path: root/devdocs/python~3.12/library%2Fitertools.html

 <span id="itertools-functions-creating-iterators-for-efficient-looping"></span><h1>itertools — Functions creating iterators for efficient looping</h1>  <p>This module implements a number of <a class="reference internal" href="../glossary#term-iterator"><span class="xref std std-term">iterator</span></a> building blocks inspired by constructs from APL, Haskell, and SML. Each has been recast in a form suitable for Python.</p> <p>The module standardizes a core set of fast, memory efficient tools that are useful by themselves or in combination. Together, they form an “iterator algebra” making it possible to construct specialized tools succinctly and efficiently in pure Python.</p> <p>For instance, SML provides a tabulation tool: <code>tabulate(f)</code> which produces a sequence <code>f(0), f(1), ...</code>. The same effect can be achieved in Python by combining <a class="reference internal" href="functions#map" title="map"><code>map()</code></a> and <a class="reference internal" href="#itertools.count" title="itertools.count"><code>count()</code></a> to form <code>map(f, count())</code>.</p> <p>These tools and their built-in counterparts also work well with the high-speed functions in the <a class="reference internal" href="operator#module-operator" title="operator: Functions corresponding to the standard operators."><code>operator</code></a> module. For example, the multiplication operator can be mapped across two vectors to form an efficient dot-product: <code>sum(starmap(operator.mul, zip(vec1, vec2, strict=True)))</code>.</p> <p><strong>Infinite iterators:</strong></p> <table class="docutils align-default">  <thead> <tr>
<th class="head"><p>Iterator</p></th> <th class="head"><p>Arguments</p></th> <th class="head"><p>Results</p></th> <th class="head"><p>Example</p></th> </tr> </thead>  <tr>
<td><p><a class="reference internal" href="#itertools.count" title="itertools.count"><code>count()</code></a></p></td> <td><p>[start[, step]]</p></td> <td><p>start, start+step, start+2*step, …</p></td> <td><p><code>count(10) --&gt; 10 11 12 13 14 ...</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.cycle" title="itertools.cycle"><code>cycle()</code></a></p></td> <td><p>p</p></td> <td><p>p0, p1, … plast, p0, p1, …</p></td> <td><p><code>cycle('ABCD') --&gt; A B C D A B C D ...</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.repeat" title="itertools.repeat"><code>repeat()</code></a></p></td> <td><p>elem [,n]</p></td> <td><p>elem, elem, elem, … endlessly or up to n times</p></td> <td><p><code>repeat(10, 3) --&gt; 10 10 10</code></p></td> </tr>  </table> <p><strong>Iterators terminating on the shortest input sequence:</strong></p> <table class="docutils align-default">  <thead> <tr>
<th class="head"><p>Iterator</p></th> <th class="head"><p>Arguments</p></th> <th class="head"><p>Results</p></th> <th class="head"><p>Example</p></th> </tr> </thead>  <tr>
<td><p><a class="reference internal" href="#itertools.accumulate" title="itertools.accumulate"><code>accumulate()</code></a></p></td> <td><p>p [,func]</p></td> <td><p>p0, p0+p1, p0+p1+p2, …</p></td> <td><p><code>accumulate([1,2,3,4,5]) --&gt; 1 3 6 10 15</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.batched" title="itertools.batched"><code>batched()</code></a></p></td> <td><p>p, n</p></td> <td><p>(p0, p1, …, p_n-1), …</p></td> <td><p><code>batched('ABCDEFG', n=3) --&gt; ABC DEF G</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.chain" title="itertools.chain"><code>chain()</code></a></p></td> <td><p>p, q, …</p></td> <td><p>p0, p1, … plast, q0, q1, …</p></td> <td><p><code>chain('ABC', 'DEF') --&gt; A B C D E F</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.chain.from_iterable" title="itertools.chain.from_iterable"><code>chain.from_iterable()</code></a></p></td> <td><p>iterable</p></td> <td><p>p0, p1, … plast, q0, q1, …</p></td> <td><p><code>chain.from_iterable(['ABC', 'DEF']) --&gt; A B C D E F</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.compress" title="itertools.compress"><code>compress()</code></a></p></td> <td><p>data, selectors</p></td> <td><p>(d[0] if s[0]), (d[1] if s[1]), …</p></td> <td><p><code>compress('ABCDEF', [1,0,1,0,1,1]) --&gt; A C E F</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.dropwhile" title="itertools.dropwhile"><code>dropwhile()</code></a></p></td> <td><p>pred, seq</p></td> <td><p>seq[n], seq[n+1], starting when pred fails</p></td> <td><p><code>dropwhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 6 4 1</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.filterfalse" title="itertools.filterfalse"><code>filterfalse()</code></a></p></td> <td><p>pred, seq</p></td> <td><p>elements of seq where pred(elem) is false</p></td> <td><p><code>filterfalse(lambda x: x%2, range(10)) --&gt; 0 2 4 6 8</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code>groupby()</code></a></p></td> <td><p>iterable[, key]</p></td> <td><p>sub-iterators grouped by value of key(v)</p></td> <td></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.islice" title="itertools.islice"><code>islice()</code></a></p></td> <td><p>seq, [start,] stop [, step]</p></td> <td><p>elements from seq[start:stop:step]</p></td> <td><p><code>islice('ABCDEFG', 2, None) --&gt; C D E F G</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.pairwise" title="itertools.pairwise"><code>pairwise()</code></a></p></td> <td><p>iterable</p></td> <td><p>(p[0], p[1]), (p[1], p[2])</p></td> <td><p><code>pairwise('ABCDEFG') --&gt; AB BC CD DE EF FG</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.starmap" title="itertools.starmap"><code>starmap()</code></a></p></td> <td><p>func, seq</p></td> <td><p>func(*seq[0]), func(*seq[1]), …</p></td> <td><p><code>starmap(pow, [(2,5), (3,2), (10,3)]) --&gt; 