Matlab programming with applications for engineers 1st edition chapman solutions manual

MATLAB Programming with Applications for Engineers 1st Edition Chapman Solutions Manual Visit to download the full and correct content document: https://testbankdeal.com/dow nload/matlab-programming-with-applications-for-engineers-1st-edition-chapman-soluti ons-manual/

6.

User-Defined Functions

6.1

Script files are just collections of MATLAB statements that are stored in a file. When a script file is executed, the result is the same as it would be if all of the commands had been typed directly into the Command Window. Script files share the Command Window’s workspace, so any variables that were defined before the script file starts are visible to the script file, and any variables created by the script file remain in the workspace after the script file finishes executing. A script file has no input arguments and returns no results, but script files can communicate with other script files through the data left behind in the workspace.

In contrast, MATLAB functions are a special type of M-file that run in their own independent workspace. They receive input data through an input argument list, and return results to the caller through an output argument list.

6.2

MATLAB programs communicate with their functions using a pass-by-value scheme. When a function call occurs, MATLAB makes a copy of the actual arguments and passes them to the function. This copying is very significant, because it means that even if the function modifies the input arguments, it won’t affect the original data in the caller. Similarly, the returned values are calculated by the function, and copied into the return variables in the calling program.

6.3

The principal advantage of the pass-by-value scheme is that any changes to input arguments within a function will not affect the input arguments in the calling program. This, along with the separate workspace for the function, eliminates unintended side effects. The disadvantage is that copying arguments, especially large arrays, can take time and memory.

6.4

A function to sort arrays in ascending or descending order, depending on the second calling parameter, is shown below: function out = ssort1(a,dir) %SSORT1 Selection sort data in ascending or descending order % Function SSORT1 sorts a numeric data set into ascending % or descending order, depending on the value of the % second parameter. If the parameter is 'up', then it % sorts the data in ascending order. If the parameter is % 'down', then it sorts the data in descending order. The % default value is 'up'. Note that 'up' and 'down' may % be abbreviated to 'u' and 'd', if desired. % % Note that the selection sort is relatively inefficient. % DO NOT USE THIS FUNCTION FOR LARGE DATA SETS. Use MATLAB's % "sort" function instead. % Define variables: % a -- Input array to sort % ii -- Index variable % iptr -- Pointer to min value % jj -- Index variable % nvals -- Number of values in "a" % out -- Sorted output array % temp -- Temp variable for swapping

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% % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,2,nargin); error(msg); % If the direction argument is missing, set it to 'up'. if nargin < 2 dir = 'up'; end % Check for the direction of the sort if dir(1) == 'u' | dir(1) == 'U' sort_up = 1; elseif dir(1) == 'd' | dir(1) == 'D' sort_up = 0; else error('Second parameter is not ''UP'' or ''DOWN''!') end % Get the length of the array to sort nvals = size(a,2); % Sort the input array for ii = 1:nvals-1 if sort_up % Sort in ascending order. % Find the minimum value in a(ii) through a(n) iptr = ii; for jj = ii+1:nvals if a(jj) < a(iptr) iptr = jj; end end else % Sort in descending order. % Find the maximum value in a(ii) through a(n) iptr = ii; for jj = ii+1:nvals if a(jj) < a(iptr) iptr = jj; end end end

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% iptr now points to the min/max value, so swap a(iptr) % with a(ii) if ii ~= iptr. if ii ~= iptr temp = a(ii); a(ii) = a(iptr); a(iptr) = temp; end end % Pass data back to caller out = a;

A script file to test this function is shown below: % Script file: test_ssort1.m % % Purpose: % To read in an input data set, sort it into ascending % order using the selection sort algorithm, and to % write the sorted data to the Command window. This % program calls function "ssort1" to do the actual % sorting. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/03/11 S. J. Chapman Original code % % Define variables: % array -- Input data array % ii -- Index variable % nvals -- Number of input values % sorted1 -- Sorted data array (up) % sorted2 -- Sorted data array (down) % Prompt for the number of values in the data set nvals = input('Enter number of values to sort: '); % Preallocate array array = zeros(1,nvals); % Get input values for ii = 1:nvals % Prompt for next value string = ['Enter value ' int2str(ii) ': array(ii) = input(string);

'];

end % Now sort the data in ascending order sorted1 = ssort1(array,'up'); % Display the sorted result.

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fprintf('\nSorted data in ascending order:\n'); for ii = 1:nvals fprintf(' %8.4f\n',sorted1(ii)); end % Now sort the data in descending order sorted2 = ssort1(array,'down'); % Display the sorted result. fprintf('\nSorted data in descending order:\n'); for ii = 1:nvals fprintf(' %8.4f\n',sorted2(ii)); end % Now sort the data in ascending order with % no second argument. sorted3 = ssort1(array); % Display the sorted result. fprintf('\nSorted data in default order:\n'); for ii = 1:nvals fprintf(' %8.4f\n',sorted3(ii)); end % The follwing call should produce an error message sorted4 = ssort1(array,'bad');

When this program is executed, the results are: » test_ssort1 Enter number of values to sort: Enter value 1: -3 Enter value 2: 5 Enter value 3: 2 Enter value 4: 2 Enter value 5: 0 Enter value 6: 1

6

Sorted data in ascending order: -3.0000 0.0000 1.0000 2.0000 2.0000 5.0000 Sorted data in descending order: -3.0000 0.0000 1.0000 2.0000 2.0000 5.0000

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Sorted data in default order: -3.0000 0.0000 1.0000 2.0000 2.0000 5.0000 ??? Error using ==> ssort1 Second parameter is not 'UP' or 'DOWN'!

Error in ==> d:\book\matlab\soln\Ex5.4\test_ssort1.m On line 66 ==>

6.5

A set of functions to perform trigonometric operations in degrees are shown below: function out = sind(x) %SIND Calculate sin(x), where x is in degrees % Function SIND calculates sin(x), where x is in degrees % % Define variables: % x -- Angle in degrees % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = sin(pi/180*x);

function out = sind(x) %COSD Calculate cos(x), where x is in degrees % Function COSD calculates cos(x), where x is in degrees % % Define variables: % x -- Angle in degrees % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = cos(pi/180*x);

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function out = tand(x) %TAND Calculate tan(x), where x is in degrees % Function TAND calculates tan(x), where x is in degrees % % Define variables: % x -- Angle in degrees % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = tan(pi/180*x);

function out = asind(x) %ASIND Calculate asin(x), where the output is in degrees % Function ASIND calculates asin(x), where the output is in degrees % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = 180/pi * asin(x);

function out = acosd(x) %ACOSD Calculate acos(x), where the output is in degrees % Function ACOSD calculates acos(x), where the output is in degrees % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments.

