8 - Chapman MATLAB Programming for Engineers

-2 kmin = 2

The maximum value is found to be 5, the minimum value −2, and the index of the minimum value is 2. Thus, vmin = v(kmin) = v(2) = -2.

For the random data vectors data1 and data2:

>>max(data1) ans =

3.9985

>>min(data1) ans =

2.0005

>>[max2,kmax2] = max(data2) max2 =

5.7316 kmax2 =

256

>>[min2,kmin2] = min(data2) min2 =

0.3558 kmin2 =

268

These results can be approximately confirmed from Figure 7.1.

For a matrix, the min and max functions return a row vector of the minimum or maximum elements of each column of the matrix.

>> B = [-1 1 7 0; -3 5 5 8; 1 4 4 -8] B =

Example 7.1 Minimum cost tank design

A cylindrical tank with a hemispherical top, as shown in Figure 7.2, is to be constructed that will hold 5.00 × 105L when filled. Determine the tank radius R and height H to minimize the tank

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cost if the cylindrical portion costs $300/m2 of surface area and the hemispherical portion costs $400/m2.

Figure 7.2: Tank configuration

Mathematical model:

Cylinder volume: Vc = πR2H

Hemisphere volume: Vh = 23 πR3

Cylinder surface area: Ac = 2πRH

Hemisphere surface area: Ah = 2πR2

Assumptions:

Tank contains no dead air space.

Concrete slab with hermetic seal is provided for the base.

Cost of the base does not change appreciably with tank dimensions.

Computational method:

Express total volume in meters cubed (note: 1m3 = 1000L) as a function of height and radius

Vtank = Vc + Vh

For Vtank = 5 × 105L = 500m3:

500 = πR2H + 23πR3

Express cost in dollars as a function of height and radius

C = 300Ac + 400Ah = 300(2πRH) + 400(2πR2)

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Method: compute H and then C for a range of values of R, then find the minimum value of C and the corresponding values of R and H.

To determine the range of R to investigate, make an approximation by assuming that H = R. Then from the tank volume:

Vtank = 500 = πR3 + 23πR3 = 53πR3

From Matlab:

>> Rest = (300/pi)^(1/3) Rest =

4.5708

We will investigate R in the range 3.0 to 7.0 meters.

Computational implementation:

Matlab script to determine the minimum cost design:

% Compute and display minimum cost, corresponding H & R [Cmin kmin] = min(C);

disp(’Minimum cost (dollars):’) disp(Cmin)

disp(’Radius R for minimum cost (m):’) disp(R(kmin))

disp(’Height H for minimum cost (m):’) disp(H(kmin))

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Running the script:

Minimum cost (dollars): 9.1394e+004

Radius R for minimum cost (m): 4.9240

Height H for minimum cost (m): 3.2816

Note that the radius corresponding to minimum cost (Rmin = 4.9240 is close to the approximate value, Rest = 4.5708 that we computed to assist in the selection of a range of R to investigate. The plot of cost C versus radius R is shown in Figure 7.3.

Figure 7.3: Tank design problem: cost versus radius

7.2Sums and Products

If x is a vector with N elements denoted x(n), n = 1, 2, . . . , N, then:

• The sum y is the scalar

N

y = x(n) = x(1) + x(2) + · · · + x(N)

n=1

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• The product y is the scalar

N

y = x(n) = x(1) · x(2) · · · x(N)

n=1

• The cumulative sum y is the vector having elements y(k), k = 1, 2, . . . , N

k

y(k) = x(n) = x(1) + x(2) + · · · + x(k)

n=1

• The cumulative product y is the vector having elements y(k), k = 1, 2, . . . , N

k

y(k) = x(n) = x(1) · x(2) · · · x(k)

n=1

If X is a matrix with M rows and N columns with elements denoted x(m, n), m = 1, 2, . . . , M, n = 1, 2, . . . , N, then

• The column sum y is the vector having elements y(n), n = 1, 2, . . . , N

M

y(n) = x(m, n) = x(1, n) + x(2, n) + · · · + x(M, n)

m=1

• The column product y is the vector having elements y(n), n = 1, 2, . . . , N

M

y(n) = x(m, n) = x(1, n) · x(2, n) · · · x(M, n)

m=1

• The cumulative column sum Y is the matrix with M rows and N columns with elements denoted y(k, n), k = 1, 2, . . . , M, n = 1, 2, . . . , N

k

y(k, n) = x(m, n) = x(1, n) + x(2, n) + · · · + x(k, n)

m=1

• The cumulative column product Y is the matrix with M rows and N columns with elements denoted y(k, n), k = 1, 2, . . . , M, n = 1, 2, . . . , N

k

y(k, n) = x(m, n) = x(1, n) · x(2, n) · · · x(k, n)

m=1

The Matlab functions implementing these mathematical functions are the following.

