Numpy#

What provide Numpy to Python ?#

ndarray multi-dimensional array object
derived objects such as masked arrays and matrices
ufunc fast array mathematical operations.
Offers some Matlab-ish capabilities within Python
Initially developed by Travis Oliphant.
Numpy 1.0 released October, 2006.
The SciPy.org website is very helpful.
NumPy fully supports an object-oriented approach.

Routines for fast operations on arrays.#

shape manipulation
sorting
I/O
FFT
basic linear algebra
basic statistical operations
random simulation
statistics
and much more…

Getting Started with NumPy#

It is handy to import everything from NumPy into a Python console:

from numpy import *

But it is easier to read and debug if you use explicit imports.

import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

import numpy as np
print(np.__version__)

1.23.5

Why Arrays ?#

Python lists are slow to process and use a lot of memory.
For tables, matrices, or volumetric data, you need lists of lists of lists… which becomes messy to program.

from random import random
from operator import truediv

l1 = [random() for i in range(1000)]
l2 = [random() for i in range(1000)]
%timeit s = sum(map(truediv,l1,l2))

27.7 µs ± 121 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

a1 = np.array(l1)
a2 = np.array(l2)
%timeit s = np.sum(a1/a2)

5.52 µs ± 29.8 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)

Numpy Arrays: The `ndarray` class.#

There are important differences between NumPy arrays and Python lists:
- NumPy arrays have a fixed size at creation.
- NumPy arrays elements are all required to be of the same data type.
- NumPy arrays operations are performed in compiled code for performance.
Most of today’s scientific/mathematical Python-based software use NumPy arrays.
NumPy gives us the code simplicity of Python, but the operation is speedily executed by pre-compiled C code.

a = np.array([0,1,2,3])  #  list
b = np.array((4,5,6,7))  #  tuple
c = np.matrix('8 9 0 1') #  string (matlab syntax)

print(a,b,c)

[0 1 2 3] [4 5 6 7] [[8 9 0 1]]

Element wise operations are the “default mode”#

a*b,a+b

(array([ 0,  5, 12, 21]), array([ 4,  6,  8, 10]))

5*a, 5+a

(array([ 0,  5, 10, 15]), array([5, 6, 7, 8]))

a @ b, np.dot(a,b)  # Matrix multiplication

(38, 38)

NumPy Arrays Properties#

a = np.array([1,2,3,4,5]) # Simple array creation

type(a) # Checking the type

numpy.ndarray

a.dtype # Print numeric type of elements

dtype('int64')

a.itemsize # Print Bytes per element

a.shape # returns a tuple listing the length along each dimension

(5,)

np.size(a), a.size # returns the entire number of elements.

(5, 5)

a.ndim  # Number of dimensions

a.nbytes # Memory used

** Always use shape or size for numpy arrays instead of len **
len gives same information only for 1d array.

Functions to allocate arrays#

x = np.zeros((2,),dtype=('i4,f4,a10'))
x

array([(0, 0., b''), (0, 0., b'')],
      dtype=[('f0', '<i4'), ('f1', '<f4'), ('f2', 'S10')])

empty, empty_like, ones, ones_like, zeros, zeros_like, full, full_like

Setting Array Elements Values#

a = np.array([1,2,3,4,5])
print(a.dtype)

int64

a[0] = 10 # Change first item value
a, a.dtype

(array([10,  2,  3,  4,  5]), dtype('int64'))

a.fill(0) # slighty faster than a[:] = 0
a

array([0, 0, 0, 0, 0])

Setting Array Elements Types#

b = np.array([1,2,3,4,5.0]) # Last item is a float
b, b.dtype

(array([1., 2., 3., 4., 5.]), dtype('float64'))

a.fill(3.0)  # assigning a float into a int array 
a[1] = 1.5   # truncates the decimal part
print(a.dtype, a)

int64 [3 1 3 3 3]

a.astype('float64') # returns a new array containing doubles

array([3., 1., 3., 3., 3.])

np.asfarray([1,2,3,4]) # Return an array converted to a float type

array([1., 2., 3., 4.])

Slicing x[lower:upper:step]#

Extracts a portion of a sequence by specifying a lower and upper bound.
The lower-bound element is included, but the upper-bound element is not included.
The default step value is 1 and can be negative.

a = np.array([10,11,12,13,14])

a[:2], a[-5:-3], a[0:2], a[-2:] # negative indices work

(array([10, 11]), array([10, 11]), array([10, 11]), array([13, 14]))

a[::2], a[::-1]

(array([10, 12, 14]), array([14, 13, 12, 11, 10]))

Exercise:#

Compute derivative of $f(x) = \sin(x)$ with finite difference method. $$ \frac{\partial f}{\partial x} \sim \frac{f(x+dx)-f(x)}{dx} $$

derivatives values are centered in-between sample points.

x, dx = np.linspace(0,4*np.pi,100, retstep=True)
y = np.sin(x)

%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = [12.,8.] # Increase plot size
plt.plot(x, np.cos(x),'b')
plt.title(r"$\rm{Derivative\ of}\ \sin(x)$");

# Compute integral of x numerically
avg_height = 0.5*(y[1:]+y[:-1])
int_sin = np.cumsum(dx*avg_height)
plt.plot(x[1:], int_sin, 'ro', x, np.cos(0)-np.cos(x));

Multidimensional array#

a = np.arange(4*3).reshape(4,3) # NumPy array
l = [[0,1,2],[3,4,5],[6,7,8],[9,10,11]] # Python List

print(a)
print(l)

[[ 0  1  2]
 [ 3  4  5]
 [ 6  7  8]
 [ 9 10 11]]
[[0, 1, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]]

l[-1][-1] # Access to last item

print(a[-1,-1])  # Indexing syntax is different with NumPy array
print(a[0,0])    # returns the first item
print(a[1,:])    # returns the second line

11
0
[3 4 5]

print(a[1]) # second line with 2d array
print(a[:,-1])  # last column

[3 4 5]
[ 2  5  8 11]

Exercise#

We compute numerically the Laplace Equation Solution using Finite Difference Method
Replace the computation of the discrete form of Laplace equation with numpy arrays $$ \Delta T_{i,j} = ( T_{i+1,j} + T_{i-1,j} + T_{i,j+1} + T_{i,j-1} - 4 T_{i,j}) $$
The function numpy.allclose can help you to compute the residual.

%%time
# Boundary conditions
Tnorth, Tsouth, Twest, Teast = 100, 20, 50, 50

# Set meshgrid
n, l = 64, 1.0
X, Y = np.meshgrid(np.linspace(0,l,n), np.linspace(0,l,n))
T = np.zeros((n,n))

# Set Boundary condition
T[-1, :] = Tnorth
T[0, :]   = Tsouth
T[:, -1] = Teast
T[:, 0]   = Twest

T_old = np.copy(T)

residual = 1.0   
istep = 0
while residual > 1e-5 :
    istep += 1
    print ((istep, residual), end="\r")
    residual = 0.0   
    for i in range(1, n-1):
        for j in range(1, n-1):
            T_old[i,j] = T[i,j]
            T[i, j] = 0.25 * (T[i+1,j] + T[i-1,j] + T[i,j+1] + T[i,j-1])
    
    residual = max(residual,np.max(np.abs((T_old-T)/T)))


print()
print("iterations = ",istep)
plt.title("Temperature")
plt.contourf(X, Y, T)
plt.colorbar();

(1, 1.0)
(2, 1.0)
(3, 1.0)
(4, 1.0)
(5, 1.0)
(6, 1.0)
(7, 1.0)
(8, 1.0)
(9, 1.0)
(10, 0.9999999999999997)
(11, 0.9999999999999902)
(12, 0.9999999999997455)
(13, 0.9999999999940791)
(14, 0.9999999998756399)
(15, 0.9999999976225641)
(16, 0.9999999583376037)
(17, 0.999999326700459)
(18, 0.9999899130081685)
(19, 0.9998592998914099)
(20, 0.9981701345865893)
(21, 0.9783799284159196)
(22, 0.8955847401702681)
(23, 0.86986471833438)
(24, 0.8098852631390057)
(25, 0.7527529110760323)
(26, 0.7272346882718936)
(27, 0.6790691522857905)
(28, 0.6320458179301557)
(29, 0.6069386094512461)
(30, 0.575385356496663)
(31, 0.5398414266844065)