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msg = nargchk(1,1,nargin); error(msg); % Calculate value out = 180/pi * acos(x);

function out = atand(x) %ATAND Calculate atan(x), where the output is in degrees % Function ATAND calculates atan(x), where the output is in degrees % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = 180/pi * atan(x);

function out = atan2d(y,x) %ATAN2D Four quadrant inverse tangent, where the output is in degrees % Function ATAN2D calculates the Four quadrant inverse tangent, where % the output is in degrees. % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(2,2,nargin); error(msg); % Calculate value out = 180/pi * atan2(y,x);

A script file to test these functions is given below: % % % % %

Script file: test_functions.m Purpose: To perform a median filter on an input data set.

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% Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/03/11 S. J. Chapman Original code % % Define variables: % ii -- Loop index % filename -- Input data file % n_ave -- Number of points to average % n_per_side -- Number of points to average per side % n_points -- Number of points in data set % slope -- Slope of the line % x -- Array of input values % y -- Array of filtered values disp('This program tests the trig functions that return answers in degrees.'); % Set the angle theta = 30 degrees, and try the forward trig functions disp(' '); disp(['Testing forward trig functions:']); disp(['sind(30) = ' num2str(sind(30))]); disp(['cosd(30) = ' num2str(cosd(30))]); disp(['tand(30) = ' num2str(tand(30))]); disp(['sind(45) = ' num2str(sind(45))]); disp(['cosd(45) = ' num2str(cosd(45))]); disp(['tand(45) = ' num2str(tand(45))]); % Test the inverse trig functions disp(' '); disp(['Testing inverse trig functions:']); disp(['asind(0) = ' num2str(asind(0))]); disp(['asind(1/sqrt(2)) = ' num2str(asind(1/sqrt(2)))]); disp(['asind(1) = ' num2str(asind(1))]); disp(['acosd(0) = ' num2str(acosd(0))]); disp(['acosd(1/sqrt(2)) = ' num2str(acosd(1/sqrt(2)))]); disp(['acosd(1) = ' num2str(acosd(1))]); disp(['atand(1) = ' num2str(atand(1))]); % Test atan2d disp(' '); disp(['Testing atan2d:']); disp(['atan2d(4,3) = ' num2str(atan2d(4,3))]);

When the script file is executed, the results are: >> test_functions This program tests the trig functions that return answers in degrees.

Testing forward trig functions: sind(30) = 0.5 cosd(30) = 0.86603 tand(30) = 0.57735 sind(45) = 0.70711

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cosd(45) = 0.70711 tand(45) = 1 Testing inverse trig functions: asind(0) = 0 asind(1/sqrt(2)) = 45 asind(1) = 90 acosd(0) = 90 acosd(1/sqrt(2)) = 45 acosd(1) = 0 atand(1) = 45 Testing atan2d: atan2d(4,3) = 53.1301

6.6

A function to convert degrees Fahrenheit to degrees Celsius is shown below: function deg_c = f_to_c(deg_f) %F_TO_C Convert degrees Fahrenheit to degrees Celsius % Function F_TO_C converts degrees Fahrenheit to degrees Celsius % % % Calling sequence: % deg_c = f_to_c(deg_f) % Define variables: % deg_f -- Input in degrees Fahrenheit % deg_c -- Output in degrees Celsius % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value deg_c = 5/9 * (deg_f - 32);

We can test this function using the freezing and boiling points of water: >> f_to_c(32) ans = 0 >> f_to_c(212) ans = 100

6.7

A function to convert degrees Celsius to degrees Fahrenheit is shown below: function deg_f = c_to_f(deg_c) %C_TO_F Convert degrees Celsius to degrees Fahrenheit % Function C_TO_F converts degrees Celsius to degrees Fahrenheit

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% % % Calling sequence: % deg_f = c_to_f(deg_c) % Define variables: % deg_c -- Input in degrees Celsius % deg_f -- Output in degrees Fahrenheit % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value deg_f = 9/5 * deg_c + 32;

We can test this function using the freezing and boiling points of water: >> f_to_f(0) ans = 100 >> c_to_f(100) ans = 0

We can also show that c_to_f and f_to_c are the inverses of each other: >> f_to_c(c_to_f(30)) ans = 30

6.8

A function to calculate the area of a triangle specified by the locations of its three vertices is shown below: function area = area2d(x1, y1, x2, y2, x3, y3) %AREA2D Calculate the area of a triangle specified by three vertices % Function AREA2D calculates the area of a triangle specified by % three vertices % % Calling sequence: % area = area2d(x1, y1, x2, y2, x3, y3) % Define variables: % x1, y1 -- Location of vertex 1 % x2, y2 -- Location of vertex 2 % x3, y3 -- Location of vertex 3 % area -- Area of triangle %

Record of revisions:

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% % %

Date ==== 07/03/11

Programmer ========== S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(6,6,nargin); error(msg); % Calculate value area = 0.5 * (x1*(y2-y3) - x2*(y1-y3) + x3*(y1-y2));

We can test this function using the specified points: >> area = area2d(0,0,10,0,15,5) area = 25.00

6.9

At this point in our studies, there is no general way to support an arbitrary number of arguments in a function. Function nargin allows a developer to know how many arguments are used in a function call, but we can only up to the number of arguments in the calling sequence1. We will design this function to support up to 6 vertices. The corresponding function is shown below: function area = area_polygon(x1, y1, x2, y2, x3, y3, x4, y4, x5, y5, x6, y6) %AREA_POLYGON Calculate the area of a polygon specified by its vertices % Function AREA_POLYGON calculates the area of a polygon specified by % its vertices % % Calling sequence: % area = area_polygon(x1, y1, x2, y2, x3, y3, x4, y4, x5, y5, x6, y6) % Define variables: % ii -- Loop index % n_vertices -- Number of vetices in polygon % x1, y1 -- Location of vertex 1 % x2, y2 -- Location of vertex 2 % x3, y3 -- Location of vertex 3 % x4, y4 -- Location of vertex 4 % x5, y5 -- Location of vertex 5 % x6, y6 -- Location of vertex 6 % area -- Area of polygon % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. % There must be at least 3 vertices, and no more than 6.

