### Numerical Python

```Numerical Python
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Contents
Efficient array computing in Python
Creating arrays
Indexing/slicing arrays
Random numbers
Linear algebra
Plotting
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Ch. 4 in the course book
www.scipy.org
The NumPy manual
The SciPy tutorial
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Numerical Python (NumPy)
NumPy enables efficient numerical computing in Python
NumPy is a package of modules, which offers efficient arrays
(contiguous storage) with associated array operations coded in C or
Fortran
There are three implementations of Numerical Python
Numeric from the mid 90s (still widely used)
numpy from 2006
We recommend to use numpy (by Travis Oliphant)
from numpy import *
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A taste of NumPy: a least-squares procedure
x = linspace(0.0, 1.0, n)
y_line = -2*x + 3
y = y_line + random.normal(0, 0.25, n)
# coordinates
# line with noise
# goal: fit a line to the data points x, y
# create and solve least squares system:
A = array([x, ones(n)])
A = A.transpose()
result = linalg.lstsq(A, y)
# result is a 4-tuple, the solution (a,b) is the 1st entry:
a, b = result[0]
plot(x, y, ’o’,
# data points w/noise
x, y_line, ’r’,
# original line
x, a*x + b, ’b’) # fitted lines
legend(’data points’, ’original line’, ’fitted line’)
hardcopy(’myplot.png’)
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Resulting plot
y = -1.86794*x + 2.92875: fit to y = -2*x + 3.0 + normal noise
3.5
data points
original line
fitted line
3
2.5
2
1.5
1
0
0.2
0.4
0.6
0.8
1
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Making arrays
>>> from numpy import *
>>> n = 4
>>> a = zeros(n)
# one-dim. array of length n
>>> print a
[ 0. 0. 0. 0.]
>>> a
array([ 0., 0., 0., 0.])
>>> p = q = 2
>>> a = zeros((p,q,3))
# p*q*3 three-dim. array
>>> print a
[[[ 0. 0. 0.]
[ 0. 0. 0.]]
[[
[
>>>
(2,
0. 0. 0.]
0. 0. 0.]]]
a.shape
2, 3)
# a’s dimension
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Making float, int, complex arrays
>>> a = zeros(3)
>>> print a.dtype # a’s data type
float64
>>> a = zeros(3, int)
>>> print a
[0 0 0]
>>> print a.dtype
int32
>>> a = zeros(3, float32)
# single precision
>>> print a
[ 0. 0. 0.]
>>> print a.dtype
float32
>>> a = zeros(3, complex)
>>> a
array([ 0.+0.j, 0.+0.j, 0.+0.j])
>>> a.dtype
dtype(’complex128’)
>>> given an array a, make a new array of same dimension
>>> and data type:
>>> x = zeros(a.shape, a.dtype)
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Array with a sequence of numbers
linspace(a, b, n) generates n uniformly spaced coordinates,
starting with a and ending with b
>>> x = linspace(-5, 5, 11)
>>> print x
[-5. -4. -3. -2. -1. 0. 1.
2.
3.
4.
5.]
A special compact syntax is also available:
>>> a = r_[-5:5:11j]
>>> print a
[-5. -4. -3. -2. -1.
# same as linspace(-5, 5, 11)
0.
1.
2.
3.
4.
5.]
arange works like range (xrange)
>>> x = arange(-5, 5, 1, float)
>>> print x # upper limit 5 is not included!!
[-5. -4. -3. -2. -1. 0. 1. 2. 3. 4.]
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Warning: arange is dangerous
arange’s upper limit may or may not be included (due to round-off
errors)
Better to use a safer method: seq(start, stop, increment)
>>> from scitools.numpyutils import seq
>>> x = seq(-5, 5, 1)
>>> print x
# upper limit always included
[-5. -4. -3. -2. -1. 0. 1. 2. 3. 4. 5.]
The package scitools is available at
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Array construction from a Python list
array(list, [datatype]) generates an array from a list:
>>> pl = [0, 1.2, 4, -9.1, 5, 8]
>>> a = array(pl)
The array elements are of the simplest possible type:
>>> z = array([1, 2, 3])
>>> print z
# array of integers
[1 2 3]
>>> z = array([1, 2, 3], float)
>>> print z
[ 1. 2. 3.]
