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Intermediate Python: Using

NumPy, SciPy and Matplotlib

Lesson 19 - Odds and Ends

1

Lambda Operator

Python also has a simple way of defining a one-line function.

These are created using the Lambda operator.

The code must be a single, valid Python statement. Looping, if-then constructs, and other control statements cannot be use in Lambdas. 2 >>> bar = lambda x,y: x + y >>> bar(2,3) 5 >>> cube_volume = lambda l, w, h: l*w*h >>> cube_volume(2, 4.5, 7) 63.0

Physical Constants

The scipy.constants module contains many

physical constants! 3

Physical Constants

4

R molar gas constant

alpha fine-structure constant

N_A Avogadro constant

k Boltzmann constant sigma Stefan-Boltzmann constant

Wien Wien displacement law constant

Rydberg Rydberg constant

m_e electron mass m_p proton mass m_n neutron mass c speed of light in vacuum mu_0 the magnetic constant epsilon_0 the electric constant (vacuum permittivity), h the Planck constant hbar the Planck constant divided by 2.

G Newtonian constant of gravitation

g standard acceleration of gravity e elementary charge Always verify that they are in the correct units!!!

Examples

>>> import scipy.constants as sc >>> sc.c

299792458.0

>>> sc.h

6.62606957e-34

>>> sc.G

6.67384e-11

>>> sc.g

9.80665

>>> sc.e

1.602176565e-19

5 >>> sc.R

8.3144621

>>> sc.N_A

6.02214129e+23

>>> sc.k

1.3806488e-23

>>> sc.sigma

5.670373e-08

>>> sc.m_e

9.10938291e-31

Constants Database

There are more constants in the constants database.

These are accecces using dictionary keys.

The methods available are:

value() # the value of the constant unit() # the units of the contant precision() # the precision of the constant

To see a list of available constants, go to: http://docs.scipy.org/doc/scipy/reference/constants.html#module-scipy.constants

6

Constants Database Example

>>> sc.value('Rydberg constant')

10973731.568539

>>> sc.unit('Rydberg constant') 'm^-1' >>> sc.precision('Rydberg constant')

5.011968778030033e-12

7

Metric Prefixes

>>> sc.yotta 1e+24
>>> sc.mega

1000000.0

>>> sc.nano 1e-09 8

Conversions

There are conversion factors to MKS units.

9 >>> sc.gram 0.001 >>> sc.lb

0.45359236999999997

>>> sc.mile

1609.3439999999998

>>> sc.foot

0.30479999999999996

>>> sc.atm

101325.0

>>> sc.psi

6894.757293168361

>>> sc.gallon

0.0037854117839999997

>>> sc.mph

0.44703999999999994

>>> sc.knot

0.5144444444444445

>>> sc.erg 1e-07 >>> sc.Btu

1055.05585262

Temperature Conversions

10 zero_Celsius zero of Celsius scale in Kelvin degree_Fahrenheit one degree Fahrenheit difference in Kelvin

C2K(C) Convert Celsius to Kelvin

K2C(K) Convert Kelvin to Celsius

F2C(F) Convert Fahrenheit to Celsius

C2F(C) Convert Celsius to Fahrenheit

F2K(F) Convert Fahrenheit to Kelvin

K2F(K) Convert Kelvin to Fahrenheit

Special Functions

The scipy.special module contains many

special functions, such as Bessel functions,

Legendre Polynomials, etc.

11

Interpolation

The scipy.interpolate module contains

functions for interpolating 1D and 2D data. 12 scipy.interpolate.interp1d()

This function takes an array of x values and an

array of y values, and then returns a function.

By passing an x value to the function the

function returns the interpolated y value.

It uses linear interpolation as the default, but

also can use other forms of interpolation including cubic splines or higher-order splines. 13 interp1d() Example 14 from scipy.interpolate import interp1d import numpy as np import matplotlib.pyplot as plt x = np.arange(0, 10) y = np.array([3.0, -4.0, -2.0, -1.0, 3.0, 6.0, 10.0, 8.0, 12.0, 20.0]) f = interp1d(x, y, kind = 'cubic') xint = 3.5 yint = f(xint) plt.plot(x, y, 'o', c = 'b') plt.plot(xint, yint, 's', c = 'r') plt.show()

Cubic Spline

Interpolated Point

Results

15

Interpolated Point

interp1d() Example 16 from scipy.interpolate import interp1d import numpy as np import matplotlib.pyplot as plt x = np.arange(0, 10) y = np.array([3.0, -4.0, -2.0, -1.0, 3.0, 6.0, 10.0, 8.0, 12.0, 20.0]) f = interp1d(x, y, kind = 'cubic') xint = np.arange(0, 9.01, 0.01) yint = f(xint) plt.plot(x, y, 'o', c = 'b') plt.plot(xint, yint, '-r') plt.show()

Cubic Spline

Interpolated Points

Results

17

Interpolated Curve

scipy.interpolate.interp2d()

This function works similarly to interp1d(), but

can interpolate values on a 2D grid.

The input values can be either regularly

spaced, or irregularly spaced. 18

Numerical Differentiation

scipy.misc.derivative(f, x, dx=dx, n = n) is a function to find the nth derivative of a function f.

The function can either be a lambda or a user

defined function. 19

Derivatives Example

from scipy.misc import derivative import numpy as np import matplotlib.pyplot as plt fig, ax = plt.subplots(3,1,sharex = True) ax[0].plot(x,f(x)) ax[0].set_ylabel(r'$f(x)$') ax[1].plot(x,first) ax[1].set_ylabel(r'$f\/\prime(x)$') ax[2].plot(x,second) ax[2].set_xlabel(r'$x$') plt.show() 20 f = lambda x : np.exp(-x)*np.sin(x) x = np.arange(0,10, 0.1) first = derivative(f,x,dx=1,n=1) second = derivative(f,x,dx=1,n=2)

Derivatives Results

21

Numerical Integration

scipy.integrate is a module that contains functions for integration.

Integration can be performed on a function

defined by a lambda.

Integration can also be performed given an

array of y values. 22

Numerical Integration

scipy.integrate is a module that contains functions for integration.

Integration can be performed on a function

defined by a lambda.

Integration can also be performed given an

array of y values. 23

Integration Example

Suppose we want to evaluate the integral.

24
2 0 sinxe xdx

Integration Example

>>> import scipy.integrate as integrate >>> import numpy as np >>> f = lambda x : np.exp(-x)*np.sin(x) >>> I = integrate.quad(f, 0, 2*np.pi) >>> print(I) (0.49906627863414593, 6.023731631928322e-15) 25

Value of integral. Estimate of absolute error.

Can Even Do Infinite Bounds

Suppose we want to evaluate the integral.

26
0 sinxe xdx

Infinite Integration Example

>>> I = integrate.quad(f, 0, float('inf')) >>> print(I) (0.5000000000000002, 1.4875911858955308e-08) 27

Note: The exact value of the integral is 0.5.

Double and Triple Integrals

There are also functions for doing double and

triple integrals. 28

Integration of Array Data

If you only have an array of y values, without

knowing the functional dependence of x and y, you can still integrate.

For this, use the scipy.integrate.simps()

function, which uses Simpson's rule.

You can either specify the x values, or just give

the x increment, dx. 29

Example of Cumulative Integration

import scipy.integrate as integrate import numpy as np x = np.arange(0, 20, 2) y = np.array([0, 3, 5, 2, 8, 9, 0, -3, 4, 9], dtype = float)

I = integrate.simps(y,x)

print(I) 30

Note that I is an array with one less element

than y.quotesdbs_dbs22.pdfusesText_28