(32, 0.5073489564930949)
(33, 0.48297721114570125)
(34, 0.4618345475862288)

(35, 0.43827557692194496)

(36, 0.4143357764645196)
(37, 0.3924382054205373)
(38, 0.3733812863051147)
(39, 0.35710865017565735)
(40, 0.3420793814531387)

(41, 0.326723185934865)
(42, 0.31152930040701554)
(43, 0.2967884812685721)
(44, 0.2841936702538212)
(45, 0.27229436409349167)
(46, 0.2606966680515375)
(47, 0.2505059654694677)
(48, 0.24105086578370669)
(49, 0.23174752119627523)
(50, 0.22268821422383456)
(51, 0.2139321391802263)
(52, 0.2056789240103906)
(53, 0.19852000459869076)
(54, 0.19154016363351395)
(55, 0.1847719042022457)
(56, 0.1782359059639316)
(57, 0.1719439656938689)
(58, 0.1661317583468397)
(59, 0.16080143658541332)
(60, 0.15562883918080994)
(61, 0.15062374814273707)
(62, 0.14579170278101808)
(63, 0.14113498009837175)
(64, 0.13701556698519357)
(65, 0.1330577816617311)
(66, 0.12921639402527813)
(67, 0.12549428484635763)
(68, 0.12189269341938404)
(69, 0.11841156181043053)
(70, 0.11504981910315674)
(71, 0.11180561442912781)
(72, 0.10867650683346695)

(73, 0.10565961910779918)
(74, 0.10277423555972015)
(75, 0.10011529446941532)
(76, 0.09753952693641722)

(77, 0.09504562060680743)
(78, 0.09264232267884208)
(79, 0.0904668914161219)
(80, 0.08835029199885054)

(81, 0.08629213569569119)
(82, 0.08429178767300372)

(83, 0.08234841271209116)
(84, 0.08046101458056627)
(85, 0.07862846977337559)
(86, 0.07684955629552748)
(87, 0.07512297810519541)
(88, 0.07344738577824392)
(89, 0.07182139389713012)
(90, 0.07024359561098874)
(91, 0.06871257476095718)
(92, 0.06722691591624932)
(93, 0.06578521262245907)
(94, 0.06438607412411373)
(95, 0.0630281307884429)
(96, 0.06171003842639934)
(97, 0.06043048167985572)
(98, 0.0592152869523178)
(99, 0.05806095286424927)
(100, 0.05697757075602701)
(101, 0.055920346653939454)
(102, 0.05488871668696673)
(103, 0.053882107696533614)
(104, 0.05289994070021914)
(105, 0.0519416339031402)
(106, 0.0510066053042519)
(107, 0.05009427494092695)
(108, 0.049204066811402214)
(109, 0.04833541051107197)
(110, 0.04748774261521061)
(111, 0.046660507837522555)
(112, 0.0458531599909676)
(113, 0.04506516277460256)

(114, 0.04429599040768293)
(115, 0.043545128130012725)
(116, 0.04281207258546671)
(117, 0.04209633210375605)

(118, 0.04139742689381655)
(119, 0.04071488916070421)
(120, 0.04004826315651514)
(121, 0.03939710517463711)
(122, 0.03876098349555084)
(123, 0.03813947829143378)

(124, 0.03755177739422651)
(125, 0.03697714917736227)

(126, 0.03641481589428299)
(127, 0.035864467615200624)
(128, 0.03532580165895137)
(129, 0.03479852258503548)
(130, 0.03428234216279196)
(131, 0.03377697932076092)
(132, 0.033282160078972826)
(133, 0.03279761746660941)
(134, 0.032323091427221125)
(135, 0.031858328713448805)
(136, 0.03140308277297653)
(137, 0.0309571136272617)
(138, 0.03052018774439911)
(139, 0.030092077907334605)
(140, 0.029672563078483802)
(141, 0.02926142826171095)
(142, 0.028858464362480605)
(143, 0.028468070612449284)
(144, 0.02808756035762739)
(145, 0.027714239842811746)
(146, 0.02734794309866435)
(147, 0.026988508171532363)
(148, 0.026635777055176644)
(149, 0.026289595619858604)
(150, 0.025949813539307552)
(151, 0.02561628421602147)
(152, 0.025288864705315075)

(153, 0.024967415638481445)
(154, 0.024651801145389205)
(155, 0.024341888776803735)
(156, 0.02403754942669084)
(157, 0.02373865725472386)
(158, 0.023448442558632248)
(159, 0.023167313848241167)
(160, 0.022890950986996342)

(161, 0.022619254071911316)
(162, 0.022352125482355056)
(163, 0.02208946983515546)

(164, 0.021831193939553087)
(165, 0.021577206752121974)
(166, 0.021327419331770053)

(167, 0.021081744794915463)
(168, 0.0208400982709233)
(169, 0.020602396857879606)
(170, 0.020368559578769898)
(171, 0.020138507338121482)
(172, 0.019912162879159593)
(173, 0.019689450741525318)
(174, 0.019470297219589903)
(175, 0.019254630321401043)
(176, 0.019042379728289205)
(177, 0.01883347675515447)
(178, 0.018627854311457605)
(179, 0.0184254468629243)
(180, 0.018226190393981823)
(181, 0.018030022370931223)
(182, 0.01783688170586273)
(183, 0.017646708721320636)
(184, 0.017459445115714647)
(185, 0.017275033929479433)
(186, 0.017093419511982084)
(187, 0.016914547489169235)
(188, 0.01673836473195097)
(189, 0.016564819325316015)
(190, 0.016393860538167694)
(191, 0.016225438793876196)
(192, 0.01605950564153134)
(193, 0.015896013727896497)

(194, 0.01573491677003853)
(195, 0.015576169528637449)
(196, 0.015419727781954395)
(197, 0.015265548300447218)
(198, 0.015113588822027244)
(199, 0.01496380802793859)
(200, 0.01481616551924994)
(201, 0.014670621793945515)

(202, 0.014527138224605437)
(203, 0.01438736890713093)

(204, 0.014250539546155195)
(205, 0.014115571242966198)
(206, 0.01398243147597854)
(207, 0.013851088398572414)
(208, 0.013721510823583295)
(209, 0.013593668208150188)
(210, 0.013467530638913205)

(211, 0.013343068817564262)
(212, 0.01322025404672726)
(213, 0.01309905821618063)
(214, 0.012979453789398838)
(215, 0.012861413790416843)
(216, 0.01274491179100515)
(217, 0.012629921898153361)
(218, 0.012516418741851219)
(219, 0.012404377463161994)
(220, 0.012293773702586103)
(221, 0.012184583588700093)
(222, 0.012076783727073161)
(223, 0.011970351189449398)
(224, 0.01186526350319165)
(225, 0.011761498640982184)
(226, 0.011659035010769501)
(227, 0.011557851445963312)
(228, 0.011457927195863356)
(229, 0.011359241916321566)
(230, 0.011261775660631194)
(231, 0.011165508870638323)
(232, 0.011070422368064244)
(233, 0.010976497346044433)
(234, 0.010883715360868395)
(235, 0.010792058323921695)

(236, 0.010701508493823068)
(237, 0.010612694239553013)
(238, 0.010524939454181407)
(239, 0.010438219705069775)
(240, 0.010352519213191077)
(241, 0.01026782248889201)

(242, 0.010184114325863348)
(243, 0.010101379795244225)
(244, 0.010019604239858305)
(245, 0.009938773268580194)
(246, 0.00985887275082239)
(247, 0.009779888811153555)
(248, 0.009701807824029046)

(249, 0.009624616408643843)
(250, 0.009548301423900236)
(251, 0.009472849963483927)
(252, 0.00939824935105715)
(253, 0.009324487135551228)