32 9 1000</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.takewhile" title="itertools.takewhile"><code>takewhile()</code></a></p></td> <td><p>pred, seq</p></td> <td><p>seq[0], seq[1], until pred fails</p></td> <td><p><code>takewhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 1 4</code></p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.tee" title="itertools.tee"><code>tee()</code></a></p></td> <td><p>it, n</p></td> <td><p>it1, it2, … itn splits one iterator into n</p></td> <td></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.zip_longest" title="itertools.zip_longest"><code>zip_longest()</code></a></p></td> <td><p>p, q, …</p></td> <td><p>(p[0], q[0]), (p[1], q[1]), …</p></td> <td><p><code>zip_longest('ABCD', 'xy', fillvalue='-') --&gt; Ax By C- D-</code></p></td> </tr>  </table> <p><strong>Combinatoric iterators:</strong></p> <table class="docutils align-default">  <thead> <tr>
<th class="head"><p>Iterator</p></th> <th class="head"><p>Arguments</p></th> <th class="head"><p>Results</p></th> </tr> </thead>  <tr>
<td><p><a class="reference internal" href="#itertools.product" title="itertools.product"><code>product()</code></a></p></td> <td><p>p, q, … [repeat=1]</p></td> <td><p>cartesian product, equivalent to a nested for-loop</p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.permutations" title="itertools.permutations"><code>permutations()</code></a></p></td> <td><p>p[, r]</p></td> <td><p>r-length tuples, all possible orderings, no repeated elements</p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.combinations" title="itertools.combinations"><code>combinations()</code></a></p></td> <td><p>p, r</p></td> <td><p>r-length tuples, in sorted order, no repeated elements</p></td> </tr> <tr>
<td><p><a class="reference internal" href="#itertools.combinations_with_replacement" title="itertools.combinations_with_replacement"><code>combinations_with_replacement()</code></a></p></td> <td><p>p, r</p></td> <td><p>r-length tuples, in sorted order, with repeated elements</p></td> </tr>  </table> <table class="docutils align-default">  <thead> <tr>
<th class="head"><p>Examples</p></th> <th class="head"><p>Results</p></th> </tr> </thead>  <tr>
<td><p><code>product('ABCD', repeat=2)</code></p></td> <td><p><code>AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD</code></p></td> </tr> <tr>
<td><p><code>permutations('ABCD', 2)</code></p></td> <td><p><code>AB AC AD BA BC BD CA CB CD DA DB DC</code></p></td> </tr> <tr>
<td><p><code>combinations('ABCD', 2)</code></p></td> <td><p><code>AB AC AD BC BD CD</code></p></td> </tr> <tr>
<td><p><code>combinations_with_replacement('ABCD', 2)</code></p></td> <td><p><code>AA AB AC AD BB BC BD CC CD DD</code></p></td> </tr>  </table> <section id="itertool-functions"> <span id="itertools-functions"></span><h2>Itertool functions</h2> <p>The following module functions all construct and return iterators. Some provide streams of infinite length, so they should only be accessed by functions or loops that truncate the stream.</p> <dl class="py function"> <dt class="sig sig-object py" id="itertools.accumulate">
<code>itertools.accumulate(iterable[, func, *, initial=None])</code> </dt> <dd>
<p>Make an iterator that returns accumulated sums, or accumulated results of other binary functions (specified via the optional <em>func</em> argument).</p> <p>If <em>func</em> is supplied, it should be a function of two arguments. Elements of the input <em>iterable</em> may be any type that can be accepted as arguments to <em>func</em>. (For example, with the default operation of addition, elements may be any addable type including <a class="reference internal" href="decimal#decimal.Decimal" title="decimal.Decimal"><code>Decimal</code></a> or <a class="reference internal" href="fractions#fractions.Fraction" title="fractions.Fraction"><code>Fraction</code></a>.)</p> <p>Usually, the number of elements output matches the input iterable. However, if the keyword argument <em>initial</em> is provided, the accumulation leads off with the <em>initial</em> value so that the output has one more element than the input iterable.</p> <p>Roughly equivalent to:</p> <pre data-language="python">def accumulate(iterable, func=operator.add, *, initial=None):
    'Return running totals'
    # accumulate([1,2,3,4,5]) --&gt; 1 3 6 10 15
    # accumulate([1,2,3,4,5], initial=100) --&gt; 100 101 103 106 110 115
    # accumulate([1,2,3,4,5], operator.mul) --&gt; 1 2 6 24 120
    it = iter(iterable)
    total = initial
    if initial is None:
        try:
            total = next(it)
        except StopIteration:
            return
    yield total
    for element in it:
        total = func(total, element)
        yield total
</pre> <p>There are a number of uses for the <em>func</em> argument. It can be set to <a class="reference internal" href="functions#min" title="min"><code>min()</code></a> for a running minimum, <a class="reference internal" href="functions#max" title="max"><code>max()</code></a> for a running maximum, or <a class="reference internal" href="operator#operator.mul" title="operator.mul"><code>operator.mul()</code></a> for a running product. Amortization tables can be built by accumulating interest and applying payments:</p> <pre data-language="pycon3">&gt;&gt;&gt; data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
&gt;&gt;&gt; list(accumulate(data, operator.mul))     # running product
[3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]
&gt;&gt;&gt; list(accumulate(data, max))              # running maximum
[3, 4, 6, 6, 6, 9, 9, 9, 9, 9]

