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<h1 class="chapter" id="sec81">Chapter&#XA0;7&#XA0;&#XA0;Iteration</h1>
<p>This chapter is about iteration, which is the ability to run
a block of statements repeatedly. We saw a kind of iteration,
using recursion, in Section&#XA0;<a href="thinkpython2006.html#recursion">5.8</a>.
We saw another kind, using a <span class="c004">for</span> loop,
in Section&#XA0;<a href="thinkpython2005.html#repetition">4.2</a>. In this chapter we&#X2019;ll see yet another
kind, using a <span class="c004">while</span> statement.
But first I want to say a little more about variable assignment.</p>
<h2 class="section" id="sec82">7.1&#XA0;&#XA0;Reassignment</h2>
<a id="hevea_default491"></a>
<a id="hevea_default492"></a>
<a id="hevea_default493"></a></p><p>As you may have discovered, it is legal to make more than one
assignment to the same variable. A new assignment makes an existing
variable refer to a new value (and stop referring to the old value).</p><pre class="verbatim">&gt;&gt;&gt; x = 5
&gt;&gt;&gt; x
&gt;&gt;&gt; x = 7
&gt;&gt;&gt; x
</pre><p>The first time we display 
<span class="c004">x</span>, its value is 5; the second time, its
value is 7.</p><p>Figure&#XA0;<a href="thinkpython2008.html#fig.assign2">7.1</a> shows what <span class="c010">reassignment</span> looks
like in a state diagram. <a id="hevea_default494"></a> <a id="hevea_default495"></a></p><p>At this point I want to address a common source of
Because Python uses the equal sign (<span class="c004">=</span>) for assignment, it is
tempting to interpret a statement like <span class="c004">a = b</span> as a
mathematical
proposition of equality; that is, the claim that <span class="c004">a</span> and
<span class="c004">b</span> are equal. But this interpretation is wrong.
<a id="hevea_default496"></a></p><p>First, equality is a symmetric relationship and assignment is not. For
example, in mathematics, if <span class="c009">a</span>=7 then 7=<span class="c009">a</span>. But in Python, the
statement <span class="c004">a = 7</span> is legal and <span class="c004">7 = a</span> is not.</p><p>Also, in mathematics, a proposition of equality is either true or
false for all time. If <span class="c009">a</span>=<span class="c009">b</span> now, then <span class="c009">a</span> will always equal <span class="c009">b</span>.
In Python, an assignment statement can make two variables equal, but
they don&#X2019;t have to stay that way:</p><pre class="verbatim">&gt;&gt;&gt; a = 5
&gt;&gt;&gt; b = a    # a and b are now equal
&gt;&gt;&gt; a = 3    # a and b are no longer equal
&gt;&gt;&gt; b
</pre><p>The third line changes the value of <span class="c004">a</span> but does not change the
value of <span class="c004">b</span>, so they are no longer equal. </p><p>Reassigning variables is often useful, but you should use it
with caution. If the values of variables change frequently, it can
make the code difficult to read and debug.</p><blockquote class="figure"><div class="center"><hr class="c019"></div>
<div class="center"><img src="thinkpython2008.png"></div>
<div class="caption"><table class="c001 cellpading0"><tr><td class="c018">Figure 7.1: State diagram.</td></tr>
</table></div>
<a id="fig.assign2"></a>
<div class="center"><hr class="c019"></div></blockquote>
<h2 class="section" id="sec83">7.2&#XA0;&#XA0;Updating variables</h2>
<a id="update"></a></p><p><a id="hevea_default497"></a>
<a id="hevea_default498"></a></p><p>A common kind of reassignment is an <span class="c010">update</span>,
where the new value of the variable depends on the old.</p><pre class="verbatim">&gt;&gt;&gt; x = x + 1
</pre><p>This means &#X201C;get the current value of <span class="c004">x</span>, add one, and then
update <span class="c004">x</span> with the new value.&#X201D;</p><p>If you try to update a variable that doesn&#X2019;t exist, you get an
error, because Python evaluates the right side before it assigns
a value to <span class="c004">x</span>:</p><pre class="verbatim">&gt;&gt;&gt; x = x + 1
NameError: name 'x' is not defined
</pre><p>Before you can update a variable, you have to <span class="c010">initialize</span>
it, usually with a simple assignment:
<a id="hevea_default499"></a></p><pre class="verbatim">&gt;&gt;&gt; x = 0
&gt;&gt;&gt; x = x + 1
</pre><p>Updating a variable by adding 1 is called an <span class="c010">increment</span>;
subtracting 1 is called a <span class="c010">decrement</span>.
