4. Calculations

4.1 Adding, subtracting, multiplying, and dividing values, and raising values to a power

4.2 Using built-in functions to compute the square root, remainder, absolute value, base of the natural logarithms raised to a power, and the logarithm

4.3 Ordering calculations

4.4 Performing statistical calculations to compute the sum, mean, standard deviation, variance, correlation, and least-squares fit

4.5 Performing statistical calculations with missing data

4.6 Calculating with matrices

4.7 Using linear algebra

4.8 Obtaining the max, min, sort, round, floor, and ceiling

4.9 Generating random numbers

4.10 Generating magic squares and calendars

4.11 Practicing calculations

top of page

4.1 Adding, subtracting, multiplying, dividing values, and raising values to a power

Code 4.1.1:

a = 1;

b = 2;

c = a + b % addition

d = 1;

e = c - d % subtraction

f = 4;

g = f * 3; % multiplication (note the use of the asterisk, *)

h = f/g % division

Output 4.1.1:

c =

e =

h =

0.3333

Code 4.1.2:

i = 2;

j = 3;

k = i^j % i raised to the j power

Output 4.1.2:

k =

Code 4.1.3:

m = 64;

n = 1/2;

p = m^n

Output 4.1.3:

p = 8

Code 4.1.4:

pp = 2^.2415

Output 4.1.4:

pp =

1.1822

top of page

4.2 Using built-in functions to compute the square root, remainder, absolute value, base of the natural logarithms raised to a power, and the logarithm

Code 4.2.1:

q = sqrt(m)

Output 4.2.1:

q = 8

Code 4.2.2:

subject_number = 7;

remainder = rem(subject_number,2)

Output 4.2.2:

remainder =

Code 4.2.3:

abs(-1)

Output 4.2.3:

ans =

Code 4.2.4:

k = exp(2)

Output 4.2.4:

k =

7.3891

Code 4.2.5:

exp(1)

Output 4.2.5:

ans =

2.7183

Code 4.2.6:

e = 12

Output 4.2.6:

e =

12.000

Code 4.2.7:

log(k)

Output 4.2.7:

ans =

Code 4.2.8:

log2(30)

Output 4.2.8:

ans =

4.9069

Code 4.2.9:

log10(30)

Output 4.2.9:

ans =

1.4771

Code 4.2.10:

log5(30)

Output 4.2.10:

>> log5(30)

??? Undefined function or variable 'log5'.

Code 4.2.11:

sqrt(-1)

Output 4.2.11:

ans =

0 + 1.0000i

Code 4.2.12:

imaginary = sqrt(1*-1)

Output 4.2.12:

imaginary =

0 + 1.0000i

Code 4.2.13:

complex = 2*imaginary

Output 4.2.13:

complex =

0 + 2.0000i

top of page

4.3 Ordering calculations

Code 4.3.1:

r = 2;

s = 3;

t = 4;

u = 5;

v = 6;

w(1) = r * s - t ^ u/v;

w(2) = r * s - (t ^ u)/v;

w(3) = r * (s - t ^ u)/v;

w(4) = r * (s - t) ^ u/v;

w(5) = (r * s) - t ^ u/v;

w(6) = (r * s - t) ^ u/v;

w(7) = (r * s - t) ^ (u/v);

w(8) = ((r * s - t) ^ u)/v;

w(9) = r * (s - t ^ u/v);

w’ % list w(1) through w(9) in column form

Output 4.3.1:

ans =

-164.6667

-340.3333

-0.3333

-164.6667

5.3333

1.7818

5.3333

-335.3333

Code 4.3.2:

w(9) = r * (s - t ^ u/v;

Output 4.3.2:

??? w(9) = r * (s - t ^ u/v;

Error: Incomplete or misformed expression or statement.

top of page

4.4 Performing statistical calculations to obtain the sum, mean, standard deviation, variance, correlation, and least-squares fit

Code 4.4.1:

r = [1:99];

sum_r = sum(r)

mean_r = mean(r)

median_r = median(r)

standard_deviation_r = std(r)

variance_r = var(r)

Output 4.4.1:

sum_r =

4950

mean_r =

median_r =

standard_deviation_r =

28.7228

variance_r =

825

Code 4.4.2:

r = [1:4; 11:14];

sum_vector = sum(r)

mean_vector = mean(r)

median_vector = median(r)

standard_deviation_vector = std(r)

variance_vector = var(r)