# Amortize a 5% loan of 1000 with 10 annual payments of 90
&gt;&gt;&gt; account_update = lambda bal, pmt: round(bal * 1.05) + pmt
&gt;&gt;&gt; list(accumulate(repeat(-90, 10), account_update, initial=1_000))
[1000, 960, 918, 874, 828, 779, 728, 674, 618, 559, 497]
</pre> <p>See <a class="reference internal" href="functools#functools.reduce" title="functools.reduce"><code>functools.reduce()</code></a> for a similar function that returns only the final accumulated value.</p> <div class="versionadded"> <p><span class="versionmodified added">New in version 3.2.</span></p> </div> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 3.3: </span>Added the optional <em>func</em> parameter.</p> </div> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 3.8: </span>Added the optional <em>initial</em> parameter.</p> </div> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.batched">
<code>itertools.batched(iterable, n)</code> </dt> <dd>
<p>Batch data from the <em>iterable</em> into tuples of length <em>n</em>. The last batch may be shorter than <em>n</em>.</p> <p>Loops over the input iterable and accumulates data into tuples up to size <em>n</em>. The input is consumed lazily, just enough to fill a batch. The result is yielded as soon as the batch is full or when the input iterable is exhausted:</p> <pre data-language="pycon3">&gt;&gt;&gt; flattened_data = ['roses', 'red', 'violets', 'blue', 'sugar', 'sweet']
&gt;&gt;&gt; unflattened = list(batched(flattened_data, 2))
&gt;&gt;&gt; unflattened
[('roses', 'red'), ('violets', 'blue'), ('sugar', 'sweet')]

