A Guide to MATLAB for Beginners and Experienced Users phần 8 pptx

Solutions to the Practice SetsSolutions to Practice Set A: Algebra and Arithmetic... Solutions to Practice Set A: Algebra and Arithmetic 221We get a one-parameter family of answers depen

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Here the vertical bar highlights the place where MATLAB believes the error is cated In this case, the actual error is earlier in the input line.

lo-Spelling Error

CAUSE: Using uppercase instead of lowercase letters in MATLAB commands, or spelling the command.

mis-SOLUTION: Fix the spelling

Consider the following accidental misspellings

Error or Warning Messages When Plotting

CAUSE: There are several possible explanations, but usually the problem is the wrong type of input for the plotting command chosen.

SOLUTION: Carefully follow the examples in the help lines of the plotting command, and

pay attention to the error and warning messages

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210 Chapter 11 Troubleshooting

0.5 1 1.5

Figure 11.1 A Spurious Graph of y = x 1/3

Figure 11.1 is not the graph of y = x 1/3on the interval−3 ≤ x ≤ 3 Actually, the

graph to the right of the origin looks correct, but to the left of the origin the graph

is certainly not right The warning message provides a clue What has happened is

that MATLAB is plotting the real part of x 1/3, but for a different branch of the

multi-valued function than the one we want MATLAB is interpreting x 1/3 for x negative

Figure 11.2 Correct Graph of y = x 1/3

A Previously Saved M-File Evaluates Differently

One of the most frustrating problems you may encounter occurs when a previouslysaved M-ﬁle, which you are sure is in good shape, won’t evaluate or evaluates incor-

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The Most Common Mistakes 211

rectly when opened in a new session

CAUSE: Change in the sequence of evaluation, or failure to initialize variables.

SOLUTION: Make sure to clear or initialize variables that are not inputs to the M-ﬁle

EXAMPLE: The problem is illustrated by the following simple, but poorly designed, program

To compute n! (n-factorial), a user writes the following script M-ﬁle:

Computer Won’t Respond

CAUSE: MATLAB is caught in a very large calculation, or some other calamity has occurred, which has caused it to fail to respond Perhaps you are using an array that is too large for your computer memory to handle.

SOLUTION: Abort the calculation with CTRL+C

If overuse of computer memory is the problem, try to redo your calculation usingsmaller arrays, e.g., by using fewer grid points in a 3D plot, or by breaking a largevectorized calculation into smaller pieces using a loop Clearing large arrays fromyour Workspace may help too

EXAMPLE: You’ll know it when you see it!

The Most Common Mistakes

The most common mistakes are all accounted for in the causes of the problems scribed earlier But to help you prevent these mistakes, we compile them here in asingle list to which you can refer periodically Doing so will help you to establish

de-“good MATLAB habits.”

• Forgetting to clear or initialize variables

• Improper use of built-in functions

• Inattention to the order of precedence of arithmetic operations

• Improper use of arithmetic symbols

• Mismatched delimiters

• Using the wrong delimiters

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• Plotting the wrong kind of object

• Using uppercase instead of lowercase letters in MATLAB commands, or

mis-spelling commands

Debugging Techniques

Now that we have discussed the most common mistakes, it’s time to discuss how todebug your M-files, and how to locate and fix those pesky problems that don’t fit intothe neat categories above

If one of your M-ﬁles is not working the way you expected, perhaps the easiest

thing you can do to debug it is to insert the command keyboard somewhere in the

middle This temporarily suspends (but does not stop) execution and returns

com-mand to the keyboard, where you are given a special prompt with a K in it You can

execute whatever commands you want at this point (for instance, to examine some of

the variables) To return to execution of the M-ﬁle, type return or dbcont, short

for “debug continue.”

A more systematic way to debug M-ﬁles is to use the debugging features of the

MATLAB M-ﬁle Editor/Debugger Start by going to the menu item Tools:Check

Code with M-Lint This checks the code of the M-ﬁle for common mistakes and

syntax errors and prints out a report specifying where potential problems are located

(You can do the same thing from the Command Window with the command mlint

followed by a space and the name of the ﬁle.) Next, use the debugger to insert

“break-points” in the ﬁle Usually you would do this with the Breakpoints menu or with the

“Set/clear breakpoint” icon at the top of the Editor/Debugger window, but you can

also do it from the Command Window with the command dbstop (See its online

help.) Once a breakpoint has been inserted in the M-ﬁle, you will see a little red dotnext to the appropriate line in the Editor/Debugger (An example is illustrated in Fig-ure 11.8 below.) Then, when you call the M-ﬁle, execution will stop at the breakpoint,

and, just as in the case of keyboard, control will return to the Command Window, where you will be given a special prompt with a K in it Again, when you are ready to resume execution of the M-file, type dbcont When you have finished the debugging process, dbclear “clears” the breakpoint from the M-file.