A two-dim. array from two one-dim. lists:
>>> x = [0, 0.5, 1]; y = [-6.1, -2, 1.2] # Python lists
>>> a = array([x, y]) # form array with x and y as rows
From array to list: alist = a.tolist()
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From “anything” to a NumPy array
Given an object a,
a = asarray(a)
converts a to a NumPy array (if possible/necessary)
Arrays can be ordered as in C (default) or Fortran:
a = asarray(a, order=’Fortran’)
isfortran(a) # returns True if a’s order is Fortran
Use asarray to, e.g., allow flexible arguments in functions:
def myfunc(some_sequence):
a = asarray(some_sequence)
return 3*a - 5
myfunc([1,2,3])
myfunc((-1,1))
myfunc(zeros(10))
myfunc(-4.5)
myfunc(6)
#
#
#
#
#
list argument
tuple argument
array argument
float argument
int argument
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Changing array dimensions
>>> a = array([0, 1.2, 4,
>>> a.shape = (2,3)
>>> print a
[[ 0.
1.2 4. ]
[-9.1 5.
8. ]]
>>> a.size
6
>>> a.shape = (a.size,)
>>> a.shape
(6,)
>>> print a
[ 0.
1.2 4. -9.1 5.
>>> a = a.reshape(2,3)
>>> a.shape
(2, 3)
-9.1, 5, 8])
# turn a into a 2x3 matrix
# turn a into a vector of length 6 again
8. ]
# same effect as setting a.shape
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Array initialization from a Python function
>>> def myfunc(i, j):
...
return (i+1)*(j+4-i)
...
>>> # make 3x6 array where a[i,j] = myfunc(i,j):
>>> a = fromfunction(myfunc, (3,6))
>>> a
array([[ 4.,
5.,
6.,
7.,
8.,
9.],
[ 6.,
8., 10., 12., 14., 16.],
[ 6.,
9., 12., 15., 18., 21.]])
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Basic array indexing
Note: all integer indices in Python start at 0!
a = linspace(-1,
a[2:4] = -1
a[-1] = a[0]
a[:]
= 0
a.fill(0)
1, 6)
# set
# set
# set
# set
a.shape = (2,3)
print a[0,1]
a[i,j] = 10
a[i][j] = 10
print a[:,k]
print a[1,:]
a[:,:] = 0
#
#
#
#
#
#
#
a[2] and a[3] equal to -1
last element equal to first one
all elements of a equal to 0
all elements of a equal to 0
turn a into a 2x3 matrix
print element (0,1)
assignment to element (i,j)
equivalent syntax (slower)
print column with index k
print second row
set all elements of a equal to 0
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>>> a = linspace(0, 29,
>>> a.shape = (5,6)
>>> a
array([[ 0., 1., 2.,
[ 6., 7., 8.,
[ 12., 13., 14.,
[ 18., 19., 20.,
[ 24., 25., 26.,
30)
3.,
9.,
15.,
21.,
27.,
4.,
10.,
16.,
22.,
28.,
5.,]
11.,]
17.,]
23.,]
29.,]])
>>> a[1:3,::2]
# a[i,j] for i=1,2 and j=0,2,4
array([[ 6.,
8., 10.],
[ 12., 14., 16.]])
>>> a[::3,2::2]
# a[i,j] for i=0,3 and j=2,4
array([[ 2.,
4.],
[ 20., 22.]])
>>> i = slice(None, None, 3);
>>> a[i,j]
array([[ 2.,
4.],
[ 20., 22.]])
j = slice(2, None, 2)
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Slices refer the array data
With a as list, a[:] makes a copy of the data
With a as array, a[:] is a reference to the data
>>> b = a[2,:]
>>> print a[2,0]
12.0
>>> b[0] = 2
>>> print a[2,0]
2.0
# extract 2nd row of a
# change in b is reflected in a!
Take a copy to avoid referencing via slices:
>>> b = a[2,:].copy()
>>> print a[2,0]
12.0
>>> b[0] = 2
# b and a are two different arrays now
>>> print a[2,0]
12.0
# a is not affected by change in b
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Loops over arrays (1)
Standard loop over each element:
for i in xrange(a.shape[0]):
for j in xrange(a.shape[1]):
a[i,j] = (i+1)*(j+1)*(j+2)
print ’a[%d,%d]=%g ’ % (i,j,a[i,j]),
print # newline after each row
A standard for loop iterates over the first index:
>>> print a
[[ 2.