&gt;&gt;&gt; for batch in batched('ABCDEFG', 3):
...     print(batch)
...
('A', 'B', 'C')
('D', 'E', 'F')
('G',)
</pre> <p>Roughly equivalent to:</p> <pre data-language="python">def batched(iterable, n):
    # batched('ABCDEFG', 3) --&gt; ABC DEF G
    if n &lt; 1:
        raise ValueError('n must be at least one')
    it = iter(iterable)
    while batch := tuple(islice(it, n)):
        yield batch
</pre> <div class="versionadded"> <p><span class="versionmodified added">New in version 3.12.</span></p> </div> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.chain">
<code>itertools.chain(*iterables)</code> </dt> <dd>
<p>Make an iterator that returns elements from the first iterable until it is exhausted, then proceeds to the next iterable, until all of the iterables are exhausted. Used for treating consecutive sequences as a single sequence. Roughly equivalent to:</p> <pre data-language="python">def chain(*iterables):
    # chain('ABC', 'DEF') --&gt; A B C D E F
    for it in iterables:
        for element in it:
            yield element
</pre> </dd>
</dl> <dl class="py method"> <dt class="sig sig-object py" id="itertools.chain.from_iterable">
<code>classmethod chain.from_iterable(iterable)</code> </dt> <dd>
<p>Alternate constructor for <a class="reference internal" href="#itertools.chain" title="itertools.chain"><code>chain()</code></a>. Gets chained inputs from a single iterable argument that is evaluated lazily. Roughly equivalent to:</p> <pre data-language="python">def from_iterable(iterables):
    # chain.from_iterable(['ABC', 'DEF']) --&gt; A B C D E F
    for it in iterables:
        for element in it:
            yield element
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.combinations">
<code>itertools.combinations(iterable, r)</code> </dt> <dd>
<p>Return <em>r</em> length subsequences of elements from the input <em>iterable</em>.</p> <p>The combination tuples are emitted in lexicographic ordering according to the order of the input <em>iterable</em>. So, if the input <em>iterable</em> is sorted, the output tuples will be produced in sorted order.</p> <p>Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeated values in each combination.</p> <p>Roughly equivalent to:</p> <pre data-language="python">def combinations(iterable, r):
    # combinations('ABCD', 2) --&gt; AB AC AD BC BD CD
    # combinations(range(4), 3) --&gt; 012 013 023 123
    pool = tuple(iterable)
    n = len(pool)
    if r &gt; n:
        return
    indices = list(range(r))
    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != i + n - r:
                break
        else:
            return
        indices[i] += 1
        for j in range(i+1, r):
            indices[j] = indices[j-1] + 1
        yield tuple(pool[i] for i in indices)
</pre> <p>The code for <a class="reference internal" href="#itertools.combinations" title="itertools.combinations"><code>combinations()</code></a> can be also expressed as a subsequence of <a class="reference internal" href="#itertools.permutations" title="itertools.permutations"><code>permutations()</code></a> after filtering entries where the elements are not in sorted order (according to their position in the input pool):</p> <pre data-language="python">def combinations(iterable, r):
    pool = tuple(iterable)
    n = len(pool)
    for indices in permutations(range(n), r):
        if sorted(indices) == list(indices):
            yield tuple(pool[i] for i in indices)
</pre> <p>The number of items returned is <code>n! / r! / (n-r)!</code> when <code>0 &lt;= r &lt;= n</code> or zero when <code>r &gt; n</code>.</p> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.combinations_with_replacement">
<code>itertools.combinations_with_replacement(iterable, r)</code> </dt> <dd>
<p>Return <em>r</em> length subsequences of elements from the input <em>iterable</em> allowing individual elements to be repeated more than once.</p> <p>The combination tuples are emitted in lexicographic ordering according to the order of the input <em>iterable</em>. So, if the input <em>iterable</em> is sorted, the output tuples will be produced in sorted order.</p> <p>Elements are treated as unique based on their position, not on their value. So if the input elements are unique, the generated combinations will also be unique.</p> <p>Roughly equivalent to:</p> <pre data-language="python">def combinations_with_replacement(iterable, r):
    # combinations_with_replacement('ABC', 2) --&gt; AA AB AC BB BC CC
    pool = tuple(iterable)
    n = len(pool)
    if not n and r:
        return
    indices = [0] * r
    yield tuple(pool[i] for i in indices)
    while True:
        for i in reversed(range(r)):
            if indices[i] != n - 1:
                break
        else:
            return
        indices[i:] = [indices[i] + 1] * (r - i)
        yield tuple(pool[i] for i in indices)
</pre> <p>The code for <a class="reference internal" href="#itertools.combinations_with_replacement" title="itertools.combinations_with_replacement"><code>combinations_with_replacement()</code></a> can be also expressed as a subsequence of <a class="reference internal" href="#itertools.product" title="itertools.product"><code>product()</code></a> after filtering entries where the elements are not in sorted order (according to their position in the input pool):</p> <pre data-language="python">def combinations_with_replacement(iterable, r):
    pool = tuple(iterable)
    n = len(pool)
    for indices in product(range(n), repeat=r):
        if sorted(indices) == list(indices):
            yield tuple(pool[i] for i in indices)
</pre> <p>The number of items returned is <code>(n+r-1)! / r! / (n-1)!</code> when <code>n &gt; 0</code>.</p> <div class="versionadded"> <p><span class="versionmodified added">New in version 3.1.</span></p> </div> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.compress">
<code>itertools.compress(data, selectors)</code> </dt> <dd>
<p>Make an iterator that filters elements from <em>data</em> returning only those that have a corresponding element in <em>selectors</em> that evaluates to <code>True</code>. Stops when either the <em>data</em> or <em>selectors</em> iterables has been exhausted. Roughly equivalent to:</p> <pre data-language="python">def compress(data, selectors):
    # compress('ABCDEF', [1,0,1,0,1,1]) --&gt; A C E F
    return (d for d, s in zip(data, selectors) if s)
</pre> <div class="versionadded"> <p><span class="versionmodified added">New in version 3.1.</span></p> </div> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.count">
<code>itertools.count(start=0, step=1)</code> </dt> <dd>
<p>Make an iterator that returns evenly spaced values starting with number <em>start</em>. Often used as an argument to <a class="reference internal" href="functions#map" title="map"><code>map()</code></a> to generate consecutive data points. Also, used with <a class="reference internal" href="functions#zip" title="zip"><code>zip()</code></a> to add sequence numbers. Roughly equivalent to:</p> <pre data-language="python">def count(start=0, step=1):
    # count(10) --&gt; 10 11 12 13 14 ...
    # count(2.5, 0.5) --&gt; 2.5 3.0 3.5 ...
    n = start
    while True:
        yield n
        n += step
</pre> <p>When counting with floating point numbers, better accuracy can sometimes be achieved by substituting multiplicative code such as: <code>(start + step * i
for i in count())</code>.</p> <div class="versionchanged"> <p><span class="versionmodified changed">Changed in version 3.1: </span>Added <em>step</em> argument and allowed non-integer arguments.</p> </div> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.cycle">
<code>itertools.cycle(iterable)</code> </dt> <dd>
<p>Make an iterator returning elements from the iterable and saving a copy of each. When the iterable is exhausted, return elements from the saved copy. Repeats indefinitely. Roughly equivalent to:</p> <pre data-language="python">def cycle(iterable):
    # cycle('ABCD') --&gt; A B C D A B C D A B C D ...
    saved = []
    for element in iterable:
        yield element
        saved.append(element)
    while saved:
        for element in saved:
              yield element
</pre> <p>Note, this member of the toolkit may require significant auxiliary storage (depending on the length of the iterable).</p> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.dropwhile">
<code>itertools.dropwhile(predicate, iterable)</code> </dt> <dd>
<p>Make an iterator that drops elements from the iterable as long as the predicate is true; afterwards, returns every element. Note, the iterator does not produce <em>any</em> output until the predicate first becomes false, so it may have a lengthy start-up time. Roughly equivalent to:</p> <pre data-language="python">def dropwhile(predicate, iterable):
    # dropwhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 6 4 1
    iterable = iter(iterable)
    for x in iterable:
        if not predicate(x):
            yield x
            break
    for x in iterable:
        yield x
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.filterfalse">
<code>itertools.filterfalse(predicate, iterable)</code> </dt> <dd>
<p>Make an iterator that filters elements from iterable returning only those for which the predicate is false. If <em>predicate</em> is <code>None</code>, return the items that are false. Roughly equivalent to:</p> <pre data-language="python">def filterfalse(predicate, iterable):
    # filterfalse(lambda x: x%2, range(10)) --&gt; 0 2 4 6 8
    if predicate is None:
        predicate = bool
    for x in iterable:
        if not predicate(x):
            yield x
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.groupby">
<code>itertools.groupby(iterable, key=None)</code> </dt> <dd>
<p>Make an iterator that returns consecutive keys and groups from the <em>iterable</em>. The <em>key</em> is a function computing a key value for each element. If not specified or is <code>None</code>, <em>key</em> defaults to an identity function and returns the element unchanged. Generally, the iterable needs to already be sorted on the same key function.</p> <p>The operation of <a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code>groupby()</code></a> is similar to the <code>uniq</code> filter in Unix. It generates a break or new group every time the value of the key function changes (which is why it is usually necessary to have sorted the data using the same key function). That behavior differs from SQL’s GROUP BY which aggregates common elements regardless of their input order.</p> <p>The returned group is itself an iterator that shares the underlying iterable with <a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code>groupby()</code></a>. Because the source is shared, when the <a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code>groupby()</code></a> object is advanced, the previous group is no longer visible. So, if that data is needed later, it should be stored as a list:</p> <pre data-language="python">groups = []
uniquekeys = []
data = sorted(data, key=keyfunc)
for k, g in groupby(data, keyfunc):
    groups.append(list(g))      # Store group iterator as a list
    uniquekeys.append(k)
</pre> <p><a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code>groupby()</code></a> is roughly equivalent to:</p> <pre data-language="python">class groupby:
    # [k for k, g in groupby('AAAABBBCCDAABBB')] --&gt; A B C D A B
    # [list(g) for k, g in groupby('AAAABBBCCD')] --&gt; AAAA BBB CC D