Output 4.4.2:

sum_vector =

12 14 16 18

mean_vector =

6 7 8 9

median_vector =

6 7 8 9

standard_deviation_vector =

7.0711 7.0711 7.0711 7.0711

variance_vector =

50 50 50 50

Code 4.4.3:

clear r % because it was used in the last example

s = [1:20];

t = [50:-1:31];

correlation_matrix = corrcoef(s,t)

r = correlation_matrix(1,2)

Output 4.4.3:

correlation_matrix =

1 -1

-1 1

r =

-1

top of page

4.5 Performing statistical calculations with missing data

Code 4.5.1:

% NaN_Calculations_01

% August 7, 2006

clc

clear all

% Review of special numbers other than NaN: pi and i.

% Reminder that the default value of i can be overwritten

% but can then be restored by clearing i

The_Special_Number_Pi = pi

The_Special_Number_Sqrt_Minus_1 = i

i = 10;

i_Redefined = i

Not_Really_The_Special_Number_Sqrt_Minus_1 = i

clear i

After_Clearing_The_Special_Number_Sqrt_Minus_1 = i

% Blank_Slate filled with NaN at first, but then gets some data

% and is finally assigned to Slate_With_Data

Blank_Slate(1:4,1:4) = NaN

Blank_Slate(1,2:4) = [7 8 9];

Blank_Slate(2,1:3) = [6 7 8];

Blank_Slate(3,3:4) = [10 11];

Blank_Slate(4,1) = [12];

Slate_With_Data = Blank_Slate

% Statistics

Column_Means = nanmean(Slate_With_Data)

Column_Standard_Seviations = nanstd(Slate_With_Data)

commandwindow

Outptu 4.5.1:

The_Special_Number_Pi =

3.1416

The_Special_Number_Sqrt_Minus_1 =

0 + 1.0000i

i_Redefined =

Not_Really_The_Special_Number_Sqrt_Minus_1 =

After_Clearing_The_Special_Number_Sqrt_Minus_1 =

0 + 1.0000i

Blank_Slate =

NaN NaN NaN NaN

Slate_With_Data =

NaN 7 8 9

6 7 8 NaN

NaN NaN 10 11

12 NaN NaN NaN

Column_Means =

9.0000 7.0000 8.6667 10.0000

Column_Standard_Seviations =

4.2426 0 1.1547 1.4142

top of page

4.6 Calculating with matrices

Code 4.6.1:

u = [1:6]

v = u + 20

Output 4.6.1:

u =

1 2 3 4 5 6

v =

21 22 23 24 25 26

Code 4.6.2:

w = v - 20

Output 4.6.2:

w =

1 2 3 4 5 6

Code 4.6.3:

x = w * 2

Output 4.6.3:

x =

2 4 6 8 10 12

Code 4.6.4:

y = x / 2

Output 4.6.4:

y =

1 2 3 4 5 6

Code 4.6.5:

Z1 = [1:6;7:12]

Z2 = Z1 + 2

Output 4.6.5:

Z1 =

1 2 3 4 5 6

7 8 9 10 11 12

Z2 =

3 4 5 6 7 8

9 10 11 12 13 14

Code 4.6.6:

Z3 = Z1 + Z2

Output 4.6.6:

Z3 =

4 6 8 10 12 14

16 18 20 22 24 26

Code 4.6.7:

Z4 = Z1 - 2

Z5 = Z1 – Z2

Output 4.6.7:

Z4 =

-1 0 1 2 3 4

5 6 7 8 9 10

Z5 =

-2 -2 -2 -2 -2 -2

Code 4.6.8:

aa = [1:4;5:8]

bb = [4:-1:1;8:-1:5]

cc = aa .* bb

Output 4.6.8:

aa =

1 2 3 4

5 6 7 8

bb =

4 3 2 1

8 7 6 5

cc =

4 6 6 4

40 42 42 40

Code 4.6.9:

dd = aa ./ bb

Output 4.6.9:

dd =

0.2500 0.6667 1.5000 4.0000

0.6250 0.8571 1.1667 1.6000

Code 4.6.10:

dd = aa .^ .25

Output 4.6.10:

dd =

1.0000 1.1892 1.3161 1.4142

1.4953 1.5651 1.6266 1.6818

top of page

4.8 Obtaining the max, min, sort, round, floor, and ceiling

Code 4.8.1:

array = [-1:.5:1];

max(array)

Output 4.8.1:

ans =

Code 4.8.2:

max(dd)