6. 12.]
[ 4. 12. 24.]]
>>> for e in a:
...
print e
...
[ 2.
6. 12.]
[ 4. 12. 24.]
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Loops over arrays (2)
View array as one-dimensional and iterate over all elements:
for e in a.ravel():
print e
Use ravel() only when reading elements, for assigning it is better
to use shape or reshape first!
For loop over all index tuples and values:
>>>
...
...
(0,
(0,
(0,
(1,
(1,
(1,
for index, value in ndenumerate(a):
print index, value
0)
1)
2)
0)
1)
2)
2.0
6.0
12.0
4.0
12.0
24.0
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Array computations
Arithmetic operations can be used with arrays:
b = 3*a - 1
# a is array, b becomes array
1) compute t1 = 3*a, 2) compute t2= t1 - 1, 3) set b = t2
Array operations are much faster than element-wise operations:
>>>
>>>
>>>
>>>
>>>
import time # module for measuring CPU time
a = linspace(0, 1, 1E+07) # create some array
t0 = time.clock()
b = 3*a -1
t1 = time.clock()
# t1-t0 is the CPU time of 3*a-1
>>> for i in xrange(a.size): b[i] = 3*a[i] - 1
>>> t2 = time.clock()
>>> print ’3*a-1: %g sec, loop: %g sec’ % (t1-t0, t2-t1)
3*a-1: 2.09 sec, loop: 31.27 sec
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Standard math functions can take array arguments
# let b be an array
c
c
c
#
= sin(b)
= arcsin(c)
= sinh(b)
same functions for the cos and tan families
c
c
c
c
=
=
=
=
b**2.5 # power function
log(b)
exp(b)
sqrt(b)
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Other useful array operations
# a is an array
a.clip(min=3, max=12)
a.mean(); mean(a)
a.var(); var(a)
a.std(); std(a)
median(a)
cov(x,y)
trapz(a)
diff(a)
#
#
#
#
clip elements
mean value
variance
standard deviation
# covariance
# Trapezoidal integration
# finite differences (da/dx)
# more Matlab-like functions:
corrcoeff, cumprod, diag, eig, eye, fliplr, flipud, max, min,
prod, ptp, rot90, squeeze, sum, svd, tri, tril, triu
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More useful array methods and attributes
>>> a = zeros(4) + 3
>>> a
array([ 3., 3., 3., 3.]) # float data
>>> a.item(2)
# more efficient than a[2]
3.0
>>> a.itemset(3,-4.5)
# more efficient than a[3]=-4.5
>>> a
array([ 3. , 3. , 3. , -4.5])
>>> a.shape = (2,2)
>>> a
array([[ 3. , 3. ],
[ 3. , -4.5]])
>>> a.ravel()
# from multi-dim to one-dim
array([ 3. , 3. , 3. , -4.5])
>>> a.ndim
# no of dimensions
2
>>> len(a.shape)
# no of dimensions
2
>>> rank(a)
# no of dimensions
2
>>> a.size
# total no of elements
4
>>> b = a.astype(int)
# change data type
>>> b
array([3, 3, 3, 3])
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Modules for curve plotting and 2D/3D visualization
Matplotlib (curve plotting, 2D scalar and vector fields)
PyX (PostScript/TeX-like drawing)
Interface to Gnuplot
Interface to Vtk
Interface to OpenDX
Interface to IDL
Interface to Grace
Interface to Matlab
Interface to R
Interface to Blender
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Curve plotting with Easyviz
Easyviz is a light-weight interface to many plotting packages, using a
Matlab-like syntax
Goal: write your program using Easyviz (“Matlab”) syntax and
postpone your choice of plotting package
Note: some powerful plotting packages (Vtk, R, matplotlib, ...) may
be troublesome to install, while Gnuplot is easily installed on all
platforms
Easyviz supports (only) the most common plotting commands
Easyviz is part of SciTools (Simula development)
from scitools.all import *
(imports all of numpy, all of easyviz, plus scitools)
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Basic Easyviz example
from scitools.all import * # import numpy and plotting
t = linspace(0, 3, 51)
# 51 points between 0 and 3
y = t**2*exp(-t**2)
