 0, 1, 2],
[ 3, 4, 5],
[ 6, 7, 8],
[ 9, 10, 11]])
In [5]: np.mean(x, axis=0)
Ou [5]: a ay([4.5, 5.5, 6.5])
In [6]:
x = x - np.mean(x, axis=0)
In [7]: x
Ou [7]:
a ay([[-4.5, -4.5, -4.5],
[-1.5, -1.5, -1.5],
[ 1.5, 1.5, 1.5],
[ 4.5, 4.5, 4.5]])
aDa a s uc u e gExample
x
=
012
345
678
9 10 11
da a
da a ype
shape
s ides
8-by e in ege
(4, 3)
(24, 8)
1234567
0 8 9 10 11
8 by es
pe elemen
3 × 8 = 24 by es
o jump one
ow down
b
Indexing ( iew)
10 11
9
9
x[:,1:] →wi h slices
12
45
78
0
0
3
3
6
6
x[:,::2]→wi h slices
wi h s eps
02
35
6 8
911
01
1
2
3
4
4
5
67
7
8
9
10
10
11
Slices a e s a :end:s ep,
any o which can be le blank
dVec o iza ion
+→
01
34
67
9 10
1
1
1
1
1
1
1
1
12
45
78
10 11
eB oadcas ing
×
3
6
0
9
12
→
00
36
612
9 18
Reduc ion
01
34
67
9 10
2
5
8
11
3
12
21
30
sum
axis 1
18 22 26
sum
axis 0
66
sum
axis (0,1)
c
Indexing (copy)
43
76
wi h a ays
wi h b oadcas ing
→
x→
,
2
1 1 0 x
,
11
22
10
10
x wi h a ays
x[0,1],x[1,2] 15
→→
0 1 1 2
,
x[x > 9] wi h masks
10 11
→
→5wi h scala s
x[1,2]
Fig. 1 | The NumPy a ay inco po a es se e al undamen al a ay concep s.
a, The NumPy a ay da a s uc u e and i s associa ed me ada a ields.
b, Indexing an a ay wi h slices and s eps. These ope a ions e u n a ‘ iew’ o
he o iginal da a. c, Indexing an a ay wi h masks, scala coo dina es o o he
a ays, so ha i e u ns a ‘copy’ o he o iginal da a. In he bo om example, an
a ay is indexed wi h o he a ays; his b oadcas s he indexing a gumen s
be o e pe o ming he lookup. d, Vec o iza ion e icien ly applies ope a ions
o g oups o elemen s. e, B oadcas ing in he mul iplica ion o wo-dimensional
a ays. , Reduc ion ope a ions ac along one o mo e axes. In his example,
an a ay is summed along selec axes o p oduce a ec o , o along wo axes
consecu i ely o p oduce a scala . g, Example NumPy code, illus a ing some o
hese concep s.
Na u e | Vol 585 | 17 Sep embe 2020 | 359
Scien i ic Py hon ecosys em
Py hon is an open-sou ce, gene al-pu pose in e p e ed p og amming
language well sui ed o s anda d p og amming asks such as cleaning
da a, in e ac ing wi h web esou ces and pa sing ex . Adding as a ay
ope a ions and linea algeb a enables scien is s o do all hei wo k
wi hin a single p og amming language—one ha has he ad an age o
being amously easy o lea n and each, as wi nessed by i s adop ion
as a p ima y lea ning language in many uni e si ies.
E en hough NumPy is no pa o Py hon’s s anda d lib a y, i ben-
e i s om a good ela ionship wi h he Py hon de elope s. O e he
yea s, he Py hon language has added new ea u es and special syn ax
so ha NumPy would ha e a mo e succinc and easie - o- ead a ay
no a ion. Howe e , because i is no pa o he s anda d lib a y, NumPy
is able o dic a e i s own elease policies and de elopmen pa e ns.
SciPy and Ma plo lib a e igh ly coupled wi h NumPy in e ms o his-
o y, de elopmen and use. SciPy p o ides undamen al algo i hms o
scien i ic compu ing, including ma hema ical, scien i ic and enginee -
ing ou ines. Ma plo lib gene a es publica ion- eady igu es and isu-
aliza ions. The combina ion o NumPy, SciPy and Ma plo lib, oge he
wi h an ad anced in e ac i e en i onmen such as IPy hon
20
o Jupy-
e 21, p o ides a solid ounda ion o a ay p og amming in Py hon. The
scien i ic Py hon ecosys em (Fig.2) builds on op o his ounda ion o
p o ide se e al, widely used echnique-speci ic lib a ies15,16,22, ha in
u n unde lie nume ous domain-speci ic p ojec s
23–28
. NumPy, a he
base o he ecosys em o a ay-awa e lib a ies, se s documen a ion
s anda ds, p o ides a ay es ing in as uc u e and adds build sup-
po o Fo an and o he compile s.