Let’s illustrate these techniques with a real example Suppose that you want to

construct a function M-ﬁle that takes as input two expressions f and g (given either

as symbolic expressions or as strings) and two numbers a and b, plots the functions

f and g between x = a and x = b, and shades the region in between them As a

ﬁrst try, you might start with the nine-line function M-ﬁle shadecurves.m given

as follows:

function shadecurves(f, g, a, b)

%SHADECURVES Draws the region between two curves

% SHADECURVES(f, g, a, b) takes strings or expressions f

% and g, interprets them as functions, plots them between

% x = a and x = b, and shades the region in between.

% Example: shadecurves(’sin(x)’, ’-sin(x)’, 0, pi)

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Debugging Techniques 213

ffun = inline(vectorize(f)); gfun = inline(vectorize(g)); xvals = a:(b - a)/50:b;

plot([xvals, xvals], [ffun(xvals), gfun(xvals)])

Let’s run the M-ﬁle with the data speciﬁed in the help lines:

Figure 11.3 A First Attempt at Shading the Region between sin x and − sin x.

The graph in Figure 11.3 was obtained by executing

>> shadecurves(’sin(x)’, ’-sin(x)’, 0, pi)

or

>> syms x; shadecurves(sin(x), -sin(x), 0, pi)

This is not really what we wanted; the ﬁgure we seek is shown in Figure 11.4

Figure 11.4 The Shaded Region between sin x and − sin x.

To begin to determine what went wrong, let’s try a different example, say

>> syms x; shadecurves(xˆ2, sqrt(x), 0, 1); axis square

Now we get the output shown in Figure 11.5

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 11.5 A First Attempt at Shading the Region between x2and√

x.

It’s not too hard to ﬁgure out why our regions aren’t shaded; that’s because we used

plot (which plots curves) instead of patch (which plots ﬁlled patches) So that

suggests we should try changing the last line of the M-ﬁle to:

patch([xvals, xvals], [ffun(xvals), gfun(xvals)])

That gives the error message:

??? Error using ==> patch

Not enough input arguments.

Error in ==> shadecurves at 9

patch([xvals, xvals], [ffun(xvals), gfun(xvals)])

so we go back and try

>> help patch

to see whether we can get the syntax right The help lines indicate that patch

re-quires a third argument, the color (in RGB coordinates) with which our patch is to beﬁlled So we change our ﬁnal line to, for instance,

patch([xvals, xvals], [ffun(xvals), gfun(xvals)], [.5,0,.7])

That gives us now as output to

>> syms x; shadecurves(xˆ2, sqrt(x), 0, 1); axis square

the picture shown in Figure 11.6

That’s better, but still not quite right, because we can see a mysterious diagonalline down the middle Not only that, but if we try

>> syms x; shadecurves(xˆ2, xˆ4, -1.5, 1.5)

we now get the bizarre picture shown in Figure 11.7

There aren’t a lot of lines in the M-ﬁle, and lines 7 and 8 seem OK, so the problem

must be with the last line We need to reread the online help for patch It indicates that patch draws a ﬁlled two-dimensional polygon deﬁned by the vectors X and

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 11.6 A Second Attempt at Shading the Region between x2and√

x.

−1.50 −1 −0.5 0 0.5 1 1.5 1

2 3 4 5 6

Figure 11.7 A First Attempt at Shading the Region between x2and x4

Y, which are its first two inputs A way to see how this is working is to changethe “50” in line 9 of the M-file to something much smaller, say 5, and then insert abreakpoint in the M-file before line 9 At this point, our M-file in the Editor/Debuggerwindow now looks like Figure 11.8 Note the large dot to the left of the last line,indicating the breakpoint When we run the M-file with the same input, we now

obtain in the Command Window a K>> prompt At this point, it is logical to try to list

the coordinates of the points that are the vertices of our ﬁlled polygon, so we try

K>> [[xvals, xvals]’, [ffun(xvals), gfun(xvals)]’]

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2 3 4 5 6

Figure 11.9 A Second Attempt at Shading the Region between x2and x4

Finally we can understand what is going on; MATLAB has “connected the dots”using the points whose coordinates were just printed out In particular, MATLAB hasdrawn a line from the point (1.5, 2.25) to the point (−1.5, 5.0625) This is not what

we wanted; we wanted MATLAB to join the point (1.5, 2.25) on the curve y = x2to

the point (1.5, 5.0625) on the curve y = x4 We can ﬁx this by reversing the order

of the x-coordinates at which we evaluate the second function g So, letting slavx

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denote xvals reversed, we correct our M-ﬁle to read:

function shadecurves(f, g, a, b)

%SHADECURVES Draws the region between two curves

% SHADECURVES(f, g, a, b) takes strings or expressions f

% and g, interprets them as functions, plots them between

% x = a and x = b, and shades the region in between.