1 Later we will learn about function varargin, which can support any number of arguments.

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msg = nargchk(6,12,nargin); error(msg); % Get the number of vertices n_vertices = nargin / 2; % Save vertices in arrays x = zeros(1,n_vertices); y = zeros(1,n_vertices); % Save values x(1) = x1; y(1) = y1; x(2) = x2; y(2) = y2; x(3) = x3; y(3) = y3; if n_vertices >= 4 x(4) = x4; y(4) = y4; end if n_vertices >= 5 x(5) = x5; y(5) = y5; end if n_vertices >= 6 x(6) = x6; y(6) = y6; end % Calculate the area area = 0; for ii = 1:n_vertices-2 area = area + 0.5 * (x(ii)*(y(ii+1)-y(ii+2)) ...

- x(ii+1)*(y(ii)-y(ii+2)) ... + x(ii+2)*(y(ii)-y(ii+1))); end

We can test this function using the specified point (0,0), (10,0), (10,10), and (0, 10), which corresponds to a square with all sides having length 10: >> area = area_polygon(0,0,10,0,10,10,0,10) area = 100.00

We can test this function using the points specified in the problem: >> area = area_polygon(0,0,10,0,10,10,0,10) area = 100.00 >> area = area_polygon(10,0,8,8,2,10,-4,5) area = 43.00

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6.10

A function to calculate the inductance of a single-phase two-wire transmission line is shown below: function ind = inductance(r, D, length) %INDUCTANCE Calculate the inductance of a transmission line % Function INDUCTANCE calculates the inductance of a % single-phase two-wire transmission line. % % Calling sequence: % ind = inductance(r, D, length) % % where % r = the radius of the conductors in meters % D = the distance between the two lines in meters % length = Length of transmission line in meters % Define variables: % ind_per_m -- Inductance per meter % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(3,3,nargin); error(msg); % Constants mu0 = pi * 4e-7;

% H/m

% Calculate the inductance ind_per_m = mu0 / pi * (1/4 + log(D/r)); % Get the inductance ind = ind_per_m * length;

We can test this function using the points specified in the problem: >> ind = inductance(0.02, 1.5, 100000) ind = 0.1827

6.11

If the diameter of a transmission line’s conductors increase, the inductance of the line will decrease.

If the diameter of the conductors are doubled, the inductance will fall to: >> ind = inductance(0.02, 1.5, 100000) ind = 0.1550

6.12

A function to calculate the capacitance of a single-phase two-wire transmission line is shown below: function cap = capacitance(r, D, length) %CAPACITANCE Calculate the capacitance of a transmission line % Function CAPACITANCE calculates the capacitance of a

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% single-phase two-wire transmission line. % % Calling sequence: % cap = capacitance(r, D, length) % % where % r = the radius of the conductors in meters % D = the distance between the two lines in meters % length = Length of transmission line in meters % Define variables: % cap_per_m -- Capacitance per meter % % % %

Record of revisions: Date Programmer ==== ========== 07/03/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(3,3,nargin); error(msg); % Constants e0 = pi * 4e-7;

% F/m

% Calculate the capacitance cap_per_m = pi * e0 / log((D-r)/r); % Get the capacitance cap = cap_per_m * length;

We can test this function using the points specified in the problem: >> cap = capacitance(0.04, 1.5, 100000) cap = 0.1097

6.13

If the distance between the two conductors increases, the inductance of the transmission line increases and the capacitance of the transmission line decreases.

6.14

A program to compare the sorting times using the selection sort of Example 6.2 and MATLAB’s built-in sort is shown below: . % % % % % % % % % % %

Script file: compare_sorts.m Purpose: To compare the sort function from Example 6.2 and the built-in MATLAB sort Record of revisions: Date Programmer ==== ========== 07/04/11 S. J. Chapman

Description of change ===================== Original code

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% Define variables: % data1 -- Array to sort % data2 -- Copy of array to sort % elapsed_time1 -- Elaosed time for ssort % elapsed_time2 -- Elaosed time for sort % Constants SIZE = 100000;

% Number of values to sort

% Set seed seed(123456); % Create a sample data set to sort. data1 = random0(1,SIZE); % Sort using ssort data2 = data1; tic; out = ssort(data2); elapsed_time1 = toc; % Sort using sort data2 = data1; tic; out = sort(data2); elapsed_time2 = toc; % Display the relative times disp(['Sort time using ssort = ' num2str(elapsed_time1)]); disp(['Sort time using sort = ' num2str(elapsed_time2)]);

When this program is executed, the results are: >> compare_sorts Sort time using ssort = 118.288 Sort time using sort = 0.013641

The built-in sorting function is dramatically faster than the selection sort of Example 6.2.

6.15

A program to compare the sorting times using the selection sort of Example 6.2 and MATLAB’s built-in sort is shown below.

Note that the sort time for the selection sort increases as the square of the number of values sorted. This may make it impossible to do the calculation for 1,000,000 samples, as specified in the original problem. This will be modified to 200,000 samples in subsequent printings. % % % % % % %

Script file: compare_sorts.m Purpose: To compare the sort function from Example 6.2 and the built-in MATLAB sort Record of revisions:

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% Date Programmer Description of change % ==== ========== ===================== % 07/04/11 S. J. Chapman Original code % % Define variables: % data1 -- Array to sort % data2 -- Copy of array to sort % elapsed_time1 -- Elaosed time for ssort % elapsed_time2 -- Elaosed time for sort % Set seed seed(123456); % Create a sample data set to sort nsamp = 10000; data1 = random0(1,nsamp); % Sort using ssort data2 = data1; tic; out = ssort(data2); elapsed_time1 = toc; % Sort using sort data2 = data1; tic; out = sort(data2); elapsed_time2 = toc; % Display the relative times disp(['Sort time for ' int2str(nsamp) ' using ssort = ' num2str(elapsed_time1)]); disp(['Sort time for ' int2str(nsamp) ' using sort = ' num2str(elapsed_time2)]); % Create a sample data set to sort nsamp = 100000; data1 = random0(1,nsamp); % Sort using ssort data2 = data1; tic; out = ssort(data2); elapsed_time1 = toc; % Sort using sort data2 = data1; tic; out = sort(data2); elapsed_time2 = toc; % Display the relative times disp(['Sort time for ' int2str(nsamp) ' using ssort = ' num2str(elapsed_time1)]);