Output 4.8.2:

ans =

1.4953 1.5651 1.6266 1.6818

Code 4.8.3:

min(array)

Output 4.8.3:

ans =

-1

Code 4.8.4:

min(dd)

Output 4.8.4:

ans =

1.0000 1.1892 1.3161 1.4142

Code 4.8.5:

r = [3 2 1]

sorted_r = sort(r)

Output 4.8.5:

r =

3 2 1

sorted_r =

1 2 3

Code 4.8.6:

rr = [rand(10,1) randperm(10)']

srr1 = sort(rr)

Output 4.8.6:

rr =

0.1870 3.0000

0.9913 2.0000

0.7120 8.0000

0.8714 10.0000

0.4796 9.0000

0.4960 4.0000

0.2875 7.0000

0.0609 5.0000

0.2625 6.0000

0.1863 1.0000

srr1 =

0.0609 1.0000

0.1863 2.0000

0.1870 3.0000

0.2625 4.0000

0.2875 5.0000

0.4796 6.0000

0.4960 7.0000

0.7120 8.0000

0.8714 9.0000

0.9913 10.0000

Code 4.8.7:

srr2 = sortrows(rr,1)

srr3 = sortrows(rr,2)

Output 4.8.7:

srr2 =

0.0609 5.0000

0.1863 1.0000

0.1870 3.0000

0.2625 6.0000

0.2875 7.0000

0.4796 9.0000

0.4960 4.0000

0.7120 8.0000

0.8714 10.0000

0.9913 2.0000

srr3 =

0.1863 1.0000

0.9913 2.0000

0.1870 3.0000

0.4960 4.0000

0.0609 5.0000

0.2625 6.0000

0.2875 7.0000

0.7120 8.0000

0.4796 9.0000

0.8714 10.0000

Code 4.8.8:

[Y,ranked_rr] = sort(rr)

Output 4.8.8:

Y =

0.0609 1.0000

0.1863 2.0000

0.1870 3.0000

0.2625 4.0000

0.2875 5.0000

0.4796 6.0000

0.4960 7.0000

0.7120 8.0000

0.8714 9.0000

0.9913 10.0000

ranked_rr =

8 10

10 2

1 1

9 6

7 8

5 9

6 7

3 3

4 5

2 4

Code 4.8.9:

round(dd)

Output 4.8.9:

ans =

1 1 1 1

1 2 2 2

Code 4.8.10:

floor(dd)

Output 4.8.10:

ans =

1 1 1 1

Code 4.8.11:

ceil(dd)

Output 4.8.11:

ans =

2 2 2 2

Code 4.8.12:

fix_dd = fix(dd)

fix_minus_dd = fix(-dd)

floor_minus_dd = floor(-dd)

Output 4.8.11:

fix_dd =

1 1 1 1

fix_minus_dd =

-1 -1 -1 -1

floor_minus_dd =

-1 -2 -2 -2

-2 -2 -2 -2

top of page

4.9 Generating random numbers

Code 4.9.1:

rand('state',sum(100*clock));

rand(2,5)

Output 4.9.1:

ans =

0.9501 0.6068 0.8913 0.4565 0.8214

0.2311 0.4860 0.7621 0.0185 0.4447

Code 4.9.2:

randn('state',sum(100*clock));

randn(2,5)

Output 4.9.2:

ans =

-0.4326 0.1253 -1.1465 1.1892 0.3273

-1.6656 0.2877 1.1909 -0.0376 0.1746

Code 4.9.3:

randn('state',sum(100*clock));

mu = 10;

stdev = 15;

new_distribution = mu + (randn(2,5)*stdev)

Output 4.9.3:

new_distribution =

-14.0613 -5.8471 -2.0764 13.2898 -22.5601

13.8596 31.2271 17.9311 -3.8285 9.1122

Code 4.9.4:

r = randperm(8)

Output 4.9.4:

r =

5 6 1 4 2 8 3 7

top of page

4.10 Generating magic squares and calendars

Code 4.10.1:

n = 4;

magic(n)

Output 4.10.1:

ans =

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

Code 4.10.2:

calendar(1776,7)

Output 4.10.2:

Jul 1776

S M Tu W Th F S

0 1 2 3 4 5 6

7 8 9 10 11 12 13

14 15 16 17 18 19 20

21 22 23 24 25 26 27

28 29 30 31 0 0 0

0 0 0 0 0 0 0