Many esea ch g oups ha e designed la ge, complex scien i ic lib a -
ies ha add applica ion-speci ic unc ionali y o he ecosys em. Fo
example, he eh -imaging lib a y29, de eloped by he E en Ho izon
Telescope collabo a ion o adio in e e ome y imaging, analysis
and simula ion, elies on many lowe -le el componen s o he scien i ic
Py hon ecosys em. In pa icula , he EHT collabo a ion used his lib a y
o he i s imaging o a black hole. Wi hin eh -imaging, NumPy a ays
a e used o s o e and manipula e nume ical da a a e e y s ep in he
p ocessing chain: om aw da a h ough calib a ion and image econ-
s uc ion. SciPy supplies ools o gene al image-p ocessing asks such
as il e ing and image alignmen , and sciki -image, an image-p ocessing
lib a y ha ex ends SciPy, p o ides highe -le el unc ionali y such
as edge il e s and Hough ans o ms. The ‘scipy.op imize’ module
pe o ms ma hema ical op imiza ion. Ne wo kX
22
, a package o com-
plex ne wo k analysis, is used o e i y image compa ison consis ency.
As opy
23,24
handles s anda d as onomical ile o ma s and compu es
ime–coo dina e ans o ma ions. Ma plo lib is used o isualize da a
and o gene a e he inal image o he black hole.
The in e ac i e en i onmen c ea ed by he a ayp og amming oun-
da ion and he su ounding ecosys em o ools—inside o IPy hon o
Jupy e —is ideally sui ed o explo a o y da a analysis. Use s can luidly
inspec , manipula e and isualize hei da a, and apidly i e a e o e ine
p og amming s a emen s. These s a emen s a e hen s i ched oge he
in o impe a i e o unc ional p og ams, o no ebooks con aining bo h
compu a ion and na a i e. Scien i ic compu ing beyond explo a o y
wo k is o en done in a ex edi o o an in eg a ed de elopmen en i-
onmen (IDE) such as Spyde . This ich and p oduc i e en i onmen
has made Py hon popula o scien i ic esea ch.
To complemen his acili y o explo a o y wo k and apid p o o-
yping, NumPy has de eloped a cul u e o using ime- es ed so wa e
enginee ing p ac ices o imp o e collabo a ion and educe e o 30. This
cul u e is no only adop ed by leade s in he p ojec bu also en husi-
as ically augh o newcome s. The NumPy eam was ea ly o adop
dis ibu ed e ision con ol and code e iew o imp o e collabo a ion
can e a
Chemis y
Biopy hon
Biology
As opy
As onomy
simpeg
Geophysics
NLTK
Linguis ics
Quan Econ
Economics
SciPy
Algo i hms
Ma plo lib
Plo s
sciki -lea n
Machine lea ning
Ne wo kX
Ne wo k analysis
pandas, s a smodels
S a is ics
sciki -image
Image p ocessing
P
syc
h
o
Py
kh
me
Q
iime2
FiPy
d
eepc
h
e
m
li
b
osa
P
y
W
a e
l
e
s
S
unP
y
Q
uTiP
y
n
ib
a
b
e
l
ye
ll
ow
b
i
c
k
mne-py
h
on
s
c
iki
-
HEP
e
h
-
i
mag
i
n
g
MDA
na
l
ys
i
s
i
is
cesium
Py
C
h on
o
Founda ion
A
pplica ion-speci ic
Domain-speci ic
Technique-speci ic
A ay P o ocolsNumPy API
Py hon
Language
IPy hon / Jupy e
In e ac i e en i onmen s
NumPy
A ays
New a ay implemen a ions
Fig. 2 | NumPy is he base o he scien i ic Py hon ecosys em. Essen ial lib a ies and p ojec s ha depend on NumPy’s API gain access o new a ay
implemen a ions ha suppo NumPy’s a ay p o ocols (Fig.3).
360 | Na u e | Vol 585 | 17 Sep embe 2020
Re iew
on code, and con inuous es ing ha uns an ex ensi e ba e y o au o-
ma ed es s o e e y p oposed change o NumPy. The p ojec also
has comp ehensi e, high-quali y documen a ion, in eg a ed wi h he
sou ce code31–33.