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disp(['Sort time for ' int2str(nsamp) ' using sort num2str(elapsed_time2)]);

= '

% Create a sample data set to sort nsamp = 200000; data1 = random0(1,nsamp); % Sort using ssort data2 = data1; tic; out = ssort(data2); elapsed_time1 = toc; % Sort using sort data2 = data1; tic; out = sort(data2); elapsed_time2 = toc; % Display the relative times disp(['Sort time for ' int2str(nsamp) ' using ssort = ' num2str(elapsed_time1)]); disp(['Sort time for ' int2str(nsamp) ' using sort = ' num2str(elapsed_time2)]);

When this program is executed, the results are: >> compare_sorts Sort time using ssort = 118.288 Sort time using sort = 0.013641

The built-in sorting function is dramatically faster than the selection sort of Example 6.2. >> compare_sorts Sort time for 10000 using ssort = 1.2153 Sort time for 10000 using sort = 0.0013928 Sort time for 100000 using ssort = 137.3888 Sort time for 100000 using sort = 0.013531 Sort time for 200000 using ssort = 631.6224 Sort time for 200000 using sort = 0.026963

6.16

A modified version of function random0 that can accept 0, 1, or 2 arguments is shown below: function ran = random0(n,m) %RANDOM0 Generate uniform random numbers in [0,1) % Function RANDOM0 generates an array of uniform % random numbers in the range [0,1). The usage % is: % % random0(n) -- Generate an n x n array % random0(n,m) -- Generate an n x m array % Define variables:

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% ii -- Index variable % ISEED -- Random number seed (global) % jj -- Index variable % m -- Number of columns % msg -- Error message % n -- Number of rows % ran -- Output array % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 02/04/10 S. J. Chapman Original code % 1. 07/04/11 S. J. Chapman Modified for 0 arguments % Declare global values global ISEED % Seed for random number generator % Check for a legal number of input arguments. msg = nargchk(0,2,nargin); error(msg); % If both arguments are missing, set to 1. % If the m argument is missing, set it to n. if nargin < 1 m = 1; n = 1; elseif nargin < 2 m = n; end % Initialize the output array ran = zeros(n,m); % Now calculate random values for ii = 1:n for jj = 1:m ISEED = mod(8121*ISEED + 28411, 134456 ); ran(ii,jj) = ISEED / 134456; end end

6.17

Function random0 has a bug under some conditions. If the global variable ISEED has not been previously defined when random0 is executed, the program will crash. This problem occurs the first time only that random0 is executed in a given MATLAB session, if function seed is not called first. A simple way to avoid this problem would be to detect if ISEED is undefined, and to supply a default value. Otherwise, the function should use the global seed supplied. A modified version of random0 that fixes this bug is shown below: function ran = random0(n,m) %RANDOM0 Generate uniform random numbers in [0,1) % Function RANDOM0 generates an array of uniform % random numbers in the range [0,1). The usage % is:

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% % random0(n) % random0(n,m)

-- Generate an n x n array -- Generate an n x m array

% Define variables: % ii -- Index variable % ISEED -- Random number seed (global) % jj -- Index variable % m -- Number of columns % msg -- Error message % n -- Number of rows % ran -- Output array % Record of revisions: % Date Programmer % ==== ========== % 02/04/10 S. J. Chapman % 1. 07/04/11 S. J. Chapman % Declare globl values global ISEED

Description of change ===================== Original code Modified to provide initial seed

% Seed for random number generator

% Check for a legal number of input arguments. msg = nargchk(1,2,nargin); error(msg); % If the m argument is missing, set it to n. if nargin < 2 m = n; end % Initialize the output array ran = zeros(n,m); % Test for missing seed, and supply a default if necessary. if isempty(ISEED) ISEED = 99999; end % Now calculate random values for ii = 1:n for jj = 1:m ISEED = mod(8121*ISEED + 28411, 134456 ); ran(ii,jj) = ISEED / 134456; end end

6.18

A function dice to simulate the roll of a fair die is shown below: function result = dice() %DICE Return a random integer between 1 and 6 % Function DICE simulates the throw of a fair % die. It returns a random integer between 1

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% and 6. % Define variables: % result -- Resulting integer % % % %

Record of revisions: Date Programmer ==== ========== 07/04/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(0,0,nargin); error(msg); % The value "6*random0(1,1)" returns a value % in the range [0,6). The "floor" function % returns {0,1,2,3,4,5} with equal probability, % so "result" returns 1-6. result = floor( 6 * random0(1,1) ) + 1;

A script file to test function dice is shown below: % Script file: test_dice.m % % Purpose: % To test the function dice. This function calls dice % 10,000 times, and examines the distribution of the % resulting values. They should be roughly uniform. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/04/11 S. J. Chapman Original code % % Define variables: % ii -- index variable % result -- Array of results % Initial values result = zeros(1,100000); for ii = 1:100000; result(ii) = dice; end % Display the first 30 values. fprintf('The first 30 values are:\n'); for ii = 1:30 fprintf('%d ',result(ii)); end fprintf('\n'); % Now calculate a histogram to determine the overall % distribution of numbers.

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hist(result,6); title('\bfHistogram of values returned from function "dice"'); xlabel('\bfValue'); ylabel('\bfCount');

When this script file is executed, the results are: » test_dice The first 30 values are: 3 1 4 6 6 4 4 4 1 6 6 4 1 2 1 3 2 6 1 2 2 6 6 5 6 3 1 6 1 5

The resulting histogram is shown below. The histogram shows that each integer between 1 and 6 is about equally likely to occur.

6.19

A function to calculate a probability from the Poisson distribution is shown below: function result = poisson(k, t, lambda) %POISSON Return a random integer between 1 and 6 % Function POISSON calculates a sample value from % the Poisson distribution for specific values of % k, t, and lambda.