    def __init__(self, iterable, key=None):
        if key is None:
            key = lambda x: x
        self.keyfunc = key
        self.it = iter(iterable)
        self.tgtkey = self.currkey = self.currvalue = object()

    def __iter__(self):
        return self

    def __next__(self):
        self.id = object()
        while self.currkey == self.tgtkey:
            self.currvalue = next(self.it)    # Exit on StopIteration
            self.currkey = self.keyfunc(self.currvalue)
        self.tgtkey = self.currkey
        return (self.currkey, self._grouper(self.tgtkey, self.id))

    def _grouper(self, tgtkey, id):
        while self.id is id and self.currkey == tgtkey:
            yield self.currvalue
            try:
                self.currvalue = next(self.it)
            except StopIteration:
                return
            self.currkey = self.keyfunc(self.currvalue)
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.islice">
<code>itertools.islice(iterable, stop)</code> </dt> <dt class="sig sig-object py"> <span class="sig-prename descclassname">itertools.</span><span class="sig-name descname">islice</span><span class="sig-paren">(</span><em class="sig-param"><span class="n">iterable</span></em>, <em class="sig-param"><span class="n">start</span></em>, <em class="sig-param"><span class="n">stop</span></em><span class="optional">[</span>, <em class="sig-param"><span class="n">step</span></em><span class="optional">]</span><span class="sig-paren">)</span>
</dt> <dd>
<p>Make an iterator that returns selected elements from the iterable. If <em>start</em> is non-zero, then elements from the iterable are skipped until start is reached. Afterward, elements are returned consecutively unless <em>step</em> is set higher than one which results in items being skipped. If <em>stop</em> is <code>None</code>, then iteration continues until the iterator is exhausted, if at all; otherwise, it stops at the specified position.</p> <p>If <em>start</em> is <code>None</code>, then iteration starts at zero. If <em>step</em> is <code>None</code>, then the step defaults to one.</p> <p>Unlike regular slicing, <a class="reference internal" href="#itertools.islice" title="itertools.islice"><code>islice()</code></a> does not support negative values for <em>start</em>, <em>stop</em>, or <em>step</em>. Can be used to extract related fields from data where the internal structure has been flattened (for example, a multi-line report may list a name field on every third line).</p> <p>Roughly equivalent to:</p> <pre data-language="python">def islice(iterable, *args):
    # islice('ABCDEFG', 2) --&gt; A B
    # islice('ABCDEFG', 2, 4) --&gt; C D
    # islice('ABCDEFG', 2, None) --&gt; C D E F G
    # islice('ABCDEFG', 0, None, 2) --&gt; A C E G
    s = slice(*args)
    start, stop, step = s.start or 0, s.stop or sys.maxsize, s.step or 1
    it = iter(range(start, stop, step))
    try:
        nexti = next(it)
    except StopIteration:
        # Consume *iterable* up to the *start* position.
        for i, element in zip(range(start), iterable):
            pass
        return
    try:
        for i, element in enumerate(iterable):
            if i == nexti:
                yield element
                nexti = next(it)
    except StopIteration:
        # Consume to *stop*.
        for i, element in zip(range(i + 1, stop), iterable):
            pass
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.pairwise">
<code>itertools.pairwise(iterable)</code> </dt> <dd>
<p>Return successive overlapping pairs taken from the input <em>iterable</em>.</p> <p>The number of 2-tuples in the output iterator will be one fewer than the number of inputs. It will be empty if the input iterable has fewer than two values.</p> <p>Roughly equivalent to:</p> <pre data-language="python">def pairwise(iterable):
    # pairwise('ABCDEFG') --&gt; AB BC CD DE EF FG
    a, b = tee(iterable)
    next(b, None)
    return zip(a, b)
</pre> <div class="versionadded"> <p><span class="versionmodified added">New in version 3.10.</span></p> </div> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.permutations">
<code>itertools.permutations(iterable, r=None)</code> </dt> <dd>
<p>Return successive <em>r</em> length permutations of elements in the <em>iterable</em>.</p> <p>If <em>r</em> is not specified or is <code>None</code>, then <em>r</em> defaults to the length of the <em>iterable</em> and all possible full-length permutations are generated.</p> <p>The permutation tuples are emitted in lexicographic order according to the order of the input <em>iterable</em>. So, if the input <em>iterable</em> is sorted, the output tuples will be produced in sorted order.</p> <p>Elements are treated as unique based on their position, not on their value. So if the input elements are unique, there will be no repeated values within a permutation.</p> <p>Roughly equivalent to:</p> <pre data-language="python">def permutations(iterable, r=None):
    # permutations('ABCD', 2) --&gt; AB AC AD BA BC BD CA CB CD DA DB DC
    # permutations(range(3)) --&gt; 012 021 102 120 201 210
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    if r &gt; n:
        return
    indices = list(range(n))
    cycles = list(range(n, n-r, -1))
    yield tuple(pool[i] for i in indices[:r])
    while n:
        for i in reversed(range(r)):
            cycles[i] -= 1
            if cycles[i] == 0:
                indices[i:] = indices[i+1:] + indices[i:i+1]
                cycles[i] = n - i
            else:
                j = cycles[i]
                indices[i], indices[-j] = indices[-j], indices[i]
                yield tuple(pool[i] for i in indices[:r])
                break
        else:
            return
</pre> <p>The code for <a class="reference internal" href="#itertools.permutations" title="itertools.permutations"><code>permutations()</code></a> can be also expressed as a subsequence of <a class="reference internal" href="#itertools.product" title="itertools.product"><code>product()</code></a>, filtered to exclude entries with repeated elements (those from the same position in the input pool):</p> <pre data-language="python">def permutations(iterable, r=None):
    pool = tuple(iterable)
    n = len(pool)
    r = n if r is None else r
    for indices in product(range(n), repeat=r):
        if len(set(indices)) == r:
            yield tuple(pool[i] for i in indices)
</pre> <p>The number of items returned is <code>n! / (n-r)!</code> when <code>0 &lt;= r &lt;= n</code> or zero when <code>r &gt; n</code>.</p> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.product">
<code>itertools.product(*iterables, repeat=1)</code> </dt> <dd>
<p>Cartesian product of input iterables.</p> <p>Roughly equivalent to nested for-loops in a generator expression. For example, <code>product(A, B)</code> returns the same as <code>((x,y) for x in A for y in B)</code>.</p> <p>The nested loops cycle like an odometer with the rightmost element advancing on every iteration. This pattern creates a lexicographic ordering so that if the input’s iterables are sorted, the product tuples are emitted in sorted order.</p> <p>To compute the product of an iterable with itself, specify the number of repetitions with the optional <em>repeat</em> keyword argument. For example, <code>product(A, repeat=4)</code> means the same as <code>product(A, A, A, A)</code>.</p> <p>This function is roughly equivalent to the following code, except that the actual implementation does not build up intermediate results in memory:</p> <pre data-language="python">def product(*args, repeat=1):
    # product('ABCD', 'xy') --&gt; Ax Ay Bx By Cx Cy Dx Dy
    # product(range(2), repeat=3) --&gt; 000 001 010 011 100 101 110 111
    pools = [tuple(pool) for pool in args] * repeat
    result = [[]]
    for pool in pools:
        result = [x+[y] for x in result for y in pool]
    for prod in result:
        yield tuple(prod)
</pre> <p>Before <a class="reference internal" href="#itertools.product" title="itertools.product"><code>product()</code></a> runs, it completely consumes the input iterables, keeping pools of values in memory to generate the products. Accordingly, it is only useful with finite inputs.</p> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.repeat">