(254, 0.009251551086570655)
(255, 0.009179429189895819)
(256, 0.009108109643086792)
(257, 0.009037580851186371)
(258, 0.00896783142251806)
(259, 0.008899260006120815)
(260, 0.008832115486814068)
(261, 0.00876568984815935)
(262, 0.008699973143390454)
(263, 0.008634955591719326)
(264, 0.008570627575166957)
(265, 0.008506979635459606)
(266, 0.008444002470989845)
(267, 0.008381686933840626)
(268, 0.008320024026871264)
(269, 0.008259004900862737)
(270, 0.008198620851723532)
(271, 0.008138863317753268)
(272, 0.008079723876961544)
(273, 0.008021194244443794)
(274, 0.007963266269810535)
(275, 0.007905931934670499)
(276, 0.007849183350162926)
(277, 0.007793012754546299)

(278, 0.007737412510830088)
(279, 0.007682375104460121)
(280, 0.007627893141048562)
(281, 0.007573959344153423)
(282, 0.0075205665530993756)
(283, 0.007467707720847371)

(284, 0.0074153759119058565)
(285, 0.007363564300283826)
(286, 0.007312266167487155)
(287, 0.007261474900556202)
(288, 0.007211183990140087)
(289, 0.007161387028614339)
(290, 0.007112077708233471)
(291, 0.007063249819323086)

(292, 0.007014897248506528)
(293, 0.006967013976969338)
(294, 0.006919594078757657)
(295, 0.006872631719110772)
(296, 0.006826121152827085)

(297, 0.006780056722665108)
(298, 0.0067344328577718035)
(299, 0.006689244072147366)
(300, 0.006644484963137547)
(301, 0.006600150209957581)
(302, 0.006556234572243802)
(303, 0.006512732888636181)
(304, 0.006469640075388475)
(305, 0.006426951125002954)
(306, 0.006384661104896593)
(307, 0.0063427651560914585)
(308, 0.006301258491930171)
(309, 0.006260136396817446)
(310, 0.006219394224986803)
(311, 0.0061790273992908306)
(312, 0.006139031410014316)
(313, 0.0060994018137132495)
(314, 0.006060134232072762)
(315, 0.006021224350790561)
(316, 0.00598266791847936)
(317, 0.005944460745593443)
(318, 0.005906598703372343)
(319, 0.005869077722809564)

(320, 0.005831893793636156)
(321, 0.005795042963327305)
(322, 0.005758521336127336)
(323, 0.005722325072091993)
(324, 0.0056864503861510235)

(325, 0.005650893547187218)
(326, 0.005616054318304152)
(327, 0.005581589574170559)
(328, 0.0055474224523471725)
(329, 0.005513549607624099)
(330, 0.005479967740687844)
(331, 0.005446673597385247)
(332, 0.005413663968000699)
(333, 0.005380935686544988)
(334, 0.0053484856300591846)

(335, 0.005316310717930253)
(336, 0.005284407911217323)
(337, 0.005252774211993267)
(338, 0.005221406662694877)
(339, 0.005190302345487994)

(340, 0.005159458381639975)
(341, 0.005128871930907377)
(342, 0.005098540190931737)
(343, 0.0050684603966477315)
(344, 0.005038629819700358)
(345, 0.0050090457678744925)
(346, 0.0049797055845321415)
(347, 0.004950606648062111)
(348, 0.004921746371335842)
(349, 0.004893122201178207)
(350, 0.004864731617842708)
(351, 0.00483657213449592)
(352, 0.004808641296717055)
(353, 0.004780936681997224)
(354, 0.004753455899254801)
(355, 0.00472619658835662)
(356, 0.004699156419644417)
(357, 0.004672333093476402)
(358, 0.00464572433976995)

(359, 0.00461932791755577)
(360, 0.004593141614539014)
(361, 0.004567163246668083)
(362, 0.004541390657709962)
(363, 0.004515821718834098)
(364, 0.004490454328202991)
(365, 0.004465286410568562)
(366, 0.004440315916877363)

(367, 0.004415540823880705)
(368, 0.00439095913375361)
(369, 0.004366568873716622)
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(1837, 4.753214404405811e-05)
(1838, 4.7411930144971405e-05)
(1839, 4.7292025455370166e-05)
(1840, 4.7172429154716065e-05)
(1841, 4.705314042379981e-05)
(1842, 4.693415844693183e-05)
(1843, 4.681548240958918e-05)
(1844, 4.6697111500449394e-05)
(1845, 4.657904490997763e-05)
(1846, 4.646128183073717e-05)
(1847, 4.63438214580132e-05)
(1848, 4.6226662989026716e-05)
(1849, 4.610980562387151e-05)
(1850, 4.599324856331871e-05)
(1851, 4.587699101241623e-05)
(1852, 4.576103217688381e-05)
(1853, 4.564537126592923e-05)
(1854, 4.5530007489426843e-05)
(1855, 4.541494006057698e-05)
(1856, 4.5300168195120094e-05)
(1857, 4.518569110945536e-05)
(1858, 4.507150802392588e-05)

(1859, 4.495761815952823e-05)
(1860, 4.484402074072779e-05)
(1861, 4.473071499295139e-05)
(1862, 4.461770014508927e-05)
(1863, 4.450497542761401e-05)
(1864, 4.439254007179556e-05)
(1865, 4.428039331361147e-05)
(1866, 4.416853438904892e-05)
(1867, 4.4056962537545375e-05)

(1868, 4.394567699963852e-05)

(1869, 4.3834677018998176e-05)
(1870, 4.372396184070236e-05)
(1871, 4.361353071170437e-05)
(1872, 4.350338288223842e-05)
(1873, 4.339351760331374e-05)
(1874, 4.328393412858972e-05)
(1875, 4.3174631713904e-05)
(1876, 4.306560961727008e-05)
(1877, 4.2956867097779835e-05)
(1878, 4.284840341841688e-05)
(1879, 4.27402178421435e-05)
(1880, 4.26323096358088e-05)
(1881, 4.252467806686164e-05)
(1882, 4.2417322405381605e-05)
(1883, 4.23102419239202e-05)
(1884, 4.220343589640373e-05)
(1885, 4.209690359875667e-05)
(1886, 4.199064430936837e-05)
(1887, 4.18846573084653e-05)
(1888, 4.177894187826511e-05)
(1889, 4.167349730250536e-05)
(1890, 4.1568322867692027e-05)
(1891, 4.1463417861533806e-05)
(1892, 4.13587815746596e-05)
(1893, 4.125441329889657e-05)
(1894, 4.11503123280497e-05)
(1895, 4.104647795821214e-05)
(1896, 4.094290948760669e-05)
(1897, 4.083960621564581e-05)
(1898, 4.0736567444492474e-05)
(1899, 4.0633792477495046e-05)
(1900, 4.0531280620748036e-05)
(1901, 4.04290311815271e-05)
(1902, 4.0327043469537196e-05)
(1903, 4.0225316795972805e-05)
(1904, 4.012385047398467e-05)

(1905, 4.002264381961515e-05)
(1906, 3.992169614913995e-05)
(1907, 3.9821006781566035e-05)
(1908, 3.9720575038316964e-05)
(1909, 3.962040024166857e-05)
(1910, 3.952048171646545e-05)
(1911, 3.942081878886917e-05)

(1912, 3.932141078744973e-05)

(1913, 3.92222570425587e-05)
(1914, 3.9123356885858666e-05)
(1915, 3.9024709651570815e-05)
(1916, 3.892631467506723e-05)
(1917, 3.8828171294118445e-05)
(1918, 3.8730278848266685e-05)
(1919, 3.863263667804319e-05)
(1920, 3.8535244126840253e-05)
(1921, 3.843810053981604e-05)
(1922, 3.834120526311205e-05)
(1923, 3.8244557645256525e-05)
(1924, 3.814815703669408e-05)
(1925, 3.8052002788690985e-05)
(1926, 3.7956094255987354e-05)
(1927, 3.786043079367285e-05)
(1928, 3.776501175859002e-05)
(1929, 3.766983651089339e-05)
(1930, 3.7574904410457295e-05)
(1931, 3.7480214820308264e-05)
(1932, 3.7385767104906115e-05)
(1933, 3.729156062998612e-05)
(1934, 3.719759476364976e-05)
(1935, 3.7103868875738584e-05)
(1936, 3.701038233689611e-05)
(1937, 3.691713452043882e-05)
(1938, 3.6824124801573994e-05)
(1939, 3.673135255599359e-05)
(1940, 3.663881716252532e-05)
(1941, 3.654651800047814e-05)
(1942, 3.645445445182511e-05)
(1943, 3.6362625899797304e-05)
(1944, 3.6271031729350266e-05)
(1945, 3.617967132700629e-05)
(1946, 3.6088544081008746e-05)
(1947, 3.599764938163239e-05)
(1948, 3.5906986620245705e-05)
(1949, 3.581655519040121e-05)

