1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
|
<HTML>
<!--
-- Copyright (c) 1996-1999
-- Silicon Graphics Computer Systems, Inc.
--
-- Permission to use, copy, modify, distribute and sell this software
-- and its documentation for any purpose is hereby granted without fee,
-- provided that the above copyright notice appears in all copies and
-- that both that copyright notice and this permission notice appear
-- in supporting documentation. Silicon Graphics makes no
-- representations about the suitability of this software for any
-- purpose. It is provided "as is" without express or implied warranty.
--
-- Copyright (c) 1994
-- Hewlett-Packard Company
--
-- Permission to use, copy, modify, distribute and sell this software
-- and its documentation for any purpose is hereby granted without fee,
-- provided that the above copyright notice appears in all copies and
-- that both that copyright notice and this permission notice appear
-- in supporting documentation. Hewlett-Packard Company makes no
-- representations about the suitability of this software for any
-- purpose. It is provided "as is" without express or implied warranty.
--
-->
<Head>
<Title>power</Title>
<!-- Generated by htmldoc -->
</HEAD>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
ALINK="#ff0000">
<IMG SRC="CorpID.gif"
ALT="SGI" HEIGHT="43" WIDTH="151">
<!--end header-->
<BR Clear>
<H1>power</H1>
<Table CellPadding=0 CellSpacing=0 width=100%>
<TR>
<TD Align=left><Img src = "algorithms.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD>
<TD Align=right><Img src = "function.gif" Alt="" WIDTH = "194" HEIGHT = "38" ></TD>
</TR>
<TR>
<TD Align=left VAlign=top><b>Category</b>: algorithms</TD>
<TD Align=right VAlign=top><b>Component type</b>: function</TD>
</TR>
</Table>
<h3>Prototype</h3>
<tt>Power</tt> is an overloaded name; there are actually two <tt>power</tt>
functions.
<pre>
template <class T, class Integer>
inline T power(T x, Integer n);
template <class T, class Integer, class <A href="MonoidOperation.html">MonoidOperation</A>>
T power(T x, Integer n, MonoidOperation op);
</pre>
<h3>Description</h3>
<tt>Power</tt> is generalized exponentiation: it raises the value <tt>x</tt>
to the power <tt>n</tt>, where <tt>n</tt> is a non-negative integer.
<P>
The first version of <tt>power</tt>
returns <tt>x</tt> raised to the <tt>n</tt>th power; that is, it returns
<tt>x * x ... * x</tt>, where <tt>x</tt> is repeated <tt>n</tt> times. <A href="#1">[1]</A> If
<tt>n == 0</tt>, then it returns <tt><A href="MonoidOperation.html">identity_element</A>(<A href="times.html">multiplies</A><T>())</tt>.
<P>
The second version of <tt>power</tt> is just like the first except
that it uses an arbitrary <A href="MonoidOperation.html">Monoid Operation</A> instead of
<tt><A href="times.html">multiplies</A><T></tt>, returning <tt><A href="MonoidOperation.html">identity_element</A>(op)</tt>
when <tt>n == 0</tt>.
<P>
<b>Important note</b>: <tt>power</tt> does not assume that multiplication is
commutative, but it does rely crucially on the fact that
multiplication is associative. If you have defined <tt>*</tt> or
<tt><A href="MonoidOperation.html">MonoidOperation</A></tt> to be a non-associative operation, then <tt>power</tt>
<i>will give you the wrong answer</i>. <A href="#2">[2]</A>
<h3>Definition</h3>
Defined in the standard header <A href="numeric">numeric</A>, and in the nonstandard
backward-compatibility header <A href="algo.h">algo.h</A>. This function is an SGI
extension; it is not part of the C++ standard.
<h3>Requirements on types</h3>
For the first version:
<UL>
<LI>
<tt><A href="times.html">multiplies</A><T></tt> is a model of <A href="MonoidOperation.html">Monoid Operation</A>.
<LI>
<tt>Integer</tt> is an integral type.
</UL>
For the second version:
<UL>
<LI>
<tt>MonoidOperation</tt> is a model of <A href="MonoidOperation.html">Monoid Operation</A>.
<LI>
<tt>T</tt> is <tt>MonoidOperation</tt>'s argument type.
<LI>
<tt>n</tt> is an integral type.
</UL>
<h3>Preconditions</h3>
<UL>
<LI>
<tt>n >= 0</tt>.
</UL>
<h3>Complexity</h3>
The number of multiplications (or, in the case of the second version,
the number of applications of <tt>MonoidOperation</tt>) is <tt>lg(n) + nu(n)</tt>
where <tt>lg</tt> is the base 2 logarithm and <tt>nu(n)</tt> is the number of
<tt>1</tt>s in the binary representation of <tt>n</tt>. <A href="#3">[3]</A>
<h3>Example</h3>
<pre>
int main() {
cout << "2 ** 30 = " << power(2, 30) << endl;
}
</pre>
<h3>Notes</h3>
<P><A name="1">[1]</A>
This is a conceptual description of what <tt>power</tt>'s return value
is; it is not how <tt>power</tt> is actually implemented. If <tt>power</tt>
were implemented that way then it would require <tt>n-1</tt> multiplications,
which would be grossly inefficient. <tt>Power</tt> is implemented using
the "Russian peasant algorithm", which requires only <tt>O(log n)</tt>
multiplications.
See section 4.6.3 of Knuth (D. E. Knuth, <i>The Art of Computer
Programming. Volume 2: Seminumerical Algorithms</i>,
Addison-Wesley, 1981) for a discussion.
<P><A name="2">[2]</A>
See the <A href="MonoidOperation.html">Monoid Operation</A> requirements for a discussion of associativity.
<P><A name="3">[3]</A>
This is in fact not the minimum possible number of multiplications:
it is possible to compute the fifteenth power of <tt>x</tt> using only
five multiplications, but <tt>power(x, 15)</tt> uses six.
<h3>See also</h3>
<A href="MonoidOperation.html">Monoid Operation</A>,
<tt><A href="times.html">multiplies</A></tt>,
<tt><A href="plus.html">plus</A></tt>
<!--start footer-->
<HR SIZE="6">
<A href="http://www.sgi.com/"><IMG SRC="surf.gif" HEIGHT="54" WIDTH="54"
ALT="[Silicon Surf]"></A>
<A HREF="index.html"><IMG SRC="stl_home.gif"
HEIGHT="54" WIDTH="54" ALT="[STL Home]"></A>
<BR>
<FONT SIZE="-2">
<A href="http://www.sgi.com/Misc/sgi_info.html" TARGET="_top">Copyright ©
1999 Silicon Graphics, Inc.</A> All Rights Reserved.</FONT>
<FONT SIZE="-3"><a href="http://www.sgi.com/Misc/external.list.html" TARGET="_top">TrademarkInformation</A>
</FONT>
<P>
</BODY>
</HTML>
|