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% Define variables: % fact -- k! (k-factorial) % result -- Resulting value from distribution % % % %

Record of revisions: Date Programmer ==== ========== 07/07/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(3,3,nargin); error(msg); % Calculate k! fact = factorial(k); % Calculate value from Poisson distribution. result = exp(-lambda*t) * (lambda*t) ^ k / fact;

A program that uses function poisson to calculate the probability of a specific number of cars passing a point on a highway in a given period of time is shown below: % Script file: traffic.m % % Purpose: % To calculate the probability of a given number of cars % passing a particular point on a highway in a given time. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/07/11 S. J. Chapman Original code % % Define variables: % k -- Actual number of cars in time t % lambda -- Expected number of cars per minute % prob -- Probability array % t -- Period of time (minutes) % Get initial values lambda = input('Enter expected number of cars/minute: '); t = input('Enter period of time in minutes: '); % Calculate values. Note that k runs from 0 to 5, while the % minimum value for a MATLAB array is 1, so we must index into % the MATLAB array using "k+1" instead of "k". for k = 0:5 prob(k+1) = poisson(k, t, lambda); end % Display results disp(['The probability of k cars passing in ' num2str(t) ' minutes is:']);

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for k = 0:5 fprintf(' end

%3d

%12.7f\n',k,prob(k+1));

% Now plot the distribution. figure(1); plot(0:5,prob,'bo','LineWidth',2); title('\bfPoisson Distribution'); xlabel('\bfValue'); ylabel('\bfProbability');

When this program is executed, the results are a shown below. Note that the plot of the probability distribution uses discrete points instead of a continuous line, since the probabilities are only defined for the integer values k = 0, 1, 2, 3, … (we can’t have 1.2 cars go by!). This plot can also be represented as a bar chart, once we learn how to create them in Chapter 6. >> traffic Enter expected number of cars/minute: 1.6 Enter period of time in minutes: 1 The probability of k cars passing in 1 minutes is: 0 0.2018965 1 0.3230344 2 0.2584275 3 0.1378280 4 0.0551312 5 0.0176420

6.20

Functions to calculate the hyperbolic sine, cosine, and tangent functions are shown below:: function out = sinh(x)

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%SIND Calculate hyperbolic sine function % Function SINH calculates the hyperbolic sine function % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/12/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = (exp(x) - exp(-x))/2; function out = cosh1(x) %COSH1 Calculate hyperbolic cosine function % Function COSH1 calculates the hyperbolic cosine function % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/12/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value out = (exp(x) + exp(-x))/2; function out = tanh1(x) %TANH1 Calculate hyperbolic tangent function % Function TANH1 calculates the hyperbolic tangent function % % Define variables: % x -- Input value % % % %

Record of revisions: Date Programmer ==== ========== 07/12/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(1,1,nargin); error(msg); % Calculate value

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out = (exp(x) - exp(-x)) ./ (exp(x) + exp(-x));

A script file to plot the hyperbolic sine, cosine, and tangent functions are shown below: % Script file: test_hyperbolic_functions.m % % Purpose: % To plot the hyperbolic functions sinh, cosh, abd tanh. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/12/11 S. J. Chapman Original code % % Define variables: % out_cosh -- Hyperbolic cosine % out_sinh -- Hyperbolic sine % out_tanh -- Hyperbolic tangent % Calculate results x = -5:0.05:5; out_cosh = cosh1(x); out_sinh = sinh1(x); out_tanh = tanh1(x);

% Display results figure(1); plot(x,out_cosh); title('\bfHyperbolic cosine'); xlabel('\bfx'); ylabel('\bfcosh(x)'); grid on; figure(2); plot(x,out_sinh); title('\bfHyperbolic sine'); xlabel('\bfx'); ylabel('\bfsinh(x)'); grid on; figure(3); plot(x,out_cosh); title('\bfHyperbolic tangent'); xlabel('\bfx'); ylabel('\bftanh(x)'); grid on;

The resulting plots are shown below:

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164 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

6.21

A function to smooth a noisy data set with a running average filter is shown below. function y = running_ave(x,n_ave) %RUNNING_AVE Function to perform a running average filter % Function RUNNING_AVE performs a running average filter % % Calling sequence: % y = running_ave(x, n_ave) % % where: % n_ave -- Number of points to average % x -- Array of input values % y -- Array of filtered values % % % %

Record of revisions: Date Programmer ==== ========== 07/12/11 S. J. Chapman

Description of change ===================== Original code

% Define local variables: % ii -- Loop index % n_per_side -- Number of points to average per side % n_points -- Number of points in data set % slope -- Slope of the line

% Check for a legal number of input arguments. msg = nargchk(1,2,nargin);

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error(msg); % Set a default number of points to average if this value % is undefined. if nargin < 2 n_ave = 7; end % Get the number of samples on either side of the sample % being averaged, dropping any fractional part. n_per_side = (n_ave-1) / 2; n_per_side = floor(n_per_side); % Get the number of points in the input structure n_points = length(x); % Now perform the running average for ii = 1:n_points % Check to see how many points we can use on either side % of the point being averaged. n_low = min([ (ii-1) n_per_side]); n_high = min([ (n_points-ii) n_per_side]); n_used = min([ n_low n_high]); % Now calculate running average y(ii) = sum(x(ii-n_used:ii+n_used)) / (2*n_used+1); end

A program to test this function is shown below. % Script file: test_running_ave.m % % Purpose: % To perform test the running average filter function. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/12/11 S. J. Chapman Original code % % Define variables: % ii -- Loop index % filename -- Input data file % n_ave -- Number of points to average % n_per_side -- Number of points to average per side % n_points -- Number of points in data set % x -- Array of input values % y -- Array of filtered values disp('This program performs a running average filter on an '); disp('input data set.'); filename = input('Enter the filename containing the data: ','s');

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n_ave = input('Enter the number of samples to average: '); % Get the number of samples on either side of the sample % being averaged, dropping any fractional part. n_per_side = (n_ave-1) / 2; n_per_side = floor(n_per_side); % Read the input data x = textread(filename,'%f'); n_points = length(x); % Get running average y = running_ave(x,n_ave); % Plot the data points as blue circles with no % connecting lines. plot(x,'bo'); hold on; % Plot the fit as a solid red line with no markers plot(y,'r-','LineWidth',2); hold off; % Add a title and legend title ('\bfRunning-Average Filter'); xlabel('\bf\itx'); ylabel('\bf\ity'); legend('Input data','Fitted line'); grid on

When this program is executed, the results are: >> test_running_ave This program performs a running average filter on an input data set.

