Informatics2-2021/Lab13

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Generate 500000 random points in the rectangle <math>[0,2]\times[0,4]</math>. Count how many of the points <math>(x,y)</math> have the property that <math>x^2>y</math>. Use this to approximate the integral <math>\int_0^2x^2 d x</math> Like in the end of the lecture.
 
Generate 500000 random points in the rectangle <math>[0,2]\times[0,4]</math>. Count how many of the points <math>(x,y)</math> have the property that <math>x^2>y</math>. Use this to approximate the integral <math>\int_0^2x^2 d x</math> Like in the end of the lecture.
  
 +
==Sorting Lambda==
 +
Write a Python program to sort a list of tuples using Lambda.
 +
Original list of tuples:
 +
[('English', 88), ('Science', 90), ('Maths', 97), ('Social sciences', 82)]
 +
Sorting the List of Tuples:
 +
[('Social sciences', 82), ('English', 88), ('Science', 90), ('Maths', 97)]
 +
 +
==Lambda Selection==
 +
Write a Python program to filter a list of integers using Lambda.
 +
Original list of integers:
 +
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
 +
Even numbers from the said list:
 +
[2, 4, 6, 8, 10]
 +
Odd numbers from the said list:
 +
[1, 3, 5, 7, 9]
 +
==Map Triples==
 +
Write a Python program to triple all numbers of a given list of integers. Use Python map.
 
=Homework 9=
 
=Homework 9=
 +
Each problem counts for 3 points
 
==Numeric integral==
 
==Numeric integral==
 
Estimate the integral of <math>e^{-x^2}</math> on the interval <math>[-2,5]</math> with the [https://en.wikipedia.org/wiki/Riemann_sum#Left_Riemann_sum left Riemann sum]!
 
Estimate the integral of <math>e^{-x^2}</math> on the interval <math>[-2,5]</math> with the [https://en.wikipedia.org/wiki/Riemann_sum#Left_Riemann_sum left Riemann sum]!
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==Gradient descent==
 
−
Let's have a vector-to-scalar function <math>f(x,y)=x^2+y^2</math>. Starting from <math>(x_0,y_0) = (-1, -1)</math> we will wind the minimum of the function.
 
−
A gradient step is when you subtract the <math>\nabla f(x,y)\cdot \epsilon</math> from the <math>(x,y)</math> point.
 
−
If you do this for small <math>\epsilon</math> many times then the point will converge a point where you cannot increase the function value any more, i.e. the gradient is zero.
 
−
This way you can find the minimum of the function (it will be <math>(x,y) = (0, 0)</math>).
 
−
* Store each step along the way, and plot them with matplotlib!
 
  
 
== Numeric derivative==
 
== Numeric derivative==
 
Plot the function <math>\sin(x)</math> and its derivative on the interval <math>[-\pi, \pi]</math>.
 
Plot the function <math>\sin(x)</math> and its derivative on the interval <math>[-\pi, \pi]</math>.
 
Calculate the derivative with [https://en.wikipedia.org/wiki/Finite_difference finite difference] method!
 
Calculate the derivative with [https://en.wikipedia.org/wiki/Finite_difference finite difference] method!

A lap jelenlegi, 2021. május 12., 12:06-kori változata


Tartalomjegyzék

Exercises

Introduction

Some exercises to get used to numpy

  1. Make a vector of length 10 with elements all zero! Then modify its 4th element to 1 (zeros)
  2. Make a 3-by-3 matrix with elements ranging from 0 up to 8 (reshape)
  3. Make a random vector of length 30 containing random number between 0 to 1! Calculate its average and standard deviation! (rand, mean, std)
    1. Make a random vector of the same length with elements between -3 and 2!
  4. Make a random unit vector in 5 dimensions! First make a random vector in 5 dimensions and then normalize it to unit length!

Monte-Carlo

Generate 500000 random points in the rectangle [0,2]\times[0,4]. Count how many of the points (x,y) have the property that x2 > y. Use this to approximate the integral \int_0^2x^2 d x Like in the end of the lecture.

Sorting Lambda

Write a Python program to sort a list of tuples using Lambda. Original list of tuples: 

[('English', 88), ('Science', 90), ('Maths', 97), ('Social sciences', 82)]

Sorting the List of Tuples: 

[('Social sciences', 82), ('English', 88), ('Science', 90), ('Maths', 97)]

Lambda Selection

Write a Python program to filter a list of integers using Lambda. Original list of integers: 

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

Even numbers from the said list: 

[2, 4, 6, 8, 10]

Odd numbers from the said list: 

[1, 3, 5, 7, 9]

Map Triples

Write a Python program to triple all numbers of a given list of integers. Use Python map.

Homework 9

Each problem counts for 3 points

Numeric integral

Estimate the integral of e^{-x^2} on the interval [ − 2,5] with the left Riemann sum!

Numeric derivative

Plot the function sin(x) and its derivative on the interval [ − π,π]. Calculate the derivative with finite difference method!

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