(1950, 3.572635448736579e-05)
(1951, 3.5636383907027264e-05)
(1952, 3.554664284854432e-05)
(1953, 3.5457130710913525e-05)
(1954, 3.5367846896554914e-05)
(1955, 3.52787908083471e-05)
(1956, 3.518996185102935e-05)
(1957, 3.510135943119995e-05)

(1958, 3.501298295731455e-05)
(1959, 3.492483183828128e-05)
(1960, 3.4836905485954076e-05)

(1961, 3.4749203313259886e-05)
(1962, 3.466172473482089e-05)
(1963, 3.45744691661734e-05)
(1964, 3.448743602579313e-05)
(1965, 3.4400624732287376e-05)
(1966, 3.431403470735557e-05)
(1967, 3.422766537282579e-05)
(1968, 3.414151615299164e-05)
(1969, 3.405558647305191e-05)
(1970, 3.396987576113542e-05)
(1971, 3.388438344502624e-05)
(1972, 3.379910895527972e-05)
(1973, 3.3714051724285604e-05)
(1974, 3.3629211184708155e-05)
(1975, 3.354458677182251e-05)
(1976, 3.346017792179885e-05)
(1977, 3.3375984073259395e-05)
(1978, 3.329200466540689e-05)
(1979, 3.320823913895831e-05)
(1980, 3.31246869369225e-05)
(1981, 3.304134750335216e-05)
(1982, 3.2958220283654136e-05)
(1983, 3.287530472489967e-05)
(1984, 3.279260027629036e-05)
(1985, 3.271010638728719e-05)
(1986, 3.262782250932304e-05)
(1987, 3.254574809626856e-05)
(1988, 3.246388260209403e-05)
(1989, 3.238222548304901e-05)
(1990, 3.230077619688209e-05)

(1991, 3.22195342020607e-05)
(1992, 3.2138498959483244e-05)
(1993, 3.2057669931231605e-05)
(1994, 3.1977046580102665e-05)
(1995, 3.189662837147596e-05)
(1996, 3.18164147712877e-05)

(1997, 3.173640524774258e-05)
(1998, 3.165659926928802e-05)
(1999, 3.157699630726023e-05)
(2000, 3.149759583323562e-05)
(2001, 3.141839732120963e-05)
(2002, 3.133940024541545e-05)

(2003, 3.126060408281416e-05)
(2004, 3.118200831075785e-05)
(2005, 3.1103612408701106e-05)
(2006, 3.1025415857109926e-05)
(2007, 3.094741813730483e-05)
(2008, 3.0869618733794884e-05)
(2009, 3.079201713100708e-05)
(2010, 3.071461281437508e-05)
(2011, 3.063740527205032e-05)
(2012, 3.056039399303269e-05)
(2013, 3.0483578467792103e-05)
(2014, 3.0406958187644653e-05)
(2015, 3.0330532645529777e-05)
(2016, 3.0254301336164658e-05)
(2017, 3.0178263755420453e-05)
(2018, 3.01024194003212e-05)
(2019, 3.0026767769198278e-05)
(2020, 2.9951308362311795e-05)
(2021, 2.9876040680760006e-05)
(2022, 2.9800964226945117e-05)
(2023, 2.9726078504883363e-05)
(2024, 2.9651383019737046e-05)
(2025, 2.957687727828022e-05)
(2026, 2.950256078811959e-05)
(2027, 2.9428433058938187e-05)
(2028, 2.9354493600782718e-05)
(2029, 2.9280741925618433e-05)
(2030, 2.920717754701675e-05)
(2031, 2.9133799979220766e-05)

(2032, 2.906060873792206e-05)
(2033, 2.8987603340570723e-05)
(2034, 2.891478330512981e-05)
(2035, 2.884214815131876e-05)
(2036, 2.8769697400767854e-05)
(2037, 2.8697430574839458e-05)
(2038, 2.8625347198204582e-05)
(2039, 2.8553446794642156e-05)
(2040, 2.848172889092652e-05)

(2041, 2.8410193014182234e-05)
(2042, 2.8338838693127357e-05)
(2043, 2.8267665457916845e-05)

(2044, 2.8196672839208468e-05)
(2045, 2.8125860369872486e-05)
(2046, 2.8055227583746456e-05)
(2047, 2.7984774015634323e-05)
(2048, 2.791449920161642e-05)
(2049, 2.7844402679359468e-05)
(2050, 2.7774483987338138e-05)
(2051, 2.7704742665456023e-05)
(2052, 2.763517825504467e-05)
(2053, 2.7565790298862547e-05)
(2054, 2.7496578339694896e-05)
(2055, 2.742754192315124e-05)
(2056, 2.735868059502143e-05)
(2057, 2.728999390251846e-05)
(2058, 2.7221481393966616e-05)
(2059, 2.7153142619577768e-05)
(2060, 2.7084977129895927e-05)
(2061, 2.7016984477350813e-05)
(2062, 2.6949164214857948e-05)
(2063, 2.6881515896905828e-05)
(2064, 2.6814039079865816e-05)
(2065, 2.6746733319970646e-05)
(2066, 2.667959817548957e-05)
(2067, 2.661263320579483e-05)
(2068, 2.6545837971205403e-05)
(2069, 2.647921203345231e-05)
(2070, 2.641275495505614e-05)
(2071, 2.634646630056936e-05)
(2072, 2.6280345634399847e-05)
(2073, 2.6214392523296395e-05)

(2074, 2.614860653448309e-05)
(2075, 2.6082987236746185e-05)
(2076, 2.6017534199345558e-05)
(2077, 2.595224699418916e-05)
(2078, 2.5887125192413874e-05)
(2079, 2.5822168367492184e-05)
(2080, 2.575737609352225e-05)
(2081, 2.5692747946780656e-05)

(2082, 2.5628283502925135e-05)
(2083, 2.5563982340256274e-05)
(2084, 2.549984403738632e-05)
(2085, 2.5435868174325856e-05)

(2086, 2.5372054332327608e-05)
(2087, 2.5308402093264296e-05)
(2088, 2.524491104087051e-05)
(2089, 2.5181580759188575e-05)
(2090, 2.511841083427636e-05)
(2091, 2.5055400852653225e-05)
(2092, 2.4992550401299303e-05)
(2093, 2.4929859069984428e-05)
(2094, 2.4867326448782246e-05)
(2095, 2.480495212760365e-05)
(2096, 2.4742735699767997e-05)
(2097, 2.4680676757809112e-05)
(2098, 2.4618774896425266e-05)
(2099, 2.455702971077014e-05)
(2100, 2.449544079722854e-05)
(2101, 2.4434007753415636e-05)
(2102, 2.437273017802091e-05)
(2103, 2.4311607670496867e-05)
(2104, 2.425063983183466e-05)
(2105, 2.4189826263786986e-05)
(2106, 2.412916656917789e-05)
(2107, 2.4068660351281032e-05)
(2108, 2.4008307216458295e-05)
(2109, 2.3948106769656615e-05)
(2110, 2.3888058617978197e-05)
(2111, 2.382816236974821e-05)
(2112, 2.376841763404834e-05)
(2113, 2.370882402149228e-05)
(2114, 2.3649381142672663e-05)
(2115, 2.3590088610023215e-05)
(2116, 2.3530946036731395e-05)
(2117, 2.347195303766903e-05)

(2118, 2.3413109227218546e-05)
(2119, 2.3354414222531913e-05)
(2120, 2.3295867640735984e-05)
(2121, 2.3237469100018383e-05)
(2122, 2.3179218220092424e-05)

(2123, 2.3121114620954795e-05)
(2124, 2.30631579245921e-05)
(2125, 2.300534775249705e-05)
(2126, 2.294768372939242e-05)