This cul u e o using bes p ac ices o p oducing eliable scien i ic
so wa e has been adop ed by he ecosys em o lib a ies ha build on
NumPy. Fo example, in a ecen awa d gi en by he Royal As onomi-
cal Socie y o As opy, hey s a e: “The As opy P ojec has p o ided
hund eds o junio scien is s wi h expe ience in p o essional-s anda d
so wa e de elopmen p ac ices including use o e sion con ol, uni
es ing, code e iew and issue acking p ocedu es. This is a i al skill
se o mode n esea che s ha is o en missing om o mal uni e si y
educa ion in physics o as onomy”
34
. Communi y membe s explici ly
wo k o add ess his lack o o mal educa ion h ough cou ses and
wo kshops35–37.
The ecen apid g ow h o da a science, machine lea ning and a i-
icial in elligence has u he and d ama ically boos ed he scien i ic
use o Py hon. Examples o i s impo an applica ions, such as he
eh -imaging lib a y, now exis in almos e e y discipline in he na u-
al and social sciences. These ools ha e become he p ima y so wa e
en i onmen in many ields. NumPy and i s ecosys em a e commonly
augh in uni e si y cou ses, boo camps and summe schools, and
a e he ocus o communi y con e ences and wo kshops wo ldwide.
NumPy and i s API ha e become uly ubiqui ous.
A ay p oli e a ion and in e ope abili y
NumPy p o ides in-memo y, mul idimensional, homogeneously yped
( ha is, single-poin e and s ided) a ays on CPUs. I uns on machines
anging om embedded de ices o he wo ld’s la ges supe compu e s,
wi h pe o mance app oaching ha o compiled languages. Fo mos
i s exis ence, NumPy add essed he as majo i y o a ay compu a-
ion use cases.
Howe e , scien i ic da ase s now ou inely exceed he memo y capac-
i y o a single machine and may be s o ed on mul iple machines o in
he cloud. In addi ion, he ecen need o accele a e deep-lea ning and
a i icial in elligence applica ions has led o he eme gence o special-
ized accele a o ha dwa e, including g aphics p ocessing uni s (GPUs),
enso p ocessing uni s (TPUs) and ield-p og ammable ga e a ays
(FPGAs). Owing o i s in-memo y da a model, NumPy is cu en ly unable
o di ec ly u ilize such s o age and specialized ha dwa e. Howe e ,
bo h dis ibu ed da a and also he pa allel execu ion o GPUs, TPUs
and FPGAs map well o he pa adigm o a ay p og amming: he e o e
leading o a gap be ween a ailable mode n ha dwa e a chi ec u es and
he ools necessa y o le e age hei compu a ional powe .
The communi y’s e o s o ill his gap led o a p oli e a ion o new
a ay implemen a ions. Fo example, each deep-lea ning amewo k
c ea ed i s own a ays; he PyTo ch
38
, Tenso low
39
, Apache MXNe
40
and JAX a ays all ha e he capabili y o un on CPUs and GPUs in a
dis ibu ed ashion, using lazy e alua ion o allow o addi ional pe -
o mance op imiza ions. SciPy and PyDa a/Spa se bo h p o ide spa se
a ays, which ypically con ain ew non-ze o alues and s o e only hose
in memo y o e iciency. In addi ion, he e a e p ojec s ha build on
NumPy a ays as da a con aine s, and ex end i s capabili ies. Dis ib-
u ed a ays a e made possible ha way by Dask, and labelled a ays—
e e ing o dimensions o an a ay by name a he han by index o
cla i y, compa e x[:,1] e sus x.loc[:,' ime']—by xa ay41.
Such lib a ies o en mimic he NumPy API, because his lowe s he
ba ie o en y o newcome s and p o ides he wide communi y wi h
a s able a ayp og amming in e ace. This, in u n, p e en s dis up i e
schisms such as he di e gence be weenNume ic and Numa ay. Bu
explo ing new ways o wo king wi h a ays is expe imen al by na u e
and, in ac , se e al p omising lib a ies (such as Theano and Ca e) ha e
al eady ceased de elopmen . And each ime ha a use decides o y a
new echnology, hey mus change impo s a emen s and ensu e ha he
new lib a y implemen s all he pa s o he NumPy API hey cu en ly use.
Ideally, ope a ing on specialized a ays using NumPy unc ions o
seman ics would simply wo k, so ha use s could w i e code once,
and would hen bene i om swi ching be ween NumPy a ays, GPU
a ays, dis ibu ed a ays and so o h as app op ia e. To suppo a ay
ope a ions be ween ex e nal a ay objec s, NumPy he e o e added
he capabili y o ac as a cen al coo dina ion mechanism wi h a well
speci ied API (Fig.2).
