Nahajate se tukaj

Domov › Uporaba › Programski jeziki in knjižnice

NumPy

Knjižnica NumPy je rezultat potreb po numeričnem računanju v Pytonu in vsebuje večino znanstvenih pomožnih knjižnic kot so

  • fft: Fast Fourrier Transform
  • random: tvorjenje naključnih števil
  • linalg: linearna algebra (Lapack, ATLAS ali MKL v ozadju + BLAS)
  • distutils: paket za pomoč pri distribuciji programov Python, ki uporabljajo numpy
  • f2py: paket za vključevanje fortranske kode v Python/numpy

NumPy vsebuje dva osnovna objekta na katerih se izvajajo algoritmi:

  • ndarray: Večdimenzijska polja
  • ufunc: Univerzalve funkcije, ki delujejo na poljih. Običajno se te funkcije aplicirajo na vsak element v polju. 

ndarray

Vsebuje ima naslednja dva argumenta pri kreiranju

  • shape: Tuple that gives the number of points in each dimension
  • dtype: Data type (int32, int64, float32, complex32,. . . ) 

Funkcije za tvorjenje ndarray so lahko

  • Empty, zeros and ones
    • zeros(shape[, dtype, order]): Returns an array filled with zeros
    • ones(shape[, dtype, order]): Returns an array filled with ones
    • empty(shape[, dtype, order]): Returns an allocated but empty array
    • something_like(a): Returns an array like a and filled with zeros, ones or not filled according to “something”
  • From existing data
    • array(object): Returns an array from a list or an existing array
    • loadtxt(fname[, dtype, comments, delimiter, . . . ]): Returns an array from a text file
    • fromstring(string[, dtype, count, sep]): Returns an array from a Python string
  • Numerical ranges
    • linspace(start, stop[, num, endpoint, retstep]): Returns an array of evenly spaced numbers over a specified interval 
    • arange([start], stop[, step, dtype]): like linspace but with the definition of a step instead of a number of points
    • logspace(start, stop[, num, endpoint, base]): Returns an array of numbers spaced evenly on a log scale
    • meshgrid(x, y): Returns coordinate matrices from two coordinate vectors
  • Matrices
    • mat(data[, dtype]): Interprets the input as a matrix
    • diag(v[, k]): Extracts a diagonal or constructs a diagonal array
    • tri(N[, M, k, dtype]): An array with ones at and below the given diagonal and zeros elsewhere
    • vander(x[, N]): Generates a Van der Monde matrix

Primeri uporabe NumPy

Kreiranje enk in ničel

>>> import numpy as np

>>> a = np.zeros(5)

>>> print ( a )

[ 0. 0. 0. 0. 0 . ]

>>> b = np.zeros ( ( 2 , 5 ) , dtype=int )

>>> print ( b )

[ [ 0 0 0 0 0]

[0 0 0 0 0 ] ]

>>> c = np.ones_like ( b )

>>> print ( c )

[ [ 1 1 1 1 1]

[1 1 1 1 1 ] ]

>>> print ( a.dtype )

float64

>>> print ( b.dtype )

int32

>>> print ( a.shape )

( 5 , )

>>> print ( b.shape )

(2 ,5)

Kreiranje iz obstoječih podatkov

>>> l =[0,1,2,3,4]

>>> print ( l )

[0 , 1 , 2 , 3 , 4]

>>> print ( type ( l ) )



>>> a = np.array ( l )

>>> print ( a )

[0 1 2 3 4]

>>> print ( type ( a ) )



>>> s = ” 1 2 3 4 5 ”

>>> b = np.fromstring (s , dtype=int , sep= ’ ’ )

>>> print ( b )

[1 2 3 4 5]

>>> print ( type ( b ) )



>>> print ( type ( s ) )

Kreiranje z numeričnimi območji

>>> a = np.linspace (0.0 ,1.0 ,10)

>>> print ( a )

[ 0. 0.11111111 0.22222222 0.33333333 0.44444444

0.55555556 0.66666667 0.77777778 0.88888889 1. ]

>>> a , dx = np.linspace (0.0 ,1.0 ,10 , endpoint=False , retstep=True )

>>> print ( a )

[ 0. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 . 9 ]

>>> print ( dx )

0.1

>>> x = np.logspace (-1 ,0 ,5) # start = 10∗∗−1 stop =10ˆ0

>>> print ( x )

[ 0.1 0.17782794 0.31622777 0.56234133 1. ]

>>> y = np.logspace (0 ,6 ,5 , base =2.0) # start = 2∗∗0 stop =2ˆ6

>>> print ( y )

[ 1. 2.82842712 8. 22.627417 64. ]

Tvorjenje matrik

>>> m = np.diag ( np.arange ( 6 ) )

>>> print (m)

[ [ 0 0 0 0 0 0]

[0 1 0 0 0 0]

[0 0 2 0 0 0]

[0 0 0 3 0 0]

[0 0 0 0 4 0]

[0 0 0 0 0 5 ] ]

>>> m = np.tri (6 , k=-2)

>>> print (m)

[ [ 0. 0. 0. 0. 0. 0. ]

[ 0. 0. 0. 0. 0. 0. ]

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

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

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

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

>>> print type (m)



>>> matrix = np.mat (m)

>>> print type ( matrix )

Manipulacija z obliko ndarray

  • reshape(shape[, order]): Returns a view of the array with a new shape
  • resize(new shape[, refcheck]): Changes shape and size of array in-place
  • transpose(*axes): Returns a view of the array with axes transposed
  • swapaxes(axis1, axis2): Returns a view of the array with axis1 and axis2 interchanged
  • flatten([order]): Returns a copy of the array collapsed into one dimension
>>> a = np.arange (10)

>>> b=a.reshape ( ( 2, 5 ) )

