How can you with paper, pen and non-graphicwriting calculator roughly localize the two positive roots?

computer science


1.      Linear equationsystem

We are going to solve the linear equationsystem Ax=b In Matlab:

a)      Count the vector x with matlab

b)      Count the residual vector r=b-Ax with matlab

c)       Why is the residual vector r not exactly 0(zero)?

2.       Non-linear scalar equation with Newtons method

We want to determine the two positive roots to the equation f(x)=0 where

with a relative inaccuracy less than 10^-8.
ß relative inaccuracy (relative wrongness?).

a)      How can you with paper, pen and non-graphicwriting calculator roughly localize the two positive roots?

b)      Display now the function f(x) in matlab in two separate windows. In the first part window, f(x) should be displayed within the interval [0,a] where a is chosen so the smaller of the two roots is visible(you should be able to see 1-2 digits in the root). In the second part window, display f(x) in a bigger interval [0,A] so the bigger root of the two roots is visible. Use the subplot commando subplot(2,1,j) to display the j-th part window. Both graphs should have suitable names.

c)       Then write matlab code to determine the roots carefully with Newton method
Also let the program write as much intermediate results that you can answer these:

d)      Which are the roots?

e)      How is quadratic convergence defined for an iterative method for equation solutions?

f)       How can we see quadratic convergence in our printing? (Easiest to see in the bigger root)

g)      How do you check that the relative inaccuracy is small enough in your calculated roots-approximations?

h)      Does the function have any negative roots? If so, how many? (Don’t forget to motivate the answers)

3.       Same non-linear scalar equation with secant method

Determine the two positive roots of the equation (f(x) in the previous assignment) with secant method.

a)      What are the roots? Are they same?

b)      Do you have quick and regular convergence?

c)       Is your convergence quadratic? Is it linear?

d)      Does your convergence match/suit the theory for the method?

e)      Which of the methods do you prefer? Why?

4.       Interpolation and linear least squares method

The table below displays the amount of time the sun is expected being up in Stockholm during different days in 2020. (First column is number of hours, second column is number of minutes and third column is the date),

We are now going to do some adaptations to these datas, with both interpolation and least square method, to

1)      Calculate the adaptations largest value and for which ‘’x-value’’ this occurs

2)      Calculate the adaptations value for Christmas eve(Swedish Christmas eve), 24/12

3)      Plot how the adaptation looks like for the whole year, together with all given points.

The different adaptations that should be done:

A)     An interpolation polynomial that passes through every point.

B)      ‘’Piece by piece’’ linear interpolation through every point.

C)      Splines-approximation through every point.

D)     A quadratic polynomial that only uses data from 1 jun to 1 aug.

E)      A least square adapted quadratic polynomial that only uses data from 1 apr to 1 sep.

F)      A least square adapted quadratic polynomial that only uses data from 1 jan to 31 dec.

G)     The function y = c1 + c2*cos(w*x) + c3*sin(w*x) least square adapted from 1 jan to 31 dec, where w=2*pi/365 (ie one year periodic time).

H)     Which adaptation did you find was the best? Does it depend on which part of the curve you were interested in?

I)        If we represent the coefficients with c_i for the polynomials

Which ‘’run-up’’ needed to calculate the most coefficients?
Which ‘’run-up’’ needed the least amount coefficients?
Four of the ‘’run-ups’’ needed three coefficients, which one or ones did you find was the best? Did the best one/ones depend on what you had to calculate?

In these questions, it is difficult to define the correct answer; the important thing is to be able to motivate your opinion. You should be able to show the teacher that you understand and know which method should be used in different situations.

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