A lap 2021. május 11., 12:24-kori változata

Tartalomjegyzék

1 Exercises
- 1.1 Introduction
- 1.2 Monte-Carlo
2 Homework 9
- 2.1 Numeric integral
- 2.2 Numeric derivative

Exercises

Introduction

Some exercises to get used to numpy

Make a vector of length 10 with elements all zero! Then modify its 4th element to 1 (zeros)
Make a 3-by-3 matrix with elements ranging from 0 up to 8 (reshape)
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!
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 $x 2 > y$ . Use this to approximate the integral $\int_0^2x^2 d x$ Like in the end of the lecture.

Homework 9

Each problem counts for 2 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!

@@ 18. sor: / 18. sor: @@
 ==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]!
-==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==
 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!

Informatics2-2021/Lab13

A lap 2021. május 11., 12:24-kori változata

Tartalomjegyzék

Exercises

Introduction

Monte-Carlo

Homework 9

Numeric integral

Numeric derivative

Nézetek

Személyes eszközök

Navigáció

Keresés

Eszközök