<a id="hevea_default500"></a>
<a id="hevea_default501"></a></p>
<h2 class="section" id="sec84">7.3&#XA0;&#XA0;The <span class="c004">while</span> statement</h2>
<a id="hevea_default502"></a>
<a id="hevea_default503"></a>
<a id="hevea_default504"></a>
<a id="hevea_default505"></a></p><p>Computers are often used to automate repetitive tasks. Repeating
identical or similar tasks without making errors is something that
computers do well and people do poorly. In a computer program,
repetition is also called <span class="c010">iteration</span>.</p><p>We have already seen two functions, <span class="c004">countdown</span> and
<code>print_n</code>, that iterate using recursion. Because iteration is so
common, Python provides language features to make it easier.
One is the <span class="c004">for</span> statement we saw in Section&#XA0;<a href="thinkpython2005.html#repetition">4.2</a>.
We&#X2019;ll get back to that later.</p><p>Another is the <span class="c004">while</span> statement. Here is a version of <span class="c004">countdown</span> that uses a <span class="c004">while</span> statement:</p><pre class="verbatim">def countdown(n):
    while n &gt; 0:
        print(n)
        n = n - 1
    print('Blastoff!')
</pre><p>You can almost read the <span class="c004">while</span> statement as if it were English.
It means, &#X201C;While <span class="c004">n</span> is greater than 0,
display the value of <span class="c004">n</span> and then decrement
<span class="c004">n</span>. When you get to 0, display the word <span class="c004">Blastoff!</span>&#X201D;
<a id="hevea_default506"></a></p><p>More formally, here is the flow of execution for a <span class="c004">while</span> statement:</p><ol class="enumerate" type=1><li class="li-enumerate">Determine whether the condition is true or false.</li><li class="li-enumerate">If false, exit the <span class="c004">while</span> statement
and continue execution at the next statement.</li><li class="li-enumerate">If the condition is true, run the
body and then go back to step 1.</li></ol><p>This type of flow is called a loop because the third step
loops back around to the top. 
<a id="hevea_default507"></a>
<a id="hevea_default508"></a>
<a id="hevea_default509"></a></p><p>The body of the loop should change the value of one or more variables
so that the condition becomes false eventually and the loop
terminates. Otherwise the loop will repeat forever, which is called
an <span class="c010">infinite loop</span>. An endless source of amusement for computer
scientists is the observation that the directions on shampoo,
&#X201C;Lather, rinse, repeat&#X201D;, are an infinite loop.
<a id="hevea_default510"></a>
<a id="hevea_default511"></a></p><p>In the case of <span class="c004">countdown</span>, we can prove that the loop
terminates: if <span class="c004">n</span> is zero or negative, the loop never runs.