<code>itertools.repeat(object[, times])</code> </dt> <dd>
<p>Make an iterator that returns <em>object</em> over and over again. Runs indefinitely unless the <em>times</em> argument is specified.</p> <p>Roughly equivalent to:</p> <pre data-language="python">def repeat(object, times=None):
    # repeat(10, 3) --&gt; 10 10 10
    if times is None:
        while True:
            yield object
    else:
        for i in range(times):
            yield object
</pre> <p>A common use for <em>repeat</em> is to supply a stream of constant values to <em>map</em> or <em>zip</em>:</p> <pre data-language="pycon3">&gt;&gt;&gt; list(map(pow, range(10), repeat(2)))
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.starmap">
<code>itertools.starmap(function, iterable)</code> </dt> <dd>
<p>Make an iterator that computes the function using arguments obtained from the iterable. Used instead of <a class="reference internal" href="functions#map" title="map"><code>map()</code></a> when argument parameters are already grouped in tuples from a single iterable (when the data has been “pre-zipped”).</p> <p>The difference between <a class="reference internal" href="functions#map" title="map"><code>map()</code></a> and <a class="reference internal" href="#itertools.starmap" title="itertools.starmap"><code>starmap()</code></a> parallels the distinction between <code>function(a,b)</code> and <code>function(*c)</code>. Roughly equivalent to:</p> <pre data-language="python">def starmap(function, iterable):
    # starmap(pow, [(2,5), (3,2), (10,3)]) --&gt; 32 9 1000
    for args in iterable:
        yield function(*args)
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.takewhile">
<code>itertools.takewhile(predicate, iterable)</code> </dt> <dd>
<p>Make an iterator that returns elements from the iterable as long as the predicate is true. Roughly equivalent to:</p> <pre data-language="python">def takewhile(predicate, iterable):
    # takewhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 1 4
    for x in iterable:
        if predicate(x):
            yield x
        else:
            break
</pre> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.tee">
<code>itertools.tee(iterable, n=2)</code> </dt> <dd>
<p>Return <em>n</em> independent iterators from a single iterable.</p> <p>The following Python code helps explain what <em>tee</em> does (although the actual implementation is more complex and uses only a single underlying <abbr title="first-in, first-out">FIFO</abbr> queue):</p> <pre data-language="python">def tee(iterable, n=2):
    it = iter(iterable)
    deques = [collections.deque() for i in range(n)]
    def gen(mydeque):
        while True:
            if not mydeque:             # when the local deque is empty
                try:
                    newval = next(it)   # fetch a new value and
                except StopIteration:
                    return
                for d in deques:        # load it to all the deques
                    d.append(newval)
            yield mydeque.popleft()
    return tuple(gen(d) for d in deques)
</pre> <p>Once a <a class="reference internal" href="#itertools.tee" title="itertools.tee"><code>tee()</code></a> has been created, the original <em>iterable</em> should not be used anywhere else; otherwise, the <em>iterable</em> could get advanced without the tee objects being informed.</p> <p><code>tee</code> iterators are not threadsafe. A <a class="reference internal" href="exceptions#RuntimeError" title="RuntimeError"><code>RuntimeError</code></a> may be raised when simultaneously using iterators returned by the same <a class="reference internal" href="#itertools.tee" title="itertools.tee"><code>tee()</code></a> call, even if the original <em>iterable</em> is threadsafe.</p> <p>This itertool may require significant auxiliary storage (depending on how much temporary data needs to be stored). In general, if one iterator uses most or all of the data before another iterator starts, it is faster to use <a class="reference internal" href="stdtypes#list" title="list"><code>list()</code></a> instead of <a class="reference internal" href="#itertools.tee" title="itertools.tee"><code>tee()</code></a>.</p> </dd>
</dl> <dl class="py function"> <dt class="sig sig-object py" id="itertools.zip_longest">
<code>itertools.zip_longest(*iterables, fillvalue=None)</code> </dt> <dd>
<p>Make an iterator that aggregates elements from each of the iterables. If the iterables are of uneven length, missing values are filled-in with <em>fillvalue</em>. Iteration continues until the longest iterable is exhausted. Roughly equivalent to:</p> <pre data-language="python">def zip_longest(*args, fillvalue=None):
    # zip_longest('ABCD', 'xy', fillvalue='-') --&gt; Ax By C- D-
    iterators = [iter(it) for it in args]
    num_active = len(iterators)
    if not num_active:
        return
    while True:
        values = []
        for i, it in enumerate(iterators):
            try:
                value = next(it)
            except StopIteration:
                num_active -= 1
                if not num_active:
                    return
                iterators[i] = repeat(fillvalue)
                value = fillvalue
            values.append(value)
        yield tuple(values)
</pre> <p>If one of the iterables is potentially infinite, then the <a class="reference internal" href="#itertools.zip_longest" title="itertools.zip_longest"><code>zip_longest()</code></a> function should be wrapped with something that limits the number of calls (for example <a class="reference internal" href="#itertools.islice" title="itertools.islice"><code>islice()</code></a> or <a class="reference internal" href="#itertools.takewhile" title="itertools.takewhile"><code>takewhile()</code></a>). If not specified, <em>fillvalue</em> defaults to <code>None</code>.</p> </dd>
</dl> </section> <section id="itertools-recipes"> <span id="id1"></span><h2>Itertools Recipes</h2> <p>This section shows recipes for creating an extended toolset using the existing itertools as building blocks.</p> <p>The primary purpose of the itertools recipes is educational. The recipes show various ways of thinking about individual tools — for example, that <code>chain.from_iterable</code> is related to the concept of flattening. The recipes also give ideas about ways that the tools can be combined — for example, how <code>compress()</code> and <code>range()</code> can work together. The recipes also show patterns for using itertools with the <a class="reference internal" href="operator#module-operator" title="operator: Functions corresponding to the standard operators."><code>operator</code></a> and <a class="reference internal" href="collections#module-collections" title="collections: Container datatypes"><code>collections</code></a> modules as well as with the built-in itertools such as <code>map()</code>, <code>filter()</code>, <code>reversed()</code>, and <code>enumerate()</code>.</p> <p>A secondary purpose of the recipes is to serve as an incubator. The <code>accumulate()</code>, <code>compress()</code>, and <code>pairwise()</code> itertools started out as recipes. Currently, the <code>sliding_window()</code> and <code>iter_index()</code> recipes are being tested to see whether they prove their worth.</p> <p>Substantially all of these recipes and many, many others can be installed from the <a class="reference external" href="https://pypi.org/project/more-itertools/">more-itertools project</a> found on the Python Package Index:</p> <pre data-language="python">python -m pip install more-itertools
</pre> <p>Many of the recipes offer the same high performance as the underlying toolset. Superior memory performance is kept by processing elements one at a time rather than bringing the whole iterable into memory all at once. Code volume is kept small by linking the tools together in a functional style which helps eliminate temporary variables. High speed is retained by preferring “vectorized” building blocks over the use of for-loops and <a class="reference internal" href="../glossary#term-generator"><span class="xref std std-term">generator</span></a>s which incur interpreter overhead.</p> <pre data-language="python">import collections
import functools
import math
import operator
import random