(2127, 2.289016547826431e-05)
(2128, 2.2832792625637875e-05)
(2129, 2.27755647970763e-05)
(2130, 2.2718481620749316e-05)
(2131, 2.266154272479451e-05)
(2132, 2.2604747737937468e-05)
(2133, 2.2548096291353104e-05)
(2134, 2.2491588016027165e-05)
(2135, 2.2435222543841832e-05)
(2136, 2.237899950912659e-05)
(2137, 2.232291854493389e-05)
(2138, 2.2266979287227695e-05)
(2139, 2.2211181371779792e-05)
(2140, 2.2155524436341254e-05)
(2141, 2.2100008118159538e-05)
(2142, 2.2044632056925564e-05)
(2143, 2.1989395892290852e-05)
(2144, 2.1934299265418308e-05)
(2145, 2.1879341818205927e-05)
(2146, 2.182452319313113e-05)
(2147, 2.1769843034491084e-05)
(2148, 2.1715300986850933e-05)
(2149, 2.1660896695973922e-05)
(2150, 2.160662980804525e-05)
(2151, 2.1552499971222558e-05)
(2152, 2.1498506833618968e-05)
(2153, 2.144465004469845e-05)
(2154, 2.1390929254965026e-05)
(2155, 2.1337344115031676e-05)
(2156, 2.1283894278101162e-05)
(2157, 2.1230579396553544e-05)
(2158, 2.1177399124271943e-05)
(2159, 2.1124353116952077e-05)
(2160, 2.1071441029465398e-05)

(2161, 2.101866251927021e-05)
(2162, 2.096601724361977e-05)
(2163, 2.091350486157767e-05)
(2164, 2.0861125031691305e-05)

(2165, 2.0808877414937548e-05)

(2166, 2.075676167270634e-05)
(2167, 2.0704777466955273e-05)
(2168, 2.0652924460054052e-05)
(2169, 2.060120231695458e-05)
(2170, 2.054961070193444e-05)
(2171, 2.0498149280301932e-05)
(2172, 2.0446817719635796e-05)
(2173, 2.0395615686218846e-05)
(2174, 2.034454284922341e-05)
(2175, 2.0293598877455114e-05)
(2176, 2.02427834410577e-05)
(2177, 2.019209621073741e-05)
(2178, 2.014153685869257e-05)
(2179, 2.009110505706291e-05)
(2180, 2.0040800479944273e-05)
(2181, 1.999062280137293e-05)
(2182, 1.994057169656519e-05)
(2183, 1.9890646841916863e-05)
(2184, 1.984084791376279e-05)
(2185, 1.979117459008135e-05)
(2186, 1.9741626550338887e-05)
(2187, 1.9692203472854417e-05)
(2188, 1.9642905038673913e-05)
(2189, 1.9593730928160083e-05)
(2190, 1.9544680824556496e-05)
(2191, 1.9495754409492666e-05)
(2192, 1.944695136763296e-05)
(2193, 1.939827138264674e-05)
(2194, 1.9349714140307352e-05)
(2195, 1.9301279326322103e-05)
(2196, 1.925296662850127e-05)
(2197, 1.9204775733813385e-05)
(2198, 1.915670633117407e-05)
(2199, 1.9108758110205875e-05)
(2200, 1.9060930760927946e-05)
(2201, 1.901322397437539e-05)
(2202, 1.8965637442598786e-05)

(2203, 1.8918170858198935e-05)
(2204, 1.887082391463628e-05)

(2205, 1.8823596306075545e-05)
(2206, 1.8776487727695153e-05)

(2207, 1.8729497875686773e-05)
(2208, 1.8682626446170394e-05)
(2209, 1.8635873137362857e-05)
(2210, 1.858923764663388e-05)
(2211, 1.8542719674068802e-05)
(2212, 1.8496318918285373e-05)
(2213, 1.8450035081080803e-05)
(2214, 1.840386786324864e-05)
(2215, 1.835781696736678e-05)
(2216, 1.831188209578327e-05)
(2217, 1.8266062953094307e-05)
(2218, 1.8220359243355615e-05)
(2219, 1.817477067178595e-05)
(2220, 1.8129296944766656e-05)
(2221, 1.8083937769221674e-05)
(2222, 1.803869285199763e-05)
(2223, 1.7993561902651353e-05)
(2224, 1.794854462942247e-05)
(2225, 1.790364074264045e-05)
(2226, 1.7858849953020493e-05)
(2227, 1.7814171971353417e-05)
(2228, 1.7769606510828345e-05)
(2229, 1.7725153283470278e-05)
(2230, 1.7680812003701688e-05)
(2231, 1.7636582385399677e-05)
(2232, 1.7592464143599118e-05)
(2233, 1.7548456994956792e-05)
(2234, 1.7504560655737867e-05)
(2235, 1.7460774843364105e-05)
(2236, 1.741709927548437e-05)
(2237, 1.737353367198727e-05)
(2238, 1.733007775159412e-05)
(2239, 1.728673123526522e-05)
(2240, 1.724349384372196e-05)
(2241, 1.7200365298530365e-05)
(2242, 1.7157345323029705e-05)

(2243, 1.7114433639854794e-05)
(2244, 1.7071629973103312e-05)
(2245, 1.702893404740643e-05)
(2246, 1.698634558823811e-05)

(2247, 1.694386432175993e-05)
(2248, 1.6901489974820712e-05)
(2249, 1.685922227495619e-05)
(2250, 1.681706095054346e-05)
(2251, 1.677500573049099e-05)
(2252, 1.6733056344238324e-05)
(2253, 1.669121252268452e-05)
(2254, 1.664947399633018e-05)
(2255, 1.660784049728954e-05)
(2256, 1.656631175836129e-05)
(2257, 1.652488751209949e-05)
(2258, 1.6483567492670777e-05)
(2259, 1.644235143508001e-05)
(2260, 1.640123907362211e-05)
(2261, 1.6360230145132226e-05)
(2262, 1.6319324385889678e-05)
(2263, 1.6278521533320283e-05)
(2264, 1.6237821325686424e-05)
(2265, 1.619722350100333e-05)
(2266, 1.615672779920558e-05)
(2267, 1.611633395997995e-05)
(2268, 1.6076041724931856e-05)
(2269, 1.60358508341802e-05)
(2270, 1.5995761030845227e-05)
(2271, 1.595577205686959e-05)
(2272, 1.5915883656422773e-05)
(2273, 1.587609557326988e-05)
(2274, 1.5836407552164162e-05)
(2275, 1.5796819338691907e-05)
(2276, 1.5757330678188945e-05)
(2277, 1.5717941318370987e-05)
(2278, 1.5678651006393156e-05)
(2279, 1.563945948993294e-05)
(2280, 1.560036651811829e-05)
(2281, 1.5561371839979965e-05)
(2282, 1.5522475205843827e-05)
(2283, 1.5483676365937966e-05)

(2284, 1.5444975072249153e-05)
(2285, 1.540637107604691e-05)
(2286, 1.5367864130513544e-05)
(2287, 1.5329453988577734e-05)
(2288, 1.5291140403842584e-05)
(2289, 1.5252923131668326e-05)
(2290, 1.521480192654187e-05)

(2291, 1.5176776544087842e-05)
(2292, 1.5138846741687076e-05)
(2293, 1.510101227569159e-05)
(2294, 1.506327290359025e-05)
(2295, 1.502562838447254e-05)
(2296, 1.498807847670776e-05)
(2297, 1.4950622939955891e-05)
(2298, 1.4913261534857865e-05)
(2299, 1.4875994021333695e-05)
(2300, 1.4838820161830722e-05)
(2301, 1.4801739718074825e-05)
(2302, 1.4764752452462391e-05)
(2303, 1.472785812821473e-05)
(2304, 1.469105650999649e-05)
(2305, 1.4654347361440526e-05)
(2306, 1.4617730448550572e-05)
(2307, 1.4581205536143413e-05)
(2308, 1.4544772391096798e-05)
(2309, 1.4508430780647735e-05)
(2310, 1.4472180471928264e-05)
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(2312, 1.4399952832938617e-05)
(2313, 1.4363975041200055e-05)
(2314, 1.43280876274805e-05)
(2315, 1.4292290361786052e-05)
(2316, 1.4256583016490834e-05)
(2317, 1.4220965362779689e-05)
(2318, 1.4185437172813118e-05)
(2319, 1.4149998219417722e-05)
(2320, 1.4114648276395213e-05)
(2321, 1.40793871179036e-05)
(2322, 1.4044214518456953e-05)
(2323, 1.4009130253543715e-05)
(2324, 1.3974134098389379e-05)
(2325, 1.3939225830275763e-05)
(2326, 1.3904405225602713e-05)
(2327, 1.3869672062362012e-05)
(2328, 1.3835026118126935e-05)
(2329, 1.3800467172371434e-05)