Otherwise, <span class="c004">n</span> gets smaller each time through the
loop, so eventually we have to get to 0.</p><p>For some other loops, it is not so easy to tell. For example:</p><pre class="verbatim">def sequence(n):
    while n != 1:
        print(n)
        if n % 2 == 0:        # n is even
        else:                 # n is odd
            n = n*3 + 1
</pre><p>The condition for this loop is <span class="c004">n != 1</span>, so the loop will continue
until <span class="c004">n</span> is <span class="c004">1</span>, which makes the condition false.</p><p>Each time through the loop, the program outputs the value of <span class="c004">n</span>
and then checks whether it is even or odd. If it is even, <span class="c004">n</span> is
divided by 2. If it is odd, the value of <span class="c004">n</span> is replaced with
<span class="c004">n*3 + 1</span>. For example, if the argument passed to <span class="c004">sequence</span>
is 3, the resulting values of <span class="c004">n</span> are 3, 10, 5, 16, 8, 4, 2, 1.</p><p>Since <span class="c004">n</span> sometimes increases and sometimes decreases, there is no
obvious proof that <span class="c004">n</span> will ever reach 1, or that the program
terminates. For some particular values of <span class="c004">n</span>, we can prove
termination. For example, if the starting value is a power of two,
<span class="c004">n</span> will be even every time through the loop
until it reaches 1. The previous example ends with such a sequence,
starting with 16.
<a id="hevea_default512"></a></p><p>The hard question is whether we can prove that this program terminates
for <em>all</em> positive values of <span class="c004">n</span>. So far, no one has
been able to prove it <em>or</em> disprove it! (See
<a href="http://en.wikipedia.org/wiki/Collatz_conjecture"><span class="c004">http://en.wikipedia.org/wiki/Collatz_conjecture</span></a>.)</p><p>As an exercise, rewrite the function <code>print_n</code> from
Section&#XA0;<a href="thinkpython2006.html#recursion">5.8</a> using iteration instead of recursion.</p>
<h2 class="section" id="sec85">7.4&#XA0;&#XA0;<span class="c004">break</span></h2>
<a id="hevea_default513"></a>
<a id="hevea_default514"></a></p><p>Sometimes you don&#X2019;t know it&#X2019;s time to end a loop until you get half
way through the body. In that case you can use the <span class="c004">break</span>
statement to jump out of the loop.</p><p>For example, suppose you want to take input from the user until they
type <span class="c004">done</span>. You could write:</p><pre class="verbatim">while True:
    line = input('&gt; ')
    if line == 'done':
        break
    print(line)
print('Done!')
</pre><p>The loop condition is <span class="c004">True</span>, which is always true, so the
loop runs until it hits the break statement.</p><p>Each time through, it prompts the user with an angle bracket.
If the user types <span class="c004">done</span>, the <span class="c004">break</span> statement exits
the loop. Otherwise the program echoes whatever the user types
and goes back to the top of the loop. Here&#X2019;s a sample run:</p><pre class="verbatim">&gt; not done
</pre><p>This way of writing <span class="c004">while</span> loops is common because you
can check the condition anywhere in the loop (not just at the
top) and you can express the stop condition affirmatively
(&#X201C;stop when this happens&#X201D;) rather than negatively (&#X201C;keep going
until that happens&#X201D;).</p>
<h2 class="section" id="sec86">7.5&#XA0;&#XA0;Square roots</h2>
<a id="squareroot"></a>
<a id="hevea_default515"></a></p><p>Loops are often used in programs that compute
numerical results by starting with an approximate answer and
iteratively improving it.
<a id="hevea_default516"></a></p><p>For example, one way of computing square roots is Newton&#X2019;s method.