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sum(x) Returns the sum of the elements in vector x, or the row vector of the sum of elements of each column in matrix x

prod(x) Returns the product of the elements in vector x, or the row vector of the product of elements of each column in matrix x

cumsum(x) Returns a vector the same size as x containing the cumulative sum of the elements in vector x, or a matrix the same size as x containing cumulative sums of values from the columns of x.

cumprod(x) Returns a vector the same size as x containing the cumulative product of the elements in vector x, or a matrix the same size as x containing cumulative products of values from the columns of x.

Example 7.2 Summation

The following is a summation identity

This identity can be checked with Matlab, for example, for N = 8:

>>N = 8;

>>n =1:8;

>>S = sum(n)

S =

36

>>N*(N+1)/2 ans =

36

Example 7.3 Factorial

The factorial of n is expressed and defined as

n! = n · (n − 1) · (n − 2) · · · 1

This can be computed as prod(1:n), as in the following examples:

>>nfact = prod(1:4) nfact =

24

>>nfact = prod(1:70) nfact =

1.1979e+100

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7.3Statistical Analysis

Many engineering measurements have an element of randomness. Statistics is a branch of applied mathematics involving the analysis, interpretation, and presentation of data including some degree of randomness or uncertainty. In descriptive statistics, the important features or properties of a set of data are summarized or described.

Continuous random variable: A variable vector x, whose elements can have any of a continuum of values for each measurement, or observation.

Population: all possible measurements of a random variable

Sample: finite number of measurements. Assume that N measurements have been made of a random variable, denoted as the vector x, with elements x(n), n = 1, . . . , N.

Frequency distribution of a random variable: indicates the relative frequency with which specific values of this random variable occur in the population.

Histogram: frequency distribution of sample data, indicating the frequency with which sample values fall within ranges of values, called bins.

• Range of values: xmin ≤ x ≤ xmax

• Number of bins: m

• Bin width: ∆x = xmax − xmin m

•Histogram value in bin k: N(k), the number of samples with value xmin + (k − 1)∆x ≤ x ≤ xmin + k∆x. This is also known as the absolute frequency.

The Matlab commands for generating and plotting a histogram are;

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For example, the following commands produce the 25-bin histograms shown in Figure 7.4, operating on the random data vectors data1 and data2.

% Histogram plots

%

subplot(2,1,1),hist(data1,25),...

title(’Histogram of data1’),...

xlabel(’x’),...

ylabel(’N’),...

subplot(2,1,2),hist(data2,25),...

title(’Histogram of data2’),...

xlabel(’x’),...

ylabel(’N’)

Note the di erences in the nature of the two distributions. Random data data1 is called a uniform distribution, as it has roughly equal, or uniform, frequency in all bins. Random data data2 has a bell-shaped distribution, called a Gaussian distribution or Normal distribution.

Relative frequency histogram: The histogram value in bin k is normalized by the total number, N, of data samples: N(k)/N.

The hist function is limited in its ability to produce relative frequency histograms. It is better to generate such plots with the bar function, defined as follows:

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Histogram of data1

Histogram of data2

Figure 7.4: Absolute frequency histograms

For example, the following commands produce relative histogram plots shown in Figure 7.5 for the random data vectors data1 and data2.

% Relative frequency histogram plots

%

n1 = length(data1); [freq1,x1] = hist(data1,25); rfreq1 = freq1/n1;

n2 = length(data2); [freq2,x2] = hist(data2,25); rfreq2 = freq2/n2;

subplot(2,1,1),bar(x1,rfreq1),...

title(’Relative histogram of data1’), grid,...

xlabel(’x’), ylabel(’relative frequency’),...

subplot(2,1,2),bar(x2,rfreq2),...

title(’Relative histogram of data2’), grid,...

xlabel(’x’), ylabel(’relative frequency’)

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Relative histogram of data1

Relative histogram of data2

Figure 7.5: Relative frequency histograms

Measures of central tendency

The mean and median describe the middle, or center, of the range of the random variable.

The sample mean of vector x having N elements is x¯

N

x¯ = 1 x(m)

N n=1

The population mean µ (mu) is the value of x¯ for an infinite number of measurements, denoted by

µ = lim x¯

N→∞

where the right hand side is read as “the limit of x bar as N approaches infinity.”

Median: Middle value of a set of random samples, sorted by value (rank ordered). If there is an even number of samples, then the median is the average of the two middle sample values.

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