To acili a e his in e ope abili y, NumPy p o ides ‘p o ocols’ (o
con ac s o ope a ion), ha allow o specialized a ays o be passed o
NumPy unc ions (Fig.3). NumPy, in u n, dispa ches ope a ions o he
o igina ing lib a y, as equi ed. O e ou hund ed o he mos popula
NumPy unc ions a e suppo ed. The p o ocols a e implemen ed by
widely used lib a ies such as Dask, CuPy, xa ay and PyDa a/Spa se.
Thanks o hese de elopmen s, use s can now, o example, scale hei
compu a ion om a single machine o dis ibu ed sys ems using Dask.
The p o ocols also compose well, allowing use s o edeploy NumPy
code a scale on dis ibu ed, mul i-GPU sys ems ia, o ins ance, CuPy
a ays embedded in Dask a ays. Using NumPy’s high-le el API, use s
can le e age highly pa allel code execu ion on mul iple sys ems wi h
millions o co es, all wi h minimal code changes42.
These a ay p o ocols a e now a key ea u e o NumPy, and a e
expec ed o only inc ease in impo ance. The NumPy de elope s—
many o whom a e au ho s o his Re iew—i e a i ely e ine and add
p o ocol designs o imp o e u ili y and simpli y adop ion.
Ou pu
a ays
Inpu
a
ays
NumPy
API
np.s ack
np. eshape
np. anspose
np.a gmin
np.mean
np.s d
np.max
np.cos
np.a c an
np.log
np.cumsum
np.di
...
NumPy a ay p o ocols
In [1]: impo numpy as np
In [2]: impo dask.a ay as da
In [3]: x = da.a ange(12)
In [4]: x = np. eshape(x, (4, 3))
In [5]: x
Ou [5]: dask.a ay<..., shape=(4, 3), ...
>
In [6]: np.mean(x, axis=0)
Ou [6]: dask.a ay<..., shape=(3,), ...>
In [7]: x = x - np.mean(x, axis=0)
In [8]: x
Ou [8]: dask.a ay<..., shape=(4, 3), ...
>
A ay
implemen a ion
NumPy
Dask
CuPy
PyDa a/
Spa se
...
...
Dask
NumP
y
CuPy
PyDa a
Spa se
...
Dask
NumPy
CuPy
PyDa a
Spa se
Fig. 3 | NumPy’s API and a ay p o ocols expose new a ays o he
ecosys em. In his example, NumPy’s ‘mean’ unc ion is called on a Dask a ay.
The call succeeds by dispa ching o he app op ia e lib a y implemen a ion (in
his case, Dask) and esul s in a new Dask a ay. Compa e his code o he
example code in Fig.1g.

Na u e | Vol 585 | 17 Sep embe 2020 | 361
Discussion
NumPy combines he exp essi e powe o a ay p og amming, he
pe o mance o C, and he eadabili y, usabili y and e sa ili y o Py hon
in a ma u e, well es ed, well documen ed and communi y-de eloped
lib a y. Lib a ies in he scien i ic Py hon ecosys em p o ide as imple-
men a ions o mos impo an algo i hms. Whe e ex eme op imiza-
ion is wa an ed, compiled languages can be used, such as Cy hon43,
Numba44 and Py h an45; hese languages ex end Py hon and ans-
pa en ly accele a e bo lenecks. Owing o NumPy’s simple memo y
model, i is easy o w i e low-le el, hand-op imized code, usually in C
o Fo an, o manipula e NumPy a ays and pass hem back o Py hon.
Fu he mo e, using a ay p o ocols, i is possible o u ilize he ull
spec um o specialized ha dwa e accele a ion wi h minimal changes
o exis ing code.
NumPy was ini ially de eloped by s uden s, acul y and esea che s
o p o ide an ad anced, open-sou ce a ay p og amming lib a y o
Py hon, which was ee o use and unencumbe ed by license se e s and
so wa e p o ec ion dongles. The e was a sense o building some hing
consequen ial oge he o he bene i o many o he s. Pa icipa ing
in such an endea ou , wi hin a welcoming communi y o like-minded
indi iduals, held a powe ul a ac ion o many ea ly con ibu o s.
These use –de elope s equen ly had o w i e code om sc a ch
o sol e hei own o hei colleagues’ p oblems—o en in low-le el
languages ha p eceded Py hon, such as Fo an46 and C. To hem,
he ad an ages o an in e ac i e, high-le el a ay lib a y we e e iden .
The design o his new ool was in o med by o he powe ul in e ac i e
p og amming languages o scien i ic compu ing such as Basis
47–50
,
Yo ick
51
, R
52
and APL
53
, as well as comme cial languages and en i on-
men s such as IDL (In e ac i e Da a Language) and MATLAB.