Suppose that you want to know the square root of <span class="c009">a</span>. If you start
with almost any estimate, <span class="c009">x</span>, you can compute a better
estimate with the following formula:</p><table class="display dcenter"><tr class="c017"><td class="dcell"><span class="c009">y</span>&#XA0;=&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell c011"><span class="c009">x</span>&#XA0;+&#XA0;<span class="c009">a</span>/<span class="c009">x</span></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell c011">2</td></tr>
</table></td><td class="dcell">&#XA0;</td></tr>
</table><p>
For example, if <span class="c009">a</span> is 4 and <span class="c009">x</span> is 3:</p><pre class="verbatim">&gt;&gt;&gt; a = 4
&gt;&gt;&gt; x = 3
&gt;&gt;&gt; y = (x + a/x) / 2
&gt;&gt;&gt; y
</pre><p>The result is closer to the correct answer (&#X221A;<span class="c016">4</span> = 2). If we
repeat the process with the new estimate, it gets even closer:</p><pre class="verbatim">&gt;&gt;&gt; x = y
&gt;&gt;&gt; y = (x + a/x) / 2
&gt;&gt;&gt; y
</pre><p>After a few more updates, the estimate is almost exact:
<a id="hevea_default517"></a></p><pre class="verbatim">&gt;&gt;&gt; x = y
&gt;&gt;&gt; y = (x + a/x) / 2
&gt;&gt;&gt; y
&gt;&gt;&gt; x = y
&gt;&gt;&gt; y = (x + a/x) / 2
&gt;&gt;&gt; y
</pre><p>In general we don&#X2019;t know ahead of time how many steps it takes
to get to the right answer, but we know when we get there
because the estimate
stops changing:</p><pre class="verbatim">&gt;&gt;&gt; x = y
&gt;&gt;&gt; y = (x + a/x) / 2
&gt;&gt;&gt; y
&gt;&gt;&gt; x = y
&gt;&gt;&gt; y = (x + a/x) / 2
&gt;&gt;&gt; y
</pre><p>When <span class="c004">y == x</span>, we can stop. Here is a loop that starts
with an initial estimate, <span class="c004">x</span>, and improves it until it
stops changing:</p><pre class="verbatim">while True:
    print(x)
    y = (x + a/x) / 2
    if y == x:
        break
</pre><p>For most values of <span class="c004">a</span> this works fine, but in general it is
dangerous to test <span class="c004">float</span> equality.
Floating-point values are only approximately right:
most rational numbers, like 1/3, and irrational numbers, like
&#X221A;<span class="c016">2</span>, can&#X2019;t be represented exactly with a <span class="c004">float</span>.
<a id="hevea_default518"></a>
<a id="hevea_default519"></a></p><p>Rather than checking whether <span class="c004">x</span> and <span class="c004">y</span> are exactly equal, it
is safer to use the built-in function <span class="c004">abs</span> to compute the
absolute value, or magnitude, of the difference between them:</p><pre class="verbatim">    if abs(y-x) &lt; epsilon:
        break
</pre><p>Where <code>epsilon</code> has a value like <span class="c004">0.0000001</span> that
determines how close is close enough.</p>
<h2 class="section" id="sec87">7.6&#XA0;&#XA0;Algorithms</h2>
<a id="hevea_default520"></a></p><p>Newton&#X2019;s method is an example of an <span class="c010">algorithm</span>: it is a
mechanical process for solving a category of problems (in this
case, computing square roots).</p><p>To understand what an algorithm is, it might help to start with
something that is not an algorithm. When you learned to multiply
single-digit numbers, you probably memorized the multiplication table.
In effect, you memorized 100 specific solutions. That kind of
knowledge is not algorithmic.</p><p>But if you were &#X201C;lazy&#X201D;, you might have learned a few
tricks. For example, to find the product of <span class="c009">n</span> and 9, you can
write <span class="c009">n</span>&#X2212;1 as the first digit and 10&#X2212;<span class="c009">n</span> as the second
digit. This trick is a general solution for multiplying any
single-digit number by 9. That&#X2019;s an algorithm!
<a id="hevea_default521"></a>
<a id="hevea_default522"></a>
<a id="hevea_default523"></a>
<a id="hevea_default524"></a></p><p>Similarly, the techniques you learned for addition with carrying,
subtraction with borrowing, and long division are all algorithms. One
of the characteristics of algorithms is that they do not require any
intelligence to carry out. They are mechanical processes where
each step follows from the last according to a simple set of rules.</p><p>Executing algorithms is boring, but designing them is interesting,
intellectually challenging, and a central part of computer science.</p><p>Some of the things that people do naturally, without difficulty or
conscious thought, are the hardest to express algorithmically.