(2330, 1.3765995003686631e-05)
(2331, 1.3731609392563825e-05)

(2332, 1.3697310119384152e-05)
(2333, 1.3663096964573016e-05)
(2334, 1.3628969710300635e-05)
(2335, 1.3594928138317174e-05)
(2336, 1.3560972031807903e-05)
(2337, 1.3527101173537619e-05)
(2338, 1.3493315347860352e-05)
(2339, 1.3459614338554641e-05)
(2340, 1.3425997930833186e-05)
(2341, 1.3392465910414978e-05)
(2342, 1.335901806321589e-05)
(2343, 1.3325654176121451e-05)
(2344, 1.3292374035286099e-05)
(2345, 1.325917742953401e-05)
(2346, 1.3226064146494057e-05)
(2347, 1.3193033975382243e-05)
(2348, 1.3160086705455593e-05)
(2349, 1.312722212632113e-05)
(2350, 1.3094440028554003e-05)
(2351, 1.3061740203388081e-05)
(2352, 1.3029122441788288e-05)
(2353, 1.2996586536459908e-05)
(2354, 1.2964132279684287e-05)
(2355, 1.2931759464709824e-05)
(2356, 1.2899467884515196e-05)
(2357, 1.2867257334436812e-05)
(2358, 1.2835127608304413e-05)
(2359, 1.2803078501841347e-05)
(2360, 1.2771109810809571e-05)
(2361, 1.2739221331009477e-05)
(2362, 1.2707412859979842e-05)
(2363, 1.267568419467924e-05)
(2364, 1.2644035133185985e-05)
(2365, 1.261246547346151e-05)
(2366, 1.2580975014895671e-05)
(2367, 1.2549563557224688e-05)
(2368, 1.2518230899140065e-05)
(2369, 1.2486976842615667e-05)
(2370, 1.2455801187652997e-05)

(2371, 1.242470373614461e-05)
(2372, 1.2393684290019408e-05)

(2373, 1.2362742652015152e-05)
(2374, 1.2331878624751057e-05)
(2375, 1.2301092011809363e-05)
(2376, 1.2270382617580591e-05)
(2377, 1.2239750246490698e-05)
(2378, 1.220919470315546e-05)
(2379, 1.2178715794234577e-05)
(2380, 1.2148313324568366e-05)
(2381, 1.2117987101195229e-05)
(2382, 1.2087736931188125e-05)
(2383, 1.205756262227247e-05)
(2384, 1.2027463981744376e-05)
(2385, 1.1997440819406225e-05)
(2386, 1.1967492943394622e-05)
(2387, 1.193762016327051e-05)
(2388, 1.1907822289246403e-05)
(2389, 1.1878099131722691e-05)
(2390, 1.1848450501905492e-05)
(2391, 1.1818876211033962e-05)
(2392, 1.1789376070998143e-05)
(2393, 1.1759949894802278e-05)
(2394, 1.1730597494401663e-05)
(2395, 1.170131868425593e-05)
(2396, 1.1672113277620947e-05)
(2397, 1.1642981089020593e-05)
(2398, 1.1613921933319624e-05)
(2399, 1.1584935625723503e-05)
(2400, 1.1556021981932743e-05)
(2401, 1.1527180819069617e-05)
(2402, 1.1498411952588347e-05)
(2403, 1.1469715200291507e-05)
(2404, 1.1441090380321548e-05)
(2405, 1.1412537310233798e-05)
(2406, 1.13840558085411e-05)
(2407, 1.135564569502257e-05)
(2408, 1.1327306788560796e-05)
(2409, 1.1299038909458812e-05)
(2410, 1.1270841878204128e-05)

(2411, 1.124271551562307e-05)
(2412, 1.1214659643189553e-05)
(2413, 1.1186674082715996e-05)
(2414, 1.1158758656662093e-05)
(2415, 1.1130913187516817e-05)
(2416, 1.1103137498570577e-05)
(2417, 1.1075431413915046e-05)
(2418, 1.1047794757052913e-05)

(2419, 1.1020227353060114e-05)
(2420, 1.0992729026423322e-05)
(2421, 1.0965299603356577e-05)
(2422, 1.0937938909484374e-05)
(2423, 1.0910646770768245e-05)
(2424, 1.0883423014587716e-05)
(2425, 1.085626746773237e-05)
(2426, 1.0829179958563924e-05)
(2427, 1.0802160314390561e-05)
(2428, 1.0775208364864487e-05)
(2429, 1.0748323937966369e-05)
(2430, 1.0721506863402849e-05)
(2431, 1.0694756971834161e-05)
(2432, 1.0668074092866386e-05)
(2433, 1.0641458057213386e-05)
(2434, 1.0614908696542205e-05)
(2435, 1.0588425842391918e-05)
(2436, 1.0562009326482386e-05)
(2437, 1.0535658981795073e-05)
(2438, 1.0509374641337505e-05)
(2439, 1.0483156137525499e-05)
(2440, 1.0457003305271397e-05)
(2441, 1.0430915978123459e-05)
(2442, 1.0404893991354085e-05)
(2443, 1.0378937179180206e-05)
(2444, 1.0353045377697026e-05)
(2445, 1.0327218423024906e-05)
(2446, 1.030145615022841e-05)
(2447, 1.027575839779401e-05)
(2448, 1.025012500160804e-05)
(2449, 1.022455579989761e-05)
(2450, 1.0199050630296766e-05)
(2451, 1.017360933154479e-05)
(2452, 1.0148231742405273e-05)
(2453, 1.0122917701665996e-05)
(2454, 1.0097667049682778e-05)
(2455, 1.0072479626217783e-05)

(2456, 1.0047355271365809e-05)
(2457, 1.0022293826789268e-05)
iterations =  2457

CPU times: user 13.5 s, sys: 60 ms, total: 13.6 s
Wall time: 13.5 s

<matplotlib.colorbar.Colorbar at 0x7f8f0c143580>

Arrays to ASCII files#

x = y = z = np.arange(0.0,5.0,1.0)

np.savetxt('test.out', (x,y,z), delimiter=',')   # X is an array
%cat test.out

000000000000000000e+00,1.000000000000000000e+00,2.000000000000000000e+00,3.000000000000000000e+00,4.000000000000000000e+00
000000000000000000e+00,1.000000000000000000e+00,2.000000000000000000e+00,3.000000000000000000e+00,4.000000000000000000e+00
000000000000000000e+00,1.000000000000000000e+00,2.000000000000000000e+00,3.000000000000000000e+00,4.000000000000000000e+00

np.savetxt('test.out', (x,y,z), fmt='%1.4e')   # use exponential notation
%cat test.out

0000e+00 1.0000e+00 2.0000e+00 3.0000e+00 4.0000e+00
0000e+00 1.0000e+00 2.0000e+00 3.0000e+00 4.0000e+00
0000e+00 1.0000e+00 2.0000e+00 3.0000e+00 4.0000e+00

Arrays from ASCII files#

np.loadtxt('test.out')

array([[0., 1., 2., 3., 4.],
       [0., 1., 2., 3., 4.],
       [0., 1., 2., 3., 4.]])

save: Save an array to a binary file in NumPy .npy format
savez : Save several arrays into an uncompressed .npz archive
savez_compressed: Save several arrays into a compressed .npz archive
load: Load arrays or pickled objects from .npy, .npz or pickled files.

