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<html><head><title>A Tour of NTL: Examples: Vectors and Matrices </title></head><body bgcolor="#fff9e6"><center><a href="tour-ex1.html"><img src="arrow1.gif" alt="[Previous]" align=bottom></a> <a href="tour-examples.html"><img src="arrow2.gif" alt="[Up]" align=bottom></a> <a href="tour-ex3.html"> <img src="arrow3.gif" alt="[Next]" align=bottom></a></center><h1> <p align=center>A Tour of NTL: Examples: Vectors and Matrices</p></h1><p> <hr> <p><p>The following routine sums up the numbers in a vector of <tt>ZZ</tt>'s.<pre>#include <NTL/vec_ZZ.h>ZZ sum(const vec_ZZ& v){ ZZ acc; acc = 0; for (long i = 0; i < v.length(); i++) acc += v[i]; return acc;}</pre><p>The class <tt>vec_ZZ</tt> is a dynamic-length array of <tt>ZZ</tt>s;more generally, NTL provides template-like macros to create dynamic-length vectors over any type T.By convention, NTL names these vec_T.The reason that macros are used instead of true templates is simple:at the present time, compiler support for templates is not entirelysatisfactory, and their use would make NTL much more difficult to port.At some point in the future, a template-version of NTL may be madeavailable.<p>Vectors in NTL are indexed from 0, but in many situationsit is convenient or more natural to index from 1.The generic vector class allows for this;the above example could be written as follows.<pre>#include <NTL/vec_ZZ.h>ZZ sum(ZZ& s, const vec_ZZ& v){ ZZ acc; acc = 0; for (long i = 1; i <= v.length(); i++) acc += v(i); return acc;}</pre><p>Note that by default, NTL does not perform range checks on vector indices.However, there is a compile-time flag that activates range checking.Therefore, it is good practice to always assume that range checking may be activated, and to not access elements that are out of range.<p> <hr> <p>The following example illustrates vector I/O,as well as changing the length of a vector.This program reads a <tt>vec_ZZ</tt>,and then creates and prints a "palindrome". <pre>#include <NTL/vec_ZZ.h>int main(){ vec_ZZ v; cin >> v; long n = v.length(); v.SetLength(2*n); long i; for (i = 0 ; i < n; i++) v[n+i] = v[n-1-i]; cout << v << "\n";}</pre><p>Notice that changing the length of a vector does not changeits contents.<p>When we compile and run this program,if we type in<pre> [1 -2 3]</pre>as input, the output is<pre> [1 -2 3 3 -2 1]</pre><p>NTL pre-defines a number of vector types.In addition, you can create your own.See <a href="vector.txt"><tt>vector.txt</tt></a> forcomplete details of NTL's generic vector mechanism.Also see <a href="vec_ZZ.txt"><tt>vec_ZZ.txt</tt></a> forcomplete details on the arithmetic operations for <tt>vec_ZZ</tt>sprovideed by NTL.<p> <hr> <p>There is also basic support for matricesin NTL.In general, the class <tt>mat_T</tt> is a specialkind of <tt>vec_vec_T</tt>, where each row is a vector of the same length.Row <tt>i</tt> of matrix <tt>M</tt>can be accessed as <tt>M[i]</tt> (indexing from 0)or as <tt>M(i)</tt> (indexing from 1).Column <tt>j</tt> of row <tt>i</tt> can be accessedas <tt>M[i][j]</tt> or <tt>M(i)(j)</tt>;for notational convenience, the latter is equivalent to <tt>M(i,j)</tt>.<p>Here is a matrix multiplication routine,which in fact is already provided by NTL.<pre>#include <NTL/mat_ZZ.h>void mul(mat_ZZ& X, const mat_ZZ& A, const mat_ZZ& B){ long n = A.NumRows(); long l = A.NumCols(); long m = B.NumCols(); if (l != B.NumRows()) Error("matrix mul: dimension mismatch"); X.SetDims(n, m); // make X have n rows and m columns long i, j, k; ZZ acc, tmp; for (i = 1; i <= n; i++) { for (j = 1; j <= m; j++) { acc = 0; for(k = 1; k <= l; k++) { mul(tmp, A(i,k), B(k,j)); add(acc, acc, tmp); } X(i,j) = acc; } }}</pre><p>In case of a dimension mismatch, the routine calls the <tt>Error</tt> function, which is a part of NTL and which simplyprints the message and aborts.That is generally how NTL deals with errors.Currently, NTL makes no use of exceptions (for the same reasonit does not use templates--see above), but a future versionmay incorporate them.<p>This routine will not work properly if <tt>X</tt> aliases <tt>A</tt> or <tt>B</tt>.The actual matrix multiplication routine in NTL takes care of this.In fact, all of NTL's routines allow outputs to alias inputs.<p>To call the multiplication routine, one can write <pre> mul(X, A, B);</pre>or one can also use the operator notation <pre> X = A * B;</pre><p>NTL provides several matrix types.See <a href="matrix.txt"><tt>matrix.txt</tt></a>for complete details on NTL's generic matrix mechanism.Also see <a href="mat_ZZ.txt"><tt>mat_ZZ.txt</tt></a> forcomplete details on the arithmetic operations for <tt>mat_ZZ</tt>sprovideed by NTL (including basic linear algebra).Also see <a href="LLL.txt"><tt>LLL.txt</tt></a> for details on routines for lattice basis reduction(as well as routines for finding the kernel and image of a matrix).<p>One thing you may have noticed by now is thatNTL code generally avoids the type<tt>int</tt>, preferring instead to use <tt>long</tt>.This seems to go against what most "style" books preach,but nevertheless seems to make the most sense in today's world.Although <tt>int</tt> was originally meant to represent the"natural" word size, this seems to no longer be the case.On 32-bit machines, <tt>int</tt> and <tt>long</tt> are the same,but on 64-bit machines, they are often different, with <tt>int</tt>'s having 32 bits and <tt>long</tt>'s having 64 bits. Moreover, on such 64-bit machines, the "natural" word size is usually 64-bits;indeed, it is often more expensive to manipulate 32-bit integers.Thus, for simplicity, efficiency, and safety, NTL uses <tt>long</tt>for all integer values.If you are used to writing <tt>int</tt> all the time,it takes a little while to get used to this.<p><center><a href="tour-ex1.html"><img src="arrow1.gif" alt="[Previous]" align=bottom></a> <a href="tour-examples.html"><img src="arrow2.gif" alt="[Up]" align=bottom></a> <a href="tour-ex3.html"> <img src="arrow3.gif" alt="[Next]" align=bottom></a></center></body></html>?? 快捷鍵說明
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