Understanding natural language is a good example. We all do it, but
so far no one has been able to explain <em>how</em> we do it, at least
not in the form of an algorithm.</p>
<h2 class="section" id="sec88">7.7&#XA0;&#XA0;Debugging</h2>
<a id="bisectbug"></a></p><p>As you start writing bigger programs, you might find yourself
spending more time debugging. More code means more chances to
make an error and more places for bugs to hide.
<a id="hevea_default525"></a>
<a id="hevea_default526"></a></p><p>One way to cut your debugging time is &#X201C;debugging by bisection&#X201D;.
For example, if there are 100 lines in your program and you
check them one at a time, it would take 100 steps.</p><p>Instead, try to break the problem in half. Look at the middle
of the program, or near it, for an intermediate value you
can check. Add a <span class="c004">print</span> statement (or something else
that has a verifiable effect) and run the program.</p><p>If the mid-point check is incorrect, there must be a problem in the
first half of the program. If it is correct, the problem is
in the second half.</p><p>Every time you perform a check like this, you halve the number of
lines you have to search. After six steps (which is fewer than 100),
you would be down to one or two lines of code, at least in theory.</p><p>In practice it is not always clear what
the &#X201C;middle of the program&#X201D; is and not always possible to
check it. It doesn&#X2019;t make sense to count lines and find the
exact midpoint. Instead, think about places
in the program where there might be errors and places where it
is easy to put a check. Then choose a spot where you
think the chances are about the same that the bug is before
or after the check.</p>
<h2 class="section" id="sec89">7.8&#XA0;&#XA0;Glossary</h2>
<dl class="description"><dt class="dt-description"><span class="c010">reassignment:</span></dt><dd class="dd-description"> Assigning a new value to a variable that
already exists.
<a id="hevea_default527"></a></dd><dt class="dt-description"><span class="c010">update:</span></dt><dd class="dd-description"> An assignment where the new value of the variable
depends on the old.
<a id="hevea_default528"></a></dd><dt class="dt-description"><span class="c010">initialization:</span></dt><dd class="dd-description"> An assignment that gives an initial value to
a variable that will be updated.
<a id="hevea_default529"></a></dd><dt class="dt-description"><span class="c010">increment:</span></dt><dd class="dd-description"> An update that increases the value of a variable
(often by one).
<a id="hevea_default530"></a></dd><dt class="dt-description"><span class="c010">decrement:</span></dt><dd class="dd-description"> An update that decreases the value of a variable.
<a id="hevea_default531"></a></dd><dt class="dt-description"><span class="c010">iteration:</span></dt><dd class="dd-description"> Repeated execution of a set of statements using
either a recursive function call or a loop.
<a id="hevea_default532"></a></dd><dt class="dt-description"><span class="c010">infinite loop:</span></dt><dd class="dd-description"> A loop in which the terminating condition is
never satisfied.
<a id="hevea_default533"></a></dd><dt class="dt-description"><span class="c010">algorithm:</span></dt><dd class="dd-description"> A general process for solving a category of
<a id="hevea_default534"></a></dd></dl>
<h2 class="section" id="sec90">7.9&#XA0;&#XA0;Exercises</h2>
<div class="theorem"><span class="c010">Exercise&#XA0;1</span>&#XA0;&#XA0;
<a id="hevea_default535"></a><p><em>Copy the loop from Section&#XA0;</em><a href="thinkpython2008.html#squareroot"><em>7.5</em></a><em>
and encapsulate it in a function called
<code>mysqrt</code> that takes <span class="c004">a</span> as a parameter, chooses a
reasonable value of <span class="c004">x</span>, and returns an estimate of the square
root of <span class="c004">a</span>. </em><a id="hevea_default536"></a></p><p><em>To test it, write a function named <code>test_square_root</code>
that prints a table like this:</em></p><pre class="verbatim"><em>a   mysqrt(a)     math.sqrt(a)  diff
-   ---------     ------------  ----
</em></pre><p><em>The first column is a number, </em><span class="c009">a</span><em>; the second column is the square
root of </em><span class="c009">a</span><em> computed with <code>mysqrt</code>; the third column is the
square root computed by <span class="c004">math.sqrt</span>; the fourth column is the
absolute value of the difference between the two estimates.
