cstut98

Computational Science CZ1103, Tutorial One (Solution)

1. Write MatLab commands for solving the following equations:

5x⁵+3x³+x=0

» v=[5,0,3,0,1,0];

» roots(v)

ans =

-0.2713 + 0.6112i

-0.2713 - 0.6112i

0.2713 + 0.6112i

0.2713 - 0.6112i

x⁹+x=9

» v=[1,0,0,0,0,0,0,0,1,-9];

» roots(v)

ans =

-1.2145 + 0.4488i

-1.2145 - 0.4488i

-0.6291 + 1.1229i

-0.6291 - 1.1229i

0.2410 + 1.2511i

0.2410 - 1.2511i

0.9749 + 0.7999i

0.9749 - 0.7999i

1.2554

3x+5y+7z=9

» A=[3,5,7;1,-8,2;7,2,-9];

» b=[9;20;-2];

» A\b

ans =

2.2897

-1.8138

1.6000

12x -8z=5

» A=[12,0,-8;1,2,1;0,9,19];

» b=[5;11;39];

» A\b

ans =

-0.0833

5.9167

-0.7500

2. Write Matlab commands for evaluating the following expressions:

x=(1-e^-p)/(1+e^-p) (Note that there is a mistake in the original handout)
y = |x| sin(x+pi/5)
z=e^-(x+iy)
w= log₁₀(x+y+z)

» x=(1-exp(-pi))/(1+exp(-pi))

x =

0.9172

» y=abs(x)*sin(x+pi*i/5)

y =

0.8766 + 0.3739i

» z=exp(-x-i*y)-sqrt(x^2+y^4)

z =

-0.6183 - 0.8628i

» w=log10(x+y+z)

w =

0.1049 - 0.1712i

3. Doubly stochastic matrix is a matrix whose row and column sums are all one. Use Matlab to do the following.

Start with a 5 by 5 magic square M, generate a doubly stochastic matrix P.
Verify that P is indeed a doubly stochastic matrix.
Generate a new matrix Q=P⁸.
Show that Q is also a doubly stochastic matrix.

» M=magic(5)

M =

17 24 1 8 15

23 5 7 14 16

4 6 13 20 22

10 12 19 21 3

11 18 25 2 9

» sum(M(1,:)) %magic sum

ans =

» P=M/65

P =

0.2615 0.3692 0.0154 0.1231 0.2308

0.3538 0.0769 0.1077 0.2154 0.2462

0.0615 0.0923 0.2000 0.3077 0.3385

0.1538 0.1846 0.2923 0.3231 0.0462

0.1692 0.2769 0.3846 0.0308 0.1385

» sum(P) %check the sum over elements in each column

ans =

1 1 1 1 1

» sum(P') %check the sum over elements in each row

ans =

1 1 1 1 1

» Q=P^8

Q =

0.2001 0.2000 0.2000 0.1999 0.2000

0.2000 0.2000 0.2000 0.2000 0.2000

0.1999 0.2000 0.2001 0.2000 0.1999

0.2000 0.2000 0.2000 0.2000 0.2000

0.2000 0.1999 0.2000 0.2000 0.2001

» sum(Q)

ans =

1.0000 1.0000 1.0000 1.0000 1.0000

» sum(Q')

ans =

1.0000 1.0000 1.0000 1.0000 1.0000

4. Construct the following matrix in Matlab

» M=[2:3:20;4:6:40;6:9:60;8:12:80;10:15:100]

M =

2 5 8 11 14 17 20

4 10 16 22 28 34 40

6 15 24 33 42 51 60

8 20 32 44 56 68 80

10 25 40 55 70 85 100