H5py#

Pythonic interface to the HDF5 binary data format. h5py user manual

import h5py as h5

with h5.File('test.h5','w') as f:
    f['x'] = x
    f['y'] = y
    f['z'] = z

with h5.File('test.h5','r') as f:
    for field in f.keys():
        print(field+':',np.array(f.get("x")))

x: [0. 1. 2. 3. 4.]
y: [0. 1. 2. 3. 4.]
z: [0. 1. 2. 3. 4.]

Slices Are References#

Slices are references to memory in the original array.
Changing values in a slice also changes the original array.

a = np.arange(10)
b = a[3:6]
b  # `b` is a view of array `a` and `a` is called base of `b`

array([3, 4, 5])

b[0] = -1
a  # you change a view the base is changed.

array([ 0,  1,  2, -1,  4,  5,  6,  7,  8,  9])

Numpy does not copy if it is not necessary to save memory.

c = a[7:8].copy() # Explicit copy of the array slice
c[0] = -1 
a

array([ 0,  1,  2, -1,  4,  5,  6,  7,  8,  9])

Fancy Indexing#

a = np.fromfunction(lambda i, j: (i+1)*10+j, (4, 5), dtype=int)
a

array([[10, 11, 12, 13, 14],
       [20, 21, 22, 23, 24],
       [30, 31, 32, 33, 34],
       [40, 41, 42, 43, 44]])

np.random.shuffle(a.flat) # shuffle modify only the first axis
a

/tmp/ipykernel_11100/865241466.py:1: UserWarning: you are shuffling a 'flatiter' object which is not a subclass of 'Sequence'; `shuffle` is not guaranteed to behave correctly. E.g., non-numpy array/tensor objects with view semantics may contain duplicates after shuffling.
  np.random.shuffle(a.flat) # shuffle modify only the first axis

array([[12, 44, 30, 24, 13],
       [20, 14, 33, 31, 43],
       [23, 10, 21, 40, 11],
       [34, 41, 22, 42, 32]])

locations = a % 3 == 0 # locations can be used as a mask
a[locations] = 0 #set to 0 only the values that are divisible by 3
a

array([[ 0, 44,  0,  0, 13],
       [20, 14,  0, 31, 43],
       [23, 10,  0, 40, 11],
       [34, 41, 22,  0, 32]])

a += a == 0
a

array([[ 1, 44,  1,  1, 13],
       [20, 14,  1, 31, 43],
       [23, 10,  1, 40, 11],
       [34, 41, 22,  1, 32]])

`numpy.take`#

a[1:3,2:5]

array([[ 1, 31, 43],
       [ 1, 40, 11]])

np.take(a,[[6,7],[10,11]])  # Use flatten array indices

array([[14,  1],
       [23, 10]])

Changing array shape#

grid = np.indices((2,3)) # Return an array representing the indices of a grid.
grid[0]

array([[0, 0, 0],
       [1, 1, 1]])

grid[1]

array([[0, 1, 2],
       [0, 1, 2]])

grid.flat[:] # Return a view of grid array

array([0, 0, 0, 1, 1, 1, 0, 1, 2, 0, 1, 2])

grid.flatten() # Return a copy

array([0, 0, 0, 1, 1, 1, 0, 1, 2, 0, 1, 2])

np.ravel(grid, order='C') # A copy is made only if needed.

array([0, 0, 0, 1, 1, 1, 0, 1, 2, 0, 1, 2])

Sorting#

a=np.array([5,3,6,1,6,7,9,0,8])
np.sort(a) # Return a copy

array([0, 1, 3, 5, 6, 6, 7, 8, 9])

array([5, 3, 6, 1, 6, 7, 9, 0, 8])

a.sort() # Change the array inplace
a

array([0, 1, 3, 5, 6, 6, 7, 8, 9])

Transpose-like operations#

a = np.array([5,3,6,1,6,7,9,0,8])
b = a
b.shape = (3,3) # b is a reference so a will be changed

array([[5, 3, 6],
       [1, 6, 7],
       [9, 0, 8]])

c = a.T # Return a view so a is not changed
np.may_share_memory(a,c)

True

c[0,0] = -1 # c is stored in same memory so change c you change a
a

array([[-1,  3,  6],
       [ 1,  6,  7],
       [ 9,  0,  8]])

c  # is a transposed view of a

array([[-1,  1,  9],
       [ 3,  6,  0],
       [ 6,  7,  8]])

b  # b is a reference to a

array([[-1,  3,  6],
       [ 1,  6,  7],
       [ 9,  0,  8]])

c.base  # When the array is not a view `base` return None

array([[-1,  3,  6],
       [ 1,  6,  7],
       [ 9,  0,  8]])

Methods Attached to NumPy Arrays#

a = np.arange(20).reshape(4,5)
np.random.shuffle(a.flat)
a

/tmp/ipykernel_11100/2362774682.py:2: UserWarning: you are shuffling a 'flatiter' object which is not a subclass of 'Sequence'; `shuffle` is not guaranteed to behave correctly. E.g., non-numpy array/tensor objects with view semantics may contain duplicates after shuffling.
  np.random.shuffle(a.flat)

array([[ 7, 14, 16, 10,  3],
       [15, 13,  1,  9,  2],
       [ 4,  6, 18, 12,  8],
       [ 0, 11,  5, 17, 19]])

a -= a.mean()
a /= a.std() # Standardize the matrix

a.std(), a.mean()

---------------------------------------------------------------------------
UFuncTypeError                            Traceback (most recent call last)
Cell In[69], line 1
----> 1 a -= a.mean()
      2 a /= a.std() # Standardize the matrix
      4 a.std(), a.mean()

UFuncTypeError: Cannot cast ufunc 'subtract' output from dtype('float64') to dtype('int64') with casting rule 'same_kind'

np.set_printoptions(precision=4)
print(a)

[[-1.6475  0.607  -1.4741  1.6475  0.2601]
 [ 0.7804 -1.1272 -0.607  -0.9538 -0.2601]
 [ 0.9538 -0.7804  0.4336  1.4741 -0.4336]
 [ 0.0867 -0.0867  1.1272 -1.3007  1.3007]]

a.argmax() # max position in the memory contiguous array

np.unravel_index(a.argmax(),a.shape) # get position in the matrix

(0, 3)

Array Operations over a given axis#

a = np.arange(20).reshape(5,4)
np.random.shuffle(a.flat)

a.sum(axis=0) # sum of each column

array([68, 39, 43, 40])

np.apply_along_axis(sum, axis=0, arr=a)

array([68, 39, 43, 40])

np.apply_along_axis(sorted, axis=0, arr=a)

array([[ 7,  0,  1,  2],
       [ 9,  3,  5,  4],
       [16, 11,  8,  6],
       [17, 12, 14, 10],
       [19, 13, 15, 18]])

You can replace the sorted builtin fonction by a user defined function.

np.empty(10)

array([ 0.1 ,  0.2 ,  0.25,  0.5 ,  1.  ,  2.  ,  2.5 ,  5.  , 10.  ,
       20.  ])

np.linspace(0,2*np.pi,10)

array([0.    , 0.6981, 1.3963, 2.0944, 2.7925, 3.4907, 4.1888, 4.8869,
       5.5851, 6.2832])

np.arange(0,2.+0.4,0.4)

array([0. , 0.4, 0.8, 1.2, 1.6, 2. ])

np.eye(4)

array([[1., 0., 0., 0.],
       [0., 1., 0., 0.],
       [0., 0., 1., 0.],
       [0., 0., 0., 1.]])

a = np.diag(range(4))
a

array([[0, 0, 0, 0],
       [0, 1, 0, 0],
       [0, 0, 2, 0],
       [0, 0, 0, 3]])

a[:,:,np.newaxis]

array([[[0],
        [0],
        [0],
        [0]],

       [[0],
        [1],
        [0],
        [0]],

       [[0],
        [0],
        [2],
        [0]],

       [[0],
        [0],
        [0],
        [3]]])

Create the following arrays#

[100 101 102 103 104 105 106 107 108 109]

Hint: numpy.arange

[-2. -1.8 -1.6 -1.4 -1.2 -1. -0.8 -0.6 -0.4 -0.2 0. 
0.2 0.4 0.6 0.8 1. 1.2 1.4 1.6 1.8]

Hint: numpy.linspace

[[ 0.001	0.00129155 0.0016681 0.00215443 0.00278256 
     0.003593810.00464159 0.00599484 0.00774264 0.01]

Hint: numpy.logspace

[[ 0. 0. -1. -1. -1.] 
 [ 0. 0.  0. -1. -1.] 
 [ 0. 0.  0.  0. -1.]
 [ 0. 0.  0.  0.  0.]
 [ 0. 0.  0.  0.  0.] 
 [ 0. 0.  0.  0.  0.] 
 [ 0. 0.  0.  0.  0.]]