Enter the filename containing the data: input3.dat Enter the number of samples to average: 7

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6.22

A function to smooth a noisy data set with a median filter is shown below. function y = median_filter(x,n_ave) %RUNNING_AVE Function to perform a median filter % Function RUNNING_AVE performs a median filter % % Calling sequence: % y = median_filter(x, n_ave) % % where: % n_ave -- Number of points to average % x -- Array of input values % y -- Array of filtered values % % % %

Record of revisions: Date Programmer ==== ========== 07/12/11 S. J. Chapman

Description of change ===================== Original code

% Define local variables: % ii -- Loop index % n_per_side -- Number of points to average per side % n_points -- Number of points in data set % slope -- Slope of the line

% Check for a legal number of input arguments. msg = nargchk(1,2,nargin);

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error(msg); % Set a default number of points to average if this value % is undefined. if nargin < 2 n_ave = 7; end % Get the number of samples on either side of the sample % being averaged, dropping any fractional part. n_per_side = (n_ave-1) / 2; n_per_side = floor(n_per_side); % Get the number of points in the input structure n_points = length(x); % Now perform the running average for ii = 1:n_points % Check to see how many points we can use on either side % of the point being averaged. n_low = min([ (ii-1) n_per_side]); n_high = min([ (n_points-ii) n_per_side]); n_used = min([ n_low n_high]); % Now calculate running average y(ii) = median(x(ii-n_used:ii+n_used)); end

A program to test this function is shown below. % Script file: test_median_filter.m % % Purpose: % To test the median filter function. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 06/30/11 S. J. Chapman Original code % % Define variables: % ii -- Loop index % filename -- Input data file % n_ave -- Number of points to average % n_per_side -- Number of points to average per side % n_points -- Number of points in data set % slope -- Slope of the line % x -- Array of input values % y -- Array of filtered values disp('This program performs a median filter on an input data set.'); filename = input('Enter the filename containing the data: ','s');

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n_ave = input('Enter the number of samples to average: '); % Get the number of samples on either side of the sample % being averaged, dropping any fractional part. n_per_side = (n_ave-1) / 2; n_per_side = floor(n_per_side); % Read the input data x = textread(filename,'%f'); n_points = length(x); % Now perform the median filter y = median_filter(x,n_ave); % Plot the data points as blue circles with no % connecting lines. plot(x,'bo'); hold on; % Plot the fit as a solid red line with no markers plot(y,'r-','LineWidth',2); hold off; % Add a title and legend title ('\bfMedian Filter'); xlabel('\bf\itx'); ylabel('\bf\ity'); legend('Input data','Fitted line'); grid on

When this program is executed, the results are: >> test_median_filter This program performs a running average filter on an input data set.

Enter the filename containing the data: input3.dat Enter the number of samples to average: 7

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6.23

A function to sort an input data set into ascending order while carrying along another array is shown below. function [out1, out2] = sort_with_carry(a,b) %SORT_WITH_CARRY Selection sort with carry in ascending order % Function SORT_WITH_CARRY sorts a numeric data set a into % ascending order, carrying along array b. Note that the % selection sort is relatively inefficient. DO NOT USE THIS % FUNCTION FOR LARGE DATA SETS. % Define variables: % a -- Input array to sort % b -- Input array to carry along % ii -- Index variable % iptr -- Pointer to min value % jj -- Index variable % nvals -- Number of values in "a" and "b" % out -- Sorted output array % out2 -- Carried output array % temp -- Temp variable for swapping % % % %

Record of revisions: Date Programmer ==== ========== 07/12/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(2,2,nargin);

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error(msg); % Check for a legal length. Both a and b must be % the same length. if length(a) ~= length(b) error('Vectors a and b must be the same length'); end % Get the length of the array to sort nvals = length(a); % Sort the input array for ii = 1:nvals-1 % Find the minimum value in a(ii) through a(n) iptr = ii; for jj = ii+1:nvals if a(jj) < a(iptr) iptr = jj; end end % iptr now points to the minimum value, so swap a(iptr) % with a(ii) if ii ~= iptr. if ii ~= iptr temp = a(ii); a(ii) = a(iptr); a(iptr) = temp; temp = b(ii); b(ii) = b(iptr); b(iptr) = temp; end end % Pass data back to caller out1 = a; out2 = b;

A program to test this function is shown below: % Script file: test_sort_with_carry.m % % Purpose: % To read in an input data set a and sort it into % ascending order while carrying along array b. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/12/11 S. J. Chapman Original code % % Define variables: % a -- Input data array to sort

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% % % % %

b ii nvals out1 out2

-- Input data array to carry -- Index variable -- Number of input values -- Sorted data array -- Carried data array

% Input arrays a = [ 1, 11, -6, 17, -23, 0, 5, 1, -1]; b = [ 31,101, 36,-17, 0, 10, -8, -1, -1]; % Now sort the data [out1, out2] = sort_with_carry(a,b); % Display the sorted result. fprintf('\nSorted data:\n'); for ii = 1:length(a) fprintf(' %8.4f %8.4f\n',out1(ii),out2(ii)); end

When this program is executed, the results are: » test_sort_with_carry Sorted data: -23.0000 0.0000 -6.0000 36.0000 -1.0000 -1.0000 0.0000 10.0000 1.0000 31.0000 1.0000 -1.0000 5.0000 -8.0000 11.0000 101.0000 17.0000 -17.0000

6.24

A program to perform sort with carry using function sortrows is shown below. % Script file: test_sortrows.m % % Purpose: % To read in an input data set a and sort it into % ascending order while carrying along array b. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/12/11 S. J. Chapman Original code % % Define variables: % a -- Input data array to sort % b -- Input data array to carry % c -- Compislte matrix % ii -- Index variable % nvals -- Number of input values

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% %

out1 out2

-- Sorted data array -- Carried data array

% Input arrays a = [ 1, 11, -6, 17, -23, 0, 5, 1, -1]; b = [ 31,101, 36,-17, 0, 10, -8, -1, -1]; c = [a' b']; % Now sort the data c = sortrows(c); % Display the sorted result. fprintf('\nSorted data:\n'); for ii = 1:length(a) fprintf(' %8.4f %8.4f\n',c(ii,1),c(ii,2)); end

