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Unformatted text preview: 1/18/2010 Function inputs may be functions. If fplot was not part of Matlab, we could easily create something very similar. function [ ] = DIYfplot( f, range ) %DIYfplot Do it yourself version of fplot. Plots a function over a range. % Inputs: f = function to be plotted % range = two element vector containing start and end of range % No Outputs (like a C++ void function) %N O (lik C id f i ) n = 100; % number of points x = linspace (range(1), range(2), n); % generate x values % generate y values y = zeros ([1 n]); % preallocate for efficiency for k = 1 : n for k = 1 : n y(k) = f(x(k)); end plot (x, y); end The function provided may be a function mfile (@ required before name); DIYfplot( @PR, [0 7]) >> grid on >> xlabel ('Mach number'); >> ylabel ('Pressure Ratio'); Built in functions are also acceptable (@ required before name): >> DIYfplot( @sin, [0 4*pi]) And so are anonymous functions (no @ required the @ is in the definition). >> f = @(x) x^3 + 2* x^2 + 7; >> DIYfplot( f, [5 5]) Note: We could also create our own version of fzero and will soon be doing exactly this. 1 1/18/2010 Another potentially useful function is shown below. It produces a table of function values. function [ ] = ftable( f, vector ) %ftable Produces table of function values. % Inputs: f = function to be tabulated % vector = vector of values for the independent variable % No Outputs (like a C++ void function) fprintf (' Independent Dependent\n'); for k = 1 : length(vector) x = vector(k); y = f(x); fprintf ('%12f%16f\n', x, y); f i f ('%12f%16f\ ' ) end end Sample usage: >> ftable (@PR, 1:0.5:6)
Independent 1.000000 1.500000 2.000000 2.500000 3.000000 3 000000 3.500000 4.000000 4.500000 5.000000 5.500000 6.000000 Dependent 1.892929 3.413275 5.640441 8.526136 12 06096 12.060965 16.242001 21.068081 26.538665 32.653474 39.412352 46.815206 Improved Loop: The loop in the function can be reduced to: for x = vector fprintf ('%12f%16f\n', x, f(x)); end 2 1/18/2010 The table below shows the situations in which a "dot" must be used to avoid having Matlab assume array arithmetic (e.g. array multiplication) . The "dot" tells Matlab to perform an element by element operation instead. Note: there is no ".+" or "." op +, * Array op Scalar OK [ [1 2 3] + 4 = [5 6 7] ] [ ] OK [1 2 3] * 2 = [2 4 6] Scalar op Array OK 9 [1 2 3] = [8 7 6] [ ] [ ] OK 2 * [1 2 3] = [2 4 6] Array op Array Array operation assumed Sizes must match Array operation assumed Sizes must be compatible Dot required Sizes must match. [2 3 4].*[1 2 3] = [2 6 12] Array operation assumed Sizes must be compatible. Dot required Dot required Sizes must match. [2 6 12] ./ [1 2 3] = [2 3 4] ERROR Dot required Sizes must match. [2 3 4].^[1 2 3] = [2 9 64] / OK [2 4 6] / 2 = [1 2 3] Array operation assumed ERROR (as sizes are incompatible) Dot required 8 ./ [1 2 4] = [8 4 2] Array operation assumed Square array required Dot required 2 .^ [1 2 3] = [2 4 8] ^ Array operation assumed Square array required Dot required [ 1 2 3] .^ 2 = [1 4 9] To evaluate B (t ) 250 for a vector of t values: 1 56.75e 0.17 t >> t = [ 0 10 20 ]; >> 0.017 * t ans = 0 0.1700 0.3400 >> exp(ans) ans = 1.0000 0.8437 0.7118 >> 56.75*ans ans = 56.7500 47.8780 40.3930 >> 1+ans ans = 57.7500 48.8780 41.3930 >> 250/ans % a scalar >> 250/ans % a scalar divided by a vector is a no no (see table) by a vector a nono ??? Error using ==> mrdivide Matrix dimensions must agree. >> 250./ans ans = 4.3290 5.1148 6.0397 >> B = @(t) 250./(1 + 56.75*exp(0,17*t)); 3 1/18/2010 Problem (text 5.14): You buy a $25,000 piece of equipment for nothing down at $5,500 per year for 6 years. What interest rate are you paying? The formula below relates the payment amount (A) to the present worth of the item (P), the number of years (n), and the interest rate (i, expressed as a fraction, 1% = 0.01) A P i (1 i ) n (1 i ) n 1 If we knew P, n, and i we could easily calculate A, but this isn't the problem. We know A, P, and n and want to calculate i. Ideally we'd manipulate the equation to obtain an expression for i in terms of A, P, and n (remember that analytic solutions are best when we can obtain them). i = somefunction (A, P, n) Since we can't do this easily we will convert the problem into root finding form