Hint: numpy.tri, numpy.zeros, numpy.transpose

[[ 0.  1.  2.  3. 4.] 
 [-1.  0.  1.  2. 3.] 
 [-1. -1.  0.  1. 2.] 
 [-1. -1. -1.  0. 1.] 
 [-1. -1. -1. -1. 0.]]

Hint: numpy.ones, numpy.diag

Compute the integral numerically with Trapezoidal rule $$ I = \int_{-\infty}^\infty e^{-v^2} dv $$ with $v \in [-10;10]$ and n=20.

Views and Memory Management#

If it exists one view of a NumPy array, it can be destroyed.

big = np.arange(1000000)
small = big[:5]
del big
small.base

array([     0,      1,      2, ..., 999997, 999998, 999999])

Array called big is still allocated.
Sometimes it is better to create a copy.

big = np.arange(1000000)
small = big[:5].copy()
del big
print(small.base)

None

Change memory alignement#

del(a)
a = np.arange(20).reshape(5,4)
print(a.flags)

  C_CONTIGUOUS : True
  F_CONTIGUOUS : False
  OWNDATA : False
  WRITEABLE : True
  ALIGNED : True
  WRITEBACKIFCOPY : False
  UPDATEIFCOPY : False

b = np.asfortranarray(a) # makes a copy
b.flags

  C_CONTIGUOUS : False
  F_CONTIGUOUS : True
  OWNDATA : True
  WRITEABLE : True
  ALIGNED : True
  WRITEBACKIFCOPY : False
  UPDATEIFCOPY : False

b.base is a

False

You can also create a fortran array with array function.

c = np.array([[1,2,3],[4,5,6]])
f = np.asfortranarray(c)

print(f.ravel(order='K')) # Return a 1D array using memory order
print(c.ravel(order='K')) # Copy is made only if necessary

[1 4 2 5 3 6]
[1 2 3 4 5 6]

Broadcasting rules#

Broadcasting rules allow you to make an outer product between two vectors: the first method involves array tiling, the second one involves broadcasting. The last method is significantly faster.

n = 1000
a = np.arange(n)
ac = a[:, np.newaxis]   # column matrix
ar = a[np.newaxis, :]   # row matrix

%timeit np.tile(a, (n,1)).T * np.tile(a, (n,1))

17.6 ms ± 177 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

%timeit ac * ar

2.7 ms ± 154 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

np.all(np.tile(a, (n,1)).T * np.tile(a, (n,1)) == ac * ar)

True

Numpy Matrix#

Specialized 2-D array that retains its 2-D nature through operations. It has certain special operators, such as $*$ (matrix multiplication) and $**$ (matrix power).

m = np.matrix('1 2; 3 4') #Matlab syntax
m

matrix([[1, 2],
        [3, 4]])

a = np.matrix([[1, 2],[ 3, 4]]) #Python syntax
a

matrix([[1, 2],
        [3, 4]])

a = np.arange(1,4)
b = np.mat(a) # 2D view, no copy!
b, np.may_share_memory(a,b)

(matrix([[1, 2, 3]]), True)

a = np.matrix([[1, 2, 3],[ 3, 4, 5]])
a * b.T # Matrix vector product

matrix([[14],
        [26]])

m * a # Matrix multiplication

matrix([[ 7, 10, 13],
        [15, 22, 29]])

StructuredArray using a compound data type specification#

data = np.zeros(4, dtype={'names':('name', 'age', 'weight'),
                          'formats':('U10', 'i4', 'f8')})
print(data.dtype)

[('name', '<U10'), ('age', '<i4'), ('weight', '<f8')]

data['name'] = ['Pierre', 'Paul', 'Jacques', 'Francois']
data['age'] = [45, 10, 71, 39]
data['weight'] = [95.0, 75.0, 88.0, 71.0]
print(data)

[('Pierre', 45, 95.) ('Paul', 10, 75.) ('Jacques', 71, 88.)
 ('Francois', 39, 71.)]

RecordArray#

data_rec = data.view(np.recarray)
data_rec.age

array([45, 10, 71, 39], dtype=int32)

NumPy Array Programming#

Array operations are fast, Python loops are slow.
Top priority: avoid loops
It’s better to do the work three times witharray operations than once with a loop.
This does require a change of habits.
This does require some experience.
NumPy’s array operations are designed to make this possible.

Fast Evaluation Of Array Expressions#

The numexpr package supplies routines for the fast evaluation of array expressions elementwise by using a vector-based virtual machine.
Expressions are cached, so reuse is fast.

Numexpr Users Guide

import numexpr as ne
import numpy as np
nrange = (2 ** np.arange(6, 24)).astype(int)

t_numpy = []
t_numexpr = []

for n in nrange:
    a = np.random.random(n)
    b = np.arange(n, dtype=np.double)
    c = np.random.random(n)
    
    c1 = ne.evaluate("a ** 2 + b ** 2 + 2 * a * b * c ", optimization='aggressive')

    t1 = %timeit -oq -n 10 a ** 2 + b ** 2 + 2 * a * b * c
    t2 = %timeit -oq -n 10 ne.re_evaluate()

    t_numpy.append(t1.best)
    t_numexpr.append(t2.best)

%matplotlib inline
%config InlineBackend.figure_format = 'retina'
import matplotlib.pyplot as plt
import seaborn; seaborn.set()

plt.loglog(nrange, t_numpy, label='numpy')
plt.loglog(nrange, t_numexpr, label='numexpr')

plt.legend(loc='lower right')
plt.xlabel('Vectors size')
plt.ylabel('Execution Time (s)');

Python notebooks

Numpy

Contents

Numpy#

What provide Numpy to Python ?#

Routines for fast operations on arrays.#

Getting Started with NumPy#

Why Arrays ?#

Numpy Arrays: The `ndarray` class.#

Element wise operations are the “default mode”#

NumPy Arrays Properties#

Functions to allocate arrays#

Setting Array Elements Values#

Setting Array Elements Types#

Slicing x[lower:upper:step]#

Exercise:#

Multidimensional array#

Exercise#

Arrays to ASCII files#

Arrays from ASCII files#

H5py#

Slices Are References#

Fancy Indexing#

`numpy.take`#

Changing array shape#

Sorting#

Transpose-like operations#

Methods Attached to NumPy Arrays#

Array Operations over a given axis#

Create the following arrays#

Views and Memory Management#

Change memory alignement#

Broadcasting rules#

Numpy Matrix#

StructuredArray using a compound data type specification#

RecordArray#

NumPy Array Programming#

Fast Evaluation Of Array Expressions#

References#

Python notebooks

Numpy

Contents

Numpy#

What provide Numpy to Python ?#

Routines for fast operations on arrays.#

Getting Started with NumPy#

Why Arrays ?#

Numpy Arrays: The ndarray class.#

Element wise operations are the “default mode”#

NumPy Arrays Properties#

Functions to allocate arrays#

Setting Array Elements Values#

Setting Array Elements Types#

Slicing x[lower:upper:step]#

Exercise:#

Multidimensional array#

Exercise#

Arrays to ASCII files#

Arrays from ASCII files#

H5py#

Slices Are References#

Fancy Indexing#

numpy.take#

Changing array shape#

Sorting#

Transpose-like operations#

Methods Attached to NumPy Arrays#

Array Operations over a given axis#

Create the following arrays#

Views and Memory Management#

Change memory alignement#

Broadcasting rules#

Numpy Matrix#

StructuredArray using a compound data type specification#

RecordArray#

NumPy Array Programming#

Fast Evaluation Of Array Expressions#

References#

Numpy Arrays: The `ndarray` class.#

`numpy.take`#