When this program is executed, the results are as shown below. The built-in function sortrows is much more efficient than sort_with_carry! >> test_sortrows Sorted data: -23.0000 0.0000 -6.0000 36.0000 -1.0000 -1.0000 0.0000 10.0000 1.0000 -1.0000 1.0000 31.0000 5.0000 -8.0000 11.0000 101.0000 17.0000 -17.0000

6.25

A program to compare sort_with_carry with sortrows is shown below. % Script file: compare_sort_with_carry.m % % Purpose: % To compare function sort_with_carry with function % sortrows. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/12/11 S. J. Chapman Original code % % Define variables: % a -- Input data array to sort % array -- Input data for sortrows % b -- Input data array to carry % ii -- Index variable % nvals -- Number of input values % out -- Output data for sortrows

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% %

out1 out2

-- Sorted data array -- Carried data array

% Create random arrays array = rand(25000,2); % Make copy for sort_with_carry a = array(:,1); b = array(:,2); % Sort using sort_with_carry tic; [out1, out2] = sort_with_carry(a,b); disp(['sort_with_carry time = ',num2str(toc)]); % Sort using sortrows tic; out = sortrows(array,1); disp(['sortrows time = ',num2str(toc)]);

When this program is executed, the results are as shown below. The built-in function sortrows is much more efficient than sort_with_carry! » compare_sort_with_carry sort_with_carry time = 8.8084 sortrows time = 0.029909

6.26

The geometry of this problem is shown below with respect to an arbitrary (x,y) reference system, where x is the distance in the North direction and y is the distance in the East direction, forming a right-hand coordinate system where  is a compass angle. Note that the angles 1 and  2 are defined clockwise from the North (x) axis (compass angles), and the bearings 1 and 2 are defined clockwise from the bow of the ship. Note that the angles will be given in degrees, which is usual aboard ships.

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With these definitions, the x location of the object can be defined as the sum of the distance from the origin to Ship 1 ( x1a ) plus the distance from Ship 1 to the object ( x1b ), and the y location of the object can be defined as the sum of the distance from the origin to Ship 1 ( y1a ) plus the distance from Ship 1 to the object ( y1b ).

x  x1a  x1b  x1a  r1 cos 1  1 

(5.1)

y  y1a  y1b  y1a  r1 sin 1  1 

(5.2)

Similarly, then the x location of the object can be defined as the sum of the distance from the origin to Ship 2 ( x2 a ) plus the distance from Ship 2 to the object ( x2 b ), and the y location of the object can be defined as the sum of the distance from the origin to Ship 2 ( y2 a ) plus the distance from Ship 2 to the object ( y2 b ).

x  x2 a  x2 b  x2 a  r2 cos  2  2 

(5.3)

y  y2 a  y2 b  y2 a  r2 sin  2  2 

(5.4)

The range r2 and bearing 2 of the target from Ship 2 can be calculated by first determining the (x,y) position of the target using Equations (5.1) and (5.2), and then calculating the range and bearing by solving Equations (5.3) and (5.4) as follows:

r2 

 x  x2 a  2   y

y2 a 

2

 y  y2 a   2  x  x2 a 

2  tan 1 

(5.5)

(5.6)

A function to calculate the range and bearing as seen by Ship 2 is shown below: function [r2, phi2] = range_bearing(x1,y1,theta1,r1,phi1, ... x2,y2,theta2) %RANGE_BEARING Calculate the range and bearing of a target from a ship % Function RANGE_BEARING calculates the range and bearing of a % target from a ship, given the location, range, and bearing of % the target from another ship. % Define variables: % phi1 -- Bearing of target from Ship 1 (degrees) % phi2 -- Bearing of target from Ship 2 (degrees) % r1 -- Range of target from Ship 1 % r2 -- Range of target from Ship 2 % theta1 -- Heading of Ship 1 (degrees) % theta2 -- Heading of Ship 2 (degrees) % x -- x-pos of target in global coordinate system % x1 -- x-pos of Ship 1

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% % % %

x2 y y1 y2

-- x-pos of Ship 2 -- y-pos of target in global coordinate system -- y-pos of Ship 1 -- y-pos of Ship 2

% % % %

Record of revisions: Date Programmer ==== ========== 07/13/11 S. J. Chapman

Description of change ===================== Original code

% Check for a legal number of input arguments. msg = nargchk(8,8,nargin); error(msg); % Convert input angles to radians for use with trig functions theta1 = theta1 * pi/180; theta2 = theta2 * pi/180; phi1 = phi1 * pi/180; % Calculate the (x,y) position of the target in the % coordinate system. x = x1 + r1 * cos( theta1 + phi1 ); y = y1 + r1 * sin( theta1 + phi1 ); % Get range to target from Ship 2 r2 = sqrt( (x - x2)^2 + (y - y2)^2 ); % Get bearing of target from ship phi2 = atan2( y - y2, x - x2 ) - theta2; % Convert output bearing to degrees phi2 = phi2 * 180/pi;

A script file to test this function is shown below: % Script file: test_range_bearing.m % % Purpose: % To test function range_bearing.m % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/13/11 S. J. Chapman Original code % % Define variables: % phi1 -- Bearing of target from Ship 1 (degrees) % phi2 -- Bearing of target from Ship 2 (degrees) % r1 -- Range of target from Ship 1 % r2 -- Range of target from Ship 2 % theta1 -- Heading of Ship 1 (degrees) % theta2 -- Heading of Ship 2 (degrees) % x1 -- x-pos of Ship 1 % x2 -- x-pos of Ship 2

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% %

y1 y2

-- y-pos of Ship 1 -- y-pos of Ship 2

% Get location of Ship 1 (x,y,theta): x1 = input('x-pos of Ship 1: '); y1 = input('y-pos of Ship 1: '); theta1 = input('Heading of Ship 1 (deg): '); % Get location of Ship 2 (x,y,theta): x2 = input('x-pos of Ship 2: '); y2 = input('y-pos of Ship 2: '); theta2 = input('Heading of Ship 2 (deg): '); % Get location of target as seen by Ship 1 r1 = input('Range to target from Ship 1: '); phi1 = input('Bearing to target from Ship 1: '); % Calculate range and bearing from Ship 2 [r2, phi2] = range_bearing( x1, y1, theta1, r1, phi1, ... x2, y2, theta2 ); % Display result fprintf('Ship 2 sees the target at a range of %.1f\n', r2); fprintf('Ship 2 sees the target at a bearing of %.1f deg\n', phi2);