P i (1 i ) n A0 (1 i ) n 1 and use numerical methods (e.g. fzero) to find the value of i that satisfies the equation. Step 1: Define the basic function. >> P = 25000; n = 6; >> calcA = @(i) P * (i .* (i + 1).^n) ./ ((i + 1).^n 1); Step 2: Plot the function to get some idea of its behaviour and the approximate location of the solution. >> fplot (calcA, [0.01 0.20]) % plot for interest rates from 1% to 20% Step 3: Define the function for root finding (the f in f(x) = 0). >> A = 5500; >> f = @(i) calcA(i) A; Step 4: Use fzero to locate the desired root. In this case the plot tells us that the Step 4: Use fzero to locate the desired root. In this case the plot tells us that the answer is somewhere between 6% and 10%. >> i = fzero(f, [0.06 0.10]) i = 0.0856 >> calcA(i) % check that answer is correct (always a good idea) ans = 5.5000e+003 4 1/18/2010 The basic idea can be used to produce a useful function mfile that calculate the interest rate for any given P, A, and n. function [ i ] = P5_14( P, A, n ) %P5_14 Calculate interest rate. % NOTE: Assumes that interest rate is >= 0.005. % Inputs: P = present value of item % A = annual payment % A l % n number of years % Outputs: i = interest rate (as a fraction) f = @(i) (P * (i .* (i + 1).^n) ./ ((i + 1).^n 1)) A; maxI = A/P; i = fzero(f [ 0 005 maxI ] ); = fzero(f, [ 0.005 maxI ] ); end The search for the root cannot be made to start at zero because f is undefined at i = 0 (as the formula for calculating A does not work for i = 0). This difficulty can be overcome by producing a calcA function that properly deals with the special case of i = 0. This function could be implemented in a separate mfile but it is also possible to place it in the same mfile as the basic function as shown below. This makes it a subfunction (interested students should read text section 3.1.3). function [ i ] = P5_14v2( P, A, n ) %P5_14v2 Calculates interest rate. % Inputs and outputs as before... % Inputs and outputs as before... f = @(i) calcA (P, n, i) A; maxI = A/P; i = fzero(f, [ 0 maxI ] ); end function [A] = calcA (P, n, i) if i == 0 A = P / n; else A = P * (i .* (i + 1).^n) ./ ((i + 1).^n 1); end end 5 1/18/2010 Problem: Find all of the points that satisfy both y = x217x+60 and y = 50sin(x/2) (i.e. find all of the intersections of the curves defined by these equations). Step 1: Plot the two curves. >> f1 = @(x) 50 * sin(0.5 * x); >> f2 = @(x) x.^2 17 * x + 60; >> x = linspace (0, 20, 100); >> y1 = f1(x); y2 = f2(x); >> plot (x, y1, 'r', x, y2, 'b') >> grid on The plot function will accept more than one pair of x and y vectors. This allows multiple plots to be placed on the same graph. Each pair of vectors may be followed by a string cntaining plotting options. Some of the permitted characters are: colour: `r' = red, `b' = blue, `k' = black, `g' = green line style: `' = solid, `:' = dotted, `' = dashed, `.' = dash dot data point markers: `x' = crosses, `o' = circles Options may be combined (e.g. `r:x' gives a dotted red plot with a crosses) 6 1/18/2010 Note: The hold command is another way of placing multiple plots on the same graph. >> plot (x, y1, 'r`) >> hold on >> plot (x, y2, 'b') >> hold off % be careful hold for a figure stays in effect until it is turned off >> grid on Step 2: Define the function for root finding (and perhaps plot it) >> f = @(x) f1(x) f2(x); >> figure (2) >> fplot (f, [0 20]) >> grid on; Step 3: Find the roots: >> x1 = fzero(f, [0 2]); >> x2 = fzero(f, [6 8]); >> x3 = fzero(f, [12 14]); >> x4 = fzero(f, [16 18]); 7 ...
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This document was uploaded on 04/14/2010.
 Winter '09

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