% Example: shadecurves(’sin(x)’, ’-sin(x)’, 0, pi)

ffun = inline(vectorize(f)); gfun = inline(vectorize(g)); xvals = a:(b - a)/50:b; slavx = b:(a - b)/50:a;

patch([xvals, slavx], [ffun(xvals), gfun(slavx)], [.5,0,.7])

Now it works properly Sample output from this M-ﬁle is shown in Figure 11.4 Try

it out on the other examples we have discussed, or on others of your choice

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Solutions to the Practice Sets

Solutions to Practice Set A: Algebra and Arithmetic

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220 Solutions to the Practice Sets

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Solutions to Practice Set A: Algebra and Arithmetic 221

We get a one-parameter family of answers depending on the variable y In fact the

three planes determined by the three linear equations are not independent, because theﬁrst equation is the sum of the second and third The locus of points that satisfy thethree equations is not a point, the intersection of three independent planes, but rather

a line, the intersection of two distinct planes Once again we check

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Let’s try simple instead.

[better, how] = simple(cos(x)ˆ2 - sin(x)ˆ2)

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But note the following:

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[ 1/3 p 1/2 / 1/3 p \] [- 1/12 %1 + - - 1/2 I 3 |1/6 %1 + 2 -|]

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This is better, but because of the wild oscillation near x = 0, neither plot nor

ezplotgives a totally accurate graph of the function

(d)

X = -1.5:0.01:1.5;

plot(X, exp(-X.ˆ2), X, X.ˆ4 - X.ˆ2)

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Let’s plot 2x and x4 and look for points of intersection We plot them ﬁrst with

ezplotjust to get a feel for the graph

ezplot(’xˆ4’); hold on; ezplot(’2ˆx’); hold off

points of intersection between−2 and 2 Let’s change to plot now and focus on the

interval between−2 and 2 We’ll plot x4dashed

X = -2:0.1:2; plot(X, 2.ˆX);

hold on; plot(X, X.ˆ4, ’ ’); hold off

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We see that there are points of intersection near−0.9 and 1.2 Are there any other

points of intersection? To the left of 0, 2x is always less than 1, whereas x4goes to

infinity as x goes to negative infinity On the other hand, both x4and 2xgo to infinity

as x goes to inﬁnity, so the graphs may cross again to the right of 6 Let’s check.

We see that they do cross again, near x = 16 If you know a little calculus, you can

show that the graphs never cross again (by taking logarithms, for example), so we

have found all the points of intersection Now let’s use fzero to ﬁnd these points

of intersection numerically This command looks for a solution near a given startingpoint To ﬁnd the three different points of intersection we will have to use threedifferent starting points The above graphical analysis suggests appropriate startingpoints

r1 = fzero(’2ˆx - xˆ4’, -0.9)

r2 = fzero(’2ˆx - xˆ4’, 1.2)

r3 = fzero(’2ˆx - xˆ4’, 16)

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Solutions to Practice Set B: Calculus, Graphics, and Linear Algebra 231

The function lambertw is a “special function” that is built into MATLAB You can learn more about it by typing help lambertw Let’s see the decimal values of the

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contour(X, Y, Y.*log(X) + X.*log(Y), [0 0], ’k’)

Warning: Log of zero.

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syms x; int(exp(sin(x)), x, 0, pi)

Warning: Explicit integral could not be found.

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MATLAB integrated this one exactly in 3(e) above; let’s integrate it numerically and

compare answers Unfortunately, the range is inﬁnite, so to use quadl we have to

approximate the interval Note that

exp(-35)

ans =

6.305116760146989e-16

which is close to the standard ﬂoating-point accuracy, so

format long; quadl(@(x) exp(-x.ˆ2), -35, 35)

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ezplot(sin(1/x), [0 1])

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Tiêu đề	Common Problems and Troubleshooting in MATLAB
Trường học	University of Science and Technology of Hanoi
Chuyên ngành	Computer Science
Thể loại	Guide
Thành phố	Hanoi

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