To test this function, we will try three simple cases. First, suppose that the two ships are in the same place and moving in the same direction. Then the object should be at the same range and bearing for both ships. >> test_range_bearing x-pos of Ship 1: 0 y-pos of Ship 1: 0 Heading of Ship 1 (deg): 0 x-pos of Ship 2: 0 y-pos of Ship 2: 0 Heading of Ship 2 (deg): 0 Range to target from Ship 1: 5 Bearing to target from Ship 1: 30 Ship 2 sees the target at a range of 5.0 Ship 2 sees the target at a bearing of 30.0 deg

This case produced the correct answer. Now, suppose that the two ships are traveling North in line abreast, with Ship 1 at (0,0) and Ship 2 at (0,5). If Ship 1 sees the object at a range of 5 and an angle of 30, then Ship 2 should see the object at the same range and an angle of -30, because the ydistance to the target is 2.5, exactly half of the distance between the two ships. >> test_range_bearing x-pos of Ship 1: 0 y-pos of Ship 1: 0 Heading of Ship 1 (deg): 0 x-pos of Ship 2: 0 y-pos of Ship 2: 5 Heading of Ship 2 (deg): 0

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Range to target from Ship 1: 5 Bearing to target from Ship 1: 30 Ship 2 sees the target at a range of 5.0 Ship 2 sees the target at a bearing of -30.0 deg

This case also produced the correct answer. Finally, suppose that the ships are in the same position, but Ship 2 has a heading of 30. In this case, the target should be at a bearing of -60 as seen by Ship 2, because the ship itself has rotated. » test_range_bearing x-pos of Ship 1: 0 y-pos of Ship 1: 0 Heading of Ship 1 (deg): 0 x-pos of Ship 2: 0 y-pos of Ship 2: 5 Heading of Ship 2 (deg): 30 Range to target from Ship 1: 5 Bearing to target from Ship 1: 30 Ship 2 sees the target at a range of 5.0 Ship 2 sees the target at a bearing of -60.0 deg

This function appears to be working properly.

6.27

A function that calculates the linear least-squares fit to an input data set is shown below: function [slope, yint] = lsqfit(x, y) %LSQFIT Calculate the linear least squares fit to a data set. % Function LSQFIT calculates the linear least squares fit line % to an input data set. Note that vectors x and y must be % the same length, and must contain at least two elements. % % The calling sequence is: % [slope,yint] = lsqfit(x,y); % % where % slope = the slope of the fitted line % yint = y-intercept of the fitted line % x = Input x values % y = Input y values % Define variables: % sum_x -- Sum of values in x % sum_x2 -- Sum of values in x.^2 % sum_xy -- Sum of values in x.*y % sum_y -- Sum of values in y % xbar -- Average of x % ybar -- Average of y % % % %

Record of revisions: Date Programmer ==== ========== 07/13/11 S. J. Chapman

Description of change ===================== Original code

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% Check for a legal number of input arguments. msg = nargchk(2,2,nargin); error(msg); % Check to make sure that the two vectors are the same length if length(x) ~= length(y) error('Vectors x and y must be the same length!'); end % Check to ensure that there are at least two elements if length(x) < 2 error('Vectors x and y must be at least 2 elements long!'); end % Now calculate sums sum_x = sum(x); sum_y = sum(y); sum_x2 = sum(x.^2); sum_xy = sum(x .* y); % Calculate slope and intercept xbar = sum_x / length(x); ybar = sum_y / length(y); slope = (sum_xy - sum_x * xbar) / (sum_x2 - sum_x * xbar); yint = ybar - slope * xbar;

A test program for this function is shown below. Note that this program also uses function polyfit to compare against. % Script file: test_lsqfit.m % % Purpose: % To test the function lsqfit. % % Record of revisions: % Date Programmer Description of change % ==== ========== ===================== % 07/13/11 S. J. Chapman Original code % % Define variables: % slope -- Slope of fitted line % x -- Input x values % x1 -- x values of fitted line % y -- Input y values % y1 -- y values of fitted line % yint -- Y-intercept of fitted line % Input data values x = [-4.91 -3.84 -2.01 0.28 1.01 3.60 y = [-8.18 -7.49 -2.83 -1.16

-2.41 1.08 4.53 -7.11 0.52

-2.62 -0.94 5.13 -6.15 0.21

-3.78 0.59 4.43 -5.62 1.73

-0.52 -1.83 ...

0.69 3.04 ...

4.12 ]; -3.30 -2.05 ...

3.96 4.26 ...

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5.75

6.67

7.70

7.31

9.05

10.95 ];

% Get fitted line [slope, yint] = lsqfit(x,y); x1(1) = min(x); x1(2) = max(x); y1 = slope*x1 + yint; % Tell user fprintf('\n%s\n','Using lsqfit.m:'); fprintf('%s%7.3f\n','Slope = ',slope); fprintf('%s%7.3f\n','Intercept = ',yint); % Plot points and the fitted line figure(1); plot(x,y,'bo','LineWidth',2); hold on; plot(x1,y1,'k-','LineWidth',2); legend ('Input data','Fitted line'); title('\bfLinear Least-Squares Fit Using Lsqfit'); xlabel('\bf\itx'); ylabel('\bf\ity'); grid on; hold off; % Now try to use function polyfit % Perform the fit p = polyfit(x,y,1); slope = p(1); y_int = p(2); % Create the fitted line x1(1) = min(x); x1(2) = max(x); y1 = polyval(p,x1); % Tell user fprintf('\n%s\n','Using polyfit.m:'); fprintf('%s%7.3f\n','Slope = ',slope); fprintf('%s%7.3f\n','Intercept = ',yint); % Plot points and the fitted line figure(2); plot(x,y,'bo','LineWidth',2); hold on; plot(x1,y1,'k-','LineWidth',2); legend ('Input data','Fitted line'); title('\bfLinear Least-Squares Fit Using Polyfit'); xlabel('\bf\itx'); ylabel('\bf\ity'); grid on; hold off;

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The resulting plots are shown below: >> test_lsqfit Using lsqfit.m: Slope = 1.858 Intercept = 0.187 Using polyfit.m: Slope = 1.844 Intercept = 0.187

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