def take(n, iterable):
    "Return first n items of the iterable as a list."
    return list(islice(iterable, n))

def prepend(value, iterable):
    "Prepend a single value in front of an iterable."
    # prepend(1, [2, 3, 4]) --&gt; 1 2 3 4
    return chain([value], iterable)

def tabulate(function, start=0):
    "Return function(0), function(1), ..."
    return map(function, count(start))

def repeatfunc(func, times=None, *args):
    """Repeat calls to func with specified arguments.

    Example:  repeatfunc(random.random)
    """
    if times is None:
        return starmap(func, repeat(args))
    return starmap(func, repeat(args, times))

def flatten(list_of_lists):
    "Flatten one level of nesting."
    return chain.from_iterable(list_of_lists)

def ncycles(iterable, n):
    "Returns the sequence elements n times."
    return chain.from_iterable(repeat(tuple(iterable), n))

def tail(n, iterable):
    "Return an iterator over the last n items."
    # tail(3, 'ABCDEFG') --&gt; E F G
    return iter(collections.deque(iterable, maxlen=n))

def consume(iterator, n=None):
    "Advance the iterator n-steps ahead. If n is None, consume entirely."
    # Use functions that consume iterators at C speed.
    if n is None:
        # feed the entire iterator into a zero-length deque
        collections.deque(iterator, maxlen=0)
    else:
        # advance to the empty slice starting at position n
        next(islice(iterator, n, n), None)

def nth(iterable, n, default=None):
    "Returns the nth item or a default value."
    return next(islice(iterable, n, None), default)

def quantify(iterable, pred=bool):
    "Given a predicate that returns True or False, count the True results."
    return sum(map(pred, iterable))

def all_equal(iterable):
    "Returns True if all the elements are equal to each other."
    g = groupby(iterable)
    return next(g, True) and not next(g, False)

def first_true(iterable, default=False, pred=None):
    """Returns the first true value in the iterable.

    If no true value is found, returns *default*

    If *pred* is not None, returns the first item
    for which pred(item) is true.

    """
    # first_true([a,b,c], x) --&gt; a or b or c or x
    # first_true([a,b], x, f) --&gt; a if f(a) else b if f(b) else x
    return next(filter(pred, iterable), default)

def unique_everseen(iterable, key=None):
    "List unique elements, preserving order. Remember all elements ever seen."
    # unique_everseen('AAAABBBCCDAABBB') --&gt; A B C D
    # unique_everseen('ABBcCAD', str.casefold) --&gt; A B c D
    seen = set()
    if key is None:
        for element in filterfalse(seen.__contains__, iterable):
            seen.add(element)
            yield element
    else:
        for element in iterable:
            k = key(element)
            if k not in seen:
                seen.add(k)
                yield element

def unique_justseen(iterable, key=None):
    "List unique elements, preserving order. Remember only the element just seen."
    # unique_justseen('AAAABBBCCDAABBB') --&gt; A B C D A B
    # unique_justseen('ABBcCAD', str.casefold) --&gt; A B c A D
    if key is None:
        return map(operator.itemgetter(0), groupby(iterable))
    return map(next, map(operator.itemgetter(1), groupby(iterable, key)))

def iter_index(iterable, value, start=0, stop=None):
    "Return indices where a value occurs in a sequence or iterable."
    # iter_index('AABCADEAF', 'A') --&gt; 0 1 4 7
    seq_index = getattr(iterable, 'index', None)
    if seq_index is None:
        # Slow path for general iterables
        it = islice(iterable, start, stop)
        for i, element in enumerate(it, start):
            if element is value or element == value:
                yield i
    else:
        # Fast path for sequences
        stop = len(iterable) if stop is None else stop
        i = start - 1
        try:
            while True:
                yield (i := seq_index(value, i+1, stop))
        except ValueError:
            pass

def sliding_window(iterable, n):
    "Collect data into overlapping fixed-length chunks or blocks."
    # sliding_window('ABCDEFG', 4) --&gt; ABCD BCDE CDEF DEFG
    it = iter(iterable)
    window = collections.deque(islice(it, n-1), maxlen=n)
    for x in it:
        window.append(x)
        yield tuple(window)

def grouper(iterable, n, *, incomplete='fill', fillvalue=None):
    "Collect data into non-overlapping fixed-length chunks or blocks."
    # grouper('ABCDEFG', 3, fillvalue='x') --&gt; ABC DEF Gxx
    # grouper('ABCDEFG', 3, incomplete='strict') --&gt; ABC DEF ValueError
    # grouper('ABCDEFG', 3, incomplete='ignore') --&gt; ABC DEF
    args = [iter(iterable)] * n
    match incomplete:
        case 'fill':
            return zip_longest(*args, fillvalue=fillvalue)
        case 'strict':
            return zip(*args, strict=True)
        case 'ignore':
            return zip(*args)
        case _:
            raise ValueError('Expected fill, strict, or ignore')

def roundrobin(*iterables):
    "Visit input iterables in a cycle until each is exhausted."
    # roundrobin('ABC', 'D', 'EF') --&gt; A D E B F C
    # Recipe credited to George Sakkis
    num_active = len(iterables)
    nexts = cycle(iter(it).__next__ for it in iterables)
    while num_active:
        try:
            for next in nexts:
                yield next()
        except StopIteration:
            # Remove the iterator we just exhausted from the cycle.
            num_active -= 1
            nexts = cycle(islice(nexts, num_active))

def partition(pred, iterable):
    """Partition entries into false entries and true entries.

    If *pred* is slow, consider wrapping it with functools.lru_cache().
    """
    # partition(is_odd, range(10)) --&gt; 0 2 4 6 8   and  1 3 5 7 9
    t1, t2 = tee(iterable)
    return filterfalse(pred, t1), filter(pred, t2)

def subslices(seq):
    "Return all contiguous non-empty subslices of a sequence."
    # subslices('ABCD') --&gt; A AB ABC ABCD B BC BCD C CD D
    slices = starmap(slice, combinations(range(len(seq) + 1), 2))
    return map(operator.getitem, repeat(seq), slices)

def iter_except(func, exception, first=None):
    """ Call a function repeatedly until an exception is raised.

    Converts a call-until-exception interface to an iterator interface.
    Like builtins.iter(func, sentinel) but uses an exception instead
    of a sentinel to end the loop.