# vectorized expression
plot(t, y)
hardcopy(’tmp1.eps’) # make PostScript image for reports
hardcopy(’tmp1.png’) # make PNG image for web pages
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.5
1
1.5
2
2.5
3
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Decorating the plot
plot(t, y)
xlabel(’t’)
ylabel(’y’)
legend(’t^2*exp(-t^2)’)
axis([0, 3, -0.05, 0.6])
# [tmin, tmax, ymin, ymax]
title(’My First Easyviz Demo’)
# or
plot(t, y, xlabel=’t’, ylabel=’y’,
legend=’t^2*exp(-t^2)’,
axis=[0, 3, -0.05, 0.6],
title=’My First Easyviz Demo’,
hardcopy=’tmp1.eps’,
show=True)
# display on the screen (default)
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The resulting plot
My First Easyviz Demo
0.6
2
2
t *exp(-t )
0.5
y
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
t
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Plotting several curves in one plot
2
2
Compare f1 (t) = t2 e−t and f2 (t) = t4 e−t for t ∈ [0, 3]
from scitools.all import *
# for curve plotting
def f1(t):
return t**2*exp(-t**2)
def f2(t):
return t**2*f1(t)
t = linspace(0, 3, 51)
y1 = f1(t)
y2 = f2(t)
plot(t, y1)
hold(’on’)
plot(t, y2)
# continue plotting in the same plot
xlabel(’t’)
ylabel(’y’)
legend(’t^2*exp(-t^2)’, ’t^4*exp(-t^2)’)
title(’Plotting two curves in the same plot’)
hardcopy(’tmp2.eps’)
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The resulting plot
Plotting two curves in the same plot
0.6
2
2
t4*exp(-t2)
t *exp(-t )
0.5
y
0.4
0.3
0.2
0.1
0
0
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0.5
1
1.5
t
2
2.5
3
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Example: plot a function given on the command line
Task: plot (e.g.) f (x) = e−0.2x sin(2πx) for x ∈ [0, 4π]
Specify f (x) and x interval as text on the command line:
Unix/DOS> python plotf.py "exp(-0.2*x)*sin(2*pi*x)" 0 4*pi
Program:
from scitools.all import *
formula = sys.argv[1]
xmin = eval(sys.argv[2])
xmax = eval(sys.argv[3])
x = linspace(xmin, xmax, 101)
y = eval(formula)
plot(x, y, title=formula)
Thanks to eval, input (text) with correct Python syntax can be
turned to running code on the fly
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Plotting 2D scalar fields
from scitools.all import *
x = y = linspace(-5, 5, 21)
xv, yv = ndgrid(x, y)
values = sin(sqrt(xv**2 + yv**2))
surf(xv, yv, values)
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
6
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4
2
0
-2
-4
-6 -6
-4
-2
0
2
4
6
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# Matlab style commands:
setp(interactive=False)
surf(xv, yv, values)
colorbar()
colormap(hot())
axis([-6,6,-6,6,-1.5,1.5])
view(35,45)
show()
# Optional Easyviz (Pythonic) short cut:
surf(xv, yv, values,
colorbar=’on’,
colormap=hot(),
axis=[-6,6,-6,6,-1.5,1.5],
view=[35,45])
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The resulting plot
1
0.8
0.6
1.5
1
0.5
0
-0.5
-1
-1.5
0.4
0.2
0
-0.2
6
-6
4
-4
2
-2
0
0
-2
2
4
-0.4
-0.6
-0.8
-1
-4
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Other commands for visualizing 2D scalar fields
contour (standard contours)), contourf (filled contours),
contour3 (elevated contours)
mesh (elevated mesh),
meshc (elevated mesh with contours in the xy plane)
surf (colored surface),
surfc (colored surface with contours in the xy plane)
pcolor (colored cells in a 2D mesh)
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Commands for visualizing 3D fields
Scalar fields:
isosurface
slice_ (colors in slice plane),
contourslice (contours in slice plane)
Vector fields:
quiver3 (arrows), (quiver for 2D vector fields)
streamline, streamtube, streamribbon (flow sheets)
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