>>> b

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

[5 , 6 , 7 , 8 , 9 ] ] )

>>> b [ 0 , 0 ] = 10

>>> print ( b )

[ [ 1 0 , 1 , 2 , 3 , 4] ,

[ 5 , 6 , 7 , 8 , 9 ] ] )

>>> print ( a )

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

>>> b.resize ( ( 5, 2 ) )

>>> print ( b )

[ [ 1 0 1]

[ 2 3]

[ 4 5]

[ 6 7]

[ 8 9 ] ]

>>> c = b.transpose ( )

>>> print ( c )

[ [ 1 0 2 4 6 8]

[ 1 3 5 7 9 ] ]

Indeksiranje polja

  • Basic indexing (returns a view)
    • x[start:stop:step]: Select a subarray from start to stop with an increment of step
    • if start or stop are negative, they are interpreted as n +
    • start or n + stop, with n being the number of elements
    • if start is not given: default is 0
    • if stop is not given: default is n (number of elements)
    • if step is not given: default is 1
  • Advanced indexing (returns a copy)
    • Integer
    • Boolean
>>> x = np.arange (10)

>>> print ( x )

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

>>> print ( x [1:7:2] )

[1 3 5]

>>> print ( x [-2:10])

[8 9]

>>> print ( x [-3:3:-1])

[7 6 5 4]

>>> print ( x [5:] )

[5 6 7 8 9]

Napredno indeksiranje

>>> x=np.linspace (0, 1 ,10)

>>> print ( x<0.5)

[ True True True True True False False False False False ]

>>> print ( np.where ( x<0.5))

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

>>> y=8∗np.ones_like ( x )

>>> y[np.where ( x<0.5)]=10

>>> print ( y )

[ 10. 10. 10. 10. 10. 8. 8. 8. 8. 8. ]

Globalne operacije z indeksi

  • min([axis, out]): Returns the minimum
  • ptp([axis, out]): Peak to peak (maximum - minimum) value
  • conj(): Complex-conjugate all elements
  • sum([axis, dtype, out]): Returns the sum of the array elements
  • prod([axis, dtype, out]): Returns the product of the array elements
  • var([axis, dtype, out, ddof]): Returns the variance of the array elements
  • std([axis, dtype, out, ddof]): Returns the standard deviation of the array elements
  • all([axis, out]): Returns True if all elements evaluate to True
  • any([axis, out]): Returns True if any of the elements of a evaluate to True
>>> x = np.arange (10)

>>> print ( x )

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

>>> print ( x.min ( ) )

0

>>> print ( x.ptp ( ) )

9

>>> print ( x.mean ( ) )

4.5

>>> print ( x.var ( ) )

8.25

>>> print ( x.std ( ) )

2.87228132327

>>> print ( x.sum ( ) )

45

Univerzalne funkcije

  • Functions that are (re)defined to work on numpy arrays
  • Mathematical functions (square root, exponential... )
  • Trigonometric functions (sin, arccos, ... )
  • Bit-twiddling functions (bitwise and, ... )
  • Comparison and boolean functions (logical and, greater, ... )
>>> x=np.linspace (0 ,1 ,10)

>>> print ( x∗∗3)

[ 0. 0.00137174 0.01097394 0.03703704 0.0877915

0.17146776 0.2962963 0.47050754 0.70233196 1. ]

>>> print (2∗x / ( x +1))

[ 0. 0.2 0.36363636 0.5 0.61538462

0.71428571 0.8 0.875 0.94117647 1. ]

>>> print ( np.sin ( x∗np.pi ) )

[ 0.00000000e+00 3.42020143e−01 6.42787610e−01 8.66025404e−01

9.84807753e−01 9.84807753e−01 8.66025404e−01 6.42787610e−01

3.42020143e−01 1.22460635e−16]

>>> print ( np.sqrt( x ) )

[ 0. 0.33333333 0.47140452 0.57735027 0.66666667

0.74535599 0.81649658 0.8819171 0.94280904 1. ]

Naprednejše funkcije: Integracija, odvajanje in linearna algebra

  • trapz(y[, x, dx, axis]): Integrates along the given axis using the composite trapezoidal rule
  • gradient(f, *varargs): Returns the gradient of an N-dimensional array
  • diff(a[, n, axis]): Calculates the n-th order discrete difference along given axis
  • dot(a, b): Dot product of two arrays
  • inner(a, b): Inner product of two arrays
  • cholesky(a): Cholesky decomposition
  • eig(a): Computes the eigenvalues and right eigenvectors of a square array
  • eigvals(a): Computes the eigenvalues of a general matrix
  • det(a): Computes the determinant of an array
  • solve(a, b): Solves a linear matrix equation, or system of linear scalar equations
  • inv(a): Computes the (multiplicative) inverse of a matrix

Primer trapezne integracije napisane kot funkcija integrate.

import numpy as np



def f1(x):

    return 1 + x**2



def f2(x):

    return np.exp(-x**2)



def integrate(f, a, b, n):

    x = a + np.arange(1, n) * (b - a)/n

    result = 0.5 * ( f(a) + f(b) ) + np.sum(f(x))

    return result * (b - a)/n



a = 0.0

b = 10.0

for n in range(4,10):

    i1 = integrate(f1, a, b, n)

    i2 = integrate(f2, a, b, n)

    print "n=%d f1=%8.5f f2=%8.5f" % (n, i1, i2)

Povzetek

Poleg predstavljenih funkcij obstajajo še druga področja za analizo, FFT, sortiranje in iskanje, nakjučna števila, polinome, finanče funkcije, …

Numpy lahko v povezavi z drugimi knjižnicami (MPI4Py, scipy, matplotlib, ...) učinkovito nadomesti matlab. Najbolje pa se numpy priučino ma konkretnih problemih.