    Priority queue iterator:
        iter_except(functools.partial(heappop, h), IndexError)

    Non-blocking dictionary iterator:
        iter_except(d.popitem, KeyError)

    Non-blocking deque iterator:
        iter_except(d.popleft, IndexError)

    Non-blocking iterator over a producer Queue:
        iter_except(q.get_nowait, Queue.Empty)

    Non-blocking set iterator:
        iter_except(s.pop, KeyError)

    """
    try:
        if first is not None:
            # For database APIs needing an initial call to db.first()
            yield first()
        while True:
            yield func()
    except exception:
        pass

def before_and_after(predicate, it):
    """ Variant of takewhile() that allows complete
        access to the remainder of the iterator.

        &gt;&gt;&gt; it = iter('ABCdEfGhI')
        &gt;&gt;&gt; all_upper, remainder = before_and_after(str.isupper, it)
        &gt;&gt;&gt; ''.join(all_upper)
        'ABC'
        &gt;&gt;&gt; ''.join(remainder)     # takewhile() would lose the 'd'
        'dEfGhI'

        Note that the true iterator must be fully consumed
        before the remainder iterator can generate valid results.
    """
    it = iter(it)
    transition = []

    def true_iterator():
        for elem in it:
            if predicate(elem):
                yield elem
            else:
                transition.append(elem)
                return

    return true_iterator(), chain(transition, it)
</pre> <p>The following recipes have a more mathematical flavor:</p> <pre data-language="python">def powerset(iterable):
    "powerset([1,2,3]) --&gt; () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)"
    s = list(iterable)
    return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

def sum_of_squares(it):
    "Add up the squares of the input values."
    # sum_of_squares([10, 20, 30]) -&gt; 1400
    return math.sumprod(*tee(it))

def reshape(matrix, cols):
    "Reshape a 2-D matrix to have a given number of columns."
    # reshape([(0, 1), (2, 3), (4, 5)], 3) --&gt;  (0, 1, 2), (3, 4, 5)
    return batched(chain.from_iterable(matrix), cols)

def transpose(matrix):
    "Swap the rows and columns of a 2-D matrix."
    # transpose([(1, 2, 3), (11, 22, 33)]) --&gt; (1, 11) (2, 22) (3, 33)
    return zip(*matrix, strict=True)

def matmul(m1, m2):
    "Multiply two matrices."
    # matmul([(7, 5), (3, 5)], [(2, 5), (7, 9)]) --&gt; (49, 80), (41, 60)
    n = len(m2[0])
    return batched(starmap(math.sumprod, product(m1, transpose(m2))), n)

def convolve(signal, kernel):
    """Discrete linear convolution of two iterables.

    The kernel is fully consumed before the calculations begin.
    The signal is consumed lazily and can be infinite.

    Convolutions are mathematically commutative.
    If the signal and kernel are swapped,
    the output will be the same.

    Article:  https://betterexplained.com/articles/intuitive-convolution/
    Video:    https://www.youtube.com/watch?v=KuXjwB4LzSA
    """
    # convolve(data, [0.25, 0.25, 0.25, 0.25]) --&gt; Moving average (blur)
    # convolve(data, [1/2, 0, -1/2]) --&gt; 1st derivative estimate
    # convolve(data, [1, -2, 1]) --&gt; 2nd derivative estimate
    kernel = tuple(kernel)[::-1]
    n = len(kernel)
    padded_signal = chain(repeat(0, n-1), signal, repeat(0, n-1))
    windowed_signal = sliding_window(padded_signal, n)
    return map(math.sumprod, repeat(kernel), windowed_signal)

def polynomial_from_roots(roots):
    """Compute a polynomial's coefficients from its roots.

       (x - 5) (x + 4) (x - 3)  expands to:   x³ -4x² -17x + 60
    """
    # polynomial_from_roots([5, -4, 3]) --&gt; [1, -4, -17, 60]
    factors = zip(repeat(1), map(operator.neg, roots))
    return list(functools.reduce(convolve, factors, [1]))

def polynomial_eval(coefficients, x):
    """Evaluate a polynomial at a specific value.

    Computes with better numeric stability than Horner's method.
    """
    # Evaluate x³ -4x² -17x + 60 at x = 2.5
    # polynomial_eval([1, -4, -17, 60], x=2.5) --&gt; 8.125
    n = len(coefficients)
    if not n:
        return type(x)(0)
    powers = map(pow, repeat(x), reversed(range(n)))
    return math.sumprod(coefficients, powers)

def polynomial_derivative(coefficients):
    """Compute the first derivative of a polynomial.

       f(x)  =  x³ -4x² -17x + 60
       f'(x) = 3x² -8x  -17
    """
    # polynomial_derivative([1, -4, -17, 60]) -&gt; [3, -8, -17]
    n = len(coefficients)
    powers = reversed(range(1, n))
    return list(map(operator.mul, coefficients, powers))

def sieve(n):
    "Primes less than n."
    # sieve(30) --&gt; 2 3 5 7 11 13 17 19 23 29
    if n &gt; 2:
        yield 2
    start = 3
    data = bytearray((0, 1)) * (n // 2)
    limit = math.isqrt(n) + 1
    for p in iter_index(data, 1, start, limit):
        yield from iter_index(data, 1, start, p*p)
        data[p*p : n : p+p] = bytes(len(range(p*p, n, p+p)))
        start = p*p
    yield from iter_index(data, 1, start)

def factor(n):
    "Prime factors of n."
    # factor(99) --&gt; 3 3 11
    # factor(1_000_000_000_000_007) --&gt; 47 59 360620266859
    # factor(1_000_000_000_000_403) --&gt; 1000000000000403
    for prime in sieve(math.isqrt(n) + 1):
        while not n % prime:
            yield prime
            n //= prime
            if n == 1:
                return
    if n &gt; 1:
        yield n

def totient(n):
    "Count of natural numbers up to n that are coprime to n."
    # https://mathworld.wolfram.com/TotientFunction.html
    # totient(12) --&gt; 4 because len([1, 5, 7, 11]) == 4
    for p in unique_justseen(factor(n)):
        n = n // p * (p - 1)
    return n
</pre> </section> <div class="_attribution">
  <p class="_attribution-p">
    &copy; 2001&ndash;2023 Python Software Foundation<br>Licensed under the PSF License.<br>
    <a href="https://docs.python.org/3.12/library/itertools.html" class="_attribution-link">https://docs.python.org/3.12/library/itertools.html</a>
  </p>
</div>