</em></p></div><div class="theorem"><span class="c010">Exercise&#XA0;2</span>&#XA0;&#XA0;
<a id="hevea_default537"></a>
<a id="hevea_default538"></a><p><em>The built-in function <span class="c004">eval</span> takes a string and evaluates
it using the Python interpreter. For example:</em></p><pre class="verbatim"><em>&gt;&gt;&gt; eval('1 + 2 * 3')
&gt;&gt;&gt; import math
&gt;&gt;&gt; eval('math.sqrt(5)')
&gt;&gt;&gt; eval('type(math.pi)')
&lt;class 'float'&gt;
</em></pre><p><em>Write a function called <code>eval_loop</code> that iteratively
prompts the user, takes the resulting input and evaluates
it using <span class="c004">eval</span>, and prints the result.</em></p><p><em>It should continue until the user enters <code>'done'</code>, and then
return the value of the last expression it evaluated.</em></p></div><div class="theorem"><span class="c010">Exercise&#XA0;3</span>&#XA0;&#XA0;
<a id="hevea_default539"></a><p><em>The mathematician Srinivasa Ramanujan found an
infinite series
that can be used to generate a numerical
approximation of </em>1 / &#X3C0;<em>:
</em><a id="hevea_default540"></a></p><table class="display dcenter"><tr class="c017"><td class="dcell"><table class="display"><tr><td class="dcell c011">1</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell c011">&#X3C0;</td></tr>
</table></td><td class="dcell">&#XA0;=&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell c011"><table class="display"><tr class="c017"><td class="dcell">2</td><td class="dcell"><span class="c008">&#X221A;</span></td><td class="dcell"><table class="c002 cellpadding0"><tr><td class="hbar"></td></tr>
<tr><td class="c012">2</td></tr>
</table></td></tr>
</table></td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell c011">9801</td></tr>
</table></td><td class="dcell">&#XA0;
</td><td class="dcell"><table class="display"><tr><td class="dcell c011">&#X221E;</td></tr>
<tr><td class="dcell c011"><span style="font-size:xx-large">&#X2211;</span></td></tr>
<tr><td class="dcell c011"><span class="c009">k</span>=0</td></tr>
</table></td><td class="dcell">&#XA0;</td><td class="dcell"><table class="display"><tr><td class="dcell c011">(4<span class="c009">k</span>)!(1103+26390<span class="c009">k</span>)</td></tr>
<tr><td class="hbar"></td></tr>
<tr><td class="dcell c011">(<span class="c009">k</span>!)<sup>4</sup>&#XA0;396<sup>4<span class="c009">k</span></sup></td></tr>
</table></td><td class="dcell">&#XA0;</td></tr>
</table><p><em>Write a function called <code>estimate_pi</code> that uses this formula
to compute and return an estimate of </em>&#X3C0;<em>. It should use a <span class="c004">while</span>
loop to compute terms of the summation until the last term is
smaller than <span class="c004">1e-15</span> (which is Python notation for </em>10<sup>&#X2212;15</sup><em>).
You can check the result by comparing it to <span class="c004">math.pi</span>.</em></p><p><em>Solution: </em><a href="http://thinkpython2.com/code/pi.py"><em><span class="c004">http://thinkpython2.com/code/pi.py</span></em></a><em>.</em></p></div>
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