Wha began as an a emp o add an a ay objec o Py hon became
he ounda ion o a ib an ecosys em o ools. Now, a la ge amoun o
scien i ic wo k depends on NumPy being co ec , as and s able. I is
no longe a small communi y p ojec , bu co e scien i ic in as uc u e.
The de elope cul u e has ma u ed: al hough ini ial de elopmen was
highly in o mal, NumPy now has a oadmap and a p ocess o p opos-
ing and discussing la ge changes. The p ojec has o mal go e nance
s uc u es and is iscally sponso ed by NumFOCUS, a nonp o i ha
p omo es open p ac ices in esea ch, da a and scien i ic compu ing.
O e he pas ew yea s, he p ojec a ac ed i s i s unded de elop-
men , sponso ed by he Moo e and Sloan Founda ions, and ecei ed
an awa d as pa o he Chan Zucke be g Ini ia i e’s Essen ials o Open
Sou ce So wa e p og amme. Wi h his unding, he p ojec was (and
is) able o ha e sus ained ocus o e mul iple mon hs o implemen
subs an ial new ea u es and imp o emen s. Tha said, he de elop-
men o NumPy s ill depends hea ily on con ibu ions made by g adu-
a e s uden s and esea che s in hei ee ime (see Supplemen a y
Me hods o mo e de ails).
NumPy is no longe me ely he ounda ional a ay lib a y unde lying
he scien i ic Py hon ecosys em, bu i has become he s anda d API o
enso compu a ion and a cen al coo dina ing mechanism be ween
a ay ypes and echnologies in Py hon. Wo k con inues o expand on
and imp o e hese in e ope abili y ea u es.
O e he nex decade, NumPy de elope s will ace se e al challenges.
New de ices will be de eloped, and exis ing specialized ha dwa e will
e ol e o mee diminishing e u ns on Moo e’s law. The e will be mo e,
and a wide a ie y o , da a science p ac i ione s, a la ge p opo ion o
whom will use NumPy. The scale o scien i ic da a ga he ing will con-
inue o inc ease, wi h he adop ion o de ices and ins umen s such
as ligh -shee mic oscopes and he La ge Synop ic Su ey Telescope
(LSST)54. New gene a ion languages, in e p e e s and compile s, such as
Rus 55, Julia56 and LLVM57, will c ea e new concep s and da a s uc u es,
and de e mine hei iabili y.
Th ough he mechanisms desc ibed in his Re iew, NumPy is poised
o emb ace such a changing landscape, and o con inue playing a
leading pa in in e ac i e scien i ic compu a ion, al hough o do so
will equi e sus ained unding om go e nmen , academia and indus-
y. Bu , impo an ly, o NumPy o mee he needs o he nex decade
o da a science, i will also need a new gene a ion o g adua e s uden s
and communi y con ibu o s o d i e i o wa d.
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Acknowledgemen s We hank R.Ba nowski, P.Dubois, M.Eickenbe g, and P.G een ield, who
sugges ed ex and p o ided help ul eedback on he manusc ip . K.J.M. and S.J. .d.W. we e
unded in pa by he Go don and Be y Moo e Founda ion h ough g an GBMF3834 and by
he Al ed P. Sloan Founda ion h ough g an 2013-10-27 o he Uni e si y o Cali o nia,
Be keley. S.J. .d.W., S.B., M.P. and W.W. we e unded in pa by he Go don and Be y Moo e
Founda ion h ough g an GBMF5447 and by he Al ed P. Sloan Founda ion h ough g an
G-2017-9960 o he Uni e si y o Cali o nia, Be keley.
Au ho con ibu ions K.J.M. and S.J. .d.W. composed he manusc ip wi h inpu om
o he s. S.B., R.G., K.S., W.W., M.B. and T.R. con ibu ed ex . All au ho s con ibu ed
subs an ial code, documen a ion and/o expe ise o he NumPy p ojec . All au ho s
e iewed he manusc ip .
Compe ing in e es s The au ho s decla e no compe ing in e es s.
Addi ional in o ma ion
Supplemen a y in o ma ion is a ailable o his pape a h ps://doi.o g/10.1038/s41586-020-
2649-2.
Co espondence and eques s o ma e ials should be add essed o K.J.M., S.J. .W. o R.G.
Pee e iew in o ma ion Na u e hanks Edoua d Duchesnay,Alan Edelman and he o he ,
anonymous, e iewe (s) o hei con ibu ion o he pee e iew o his wo k.
Rep in s and pe missions in o ma ion is a ailable a h p://www.na u e.com/ ep in s.
Publishe ’s no e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in
published maps and ins i u ional a ilia ions.
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