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<div class="title">LU LU Decomposition for Matrices </div> </div>
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<div class="textblock"><p>Section: <a class="el" href="sec_transforms.html">Transforms/Decompositions</a> </p>
<h1><a class="anchor" id="Usage"></a>
Usage</h1>
<p>Computes the LU decomposition for a matrix. The form of the command depends on the type of the argument. For full (non-sparse) matrices, the primary form for <code>lu</code> is </p>
<pre class="fragment"> [L,U,P] = lu(A),
</pre><p> where <code>L</code> is lower triangular, <code>U</code> is upper triangular, and <code>P</code> is a permutation matrix such that <code>L*U = P*A</code>. The second form is </p>
<pre class="fragment"> [V,U] = lu(A),
</pre><p> where <code>V</code> is <code>P'*L</code> (a row-permuted lower triangular matrix), and <code>U</code> is upper triangular. For sparse, square matrices, the LU decomposition has the following form: </p>
<pre class="fragment"> [L,U,P,Q,R] = lu(A),
</pre><p> where <code>A</code> is a sparse matrix of either <code>double</code> or <code>dcomplex</code> type. The matrices are such that <code>L*U=P*R*A*Q</code>, where <code>L</code> is a lower triangular matrix, <code>U</code> is upper triangular, <code>P</code> and <code>Q</code> are permutation vectors and <code>R</code> is a diagonal matrix of row scaling factors. The decomposition is computed using UMFPACK for sparse matrices, and LAPACK for dense matrices. </p>
<h1><a class="anchor" id="Example"></a>
Example</h1>
<p>First, we compute the LU decomposition of a dense matrix.</p>
<pre class="fragment">--> a = float([1,2,3;4,5,8;10,12,3])
a =
1 2 3
4 5 8
10 12 3
--> [l,u,p] = lu(a)
l =
1.0000 0 0
0.1000 1.0000 0
0.4000 0.2500 1.0000
u =
10.0000 12.0000 3.0000
0 0.8000 2.7000
0 0 6.1250
p =
0 0 1
1 0 0
0 1 0
--> l*u
ans =
10 12 3
1 2 3
4 5 8
--> p*a
ans =
10 12 3
1 2 3
4 5 8
</pre><p>Now we repeat the exercise with a sparse matrix, and demonstrate the use of the permutation vectors.</p>
<pre class="fragment">--> a = sparse([1,0,0,4;3,2,0,0;0,0,0,1;4,3,2,4])
a =
1 1 1
2 1 3
4 1 4
2 2 2
4 2 3
4 3 2
1 4 4
3 4 1
4 4 4
--> [l,u,p,q,r] = lu(a)
l =
1 1 1
2 2 1
3 3 1
4 4 1
u =
1 1 0.153846
1 2 0.230769
2 2 0.4
1 3 0.307692
2 3 0.6
3 3 0.2
1 4 0.307692
3 4 0.8
4 4 1
p =
4
2
1
3
q =
3
2
1
4
r =
1 1 0.2
2 2 0.2
3 3 1
4 4 0.0769231
--> full(l*a)
ans =
1 0 0 4
3 2 0 0
0 0 0 1
4 3 2 4
--> b = r*a
b =
1 1 0.2
2 1 0.6
3 1 0
4 1 0.307692
1 2 0
2 2 0.4
3 2 0
4 2 0.230769
1 3 0
2 3 0
3 3 0
4 3 0.153846
1 4 0.8
2 4 0
3 4 1
4 4 0.307692
--> full(b(p,q))
ans =
0.1538 0.2308 0.3077 0.3077
0 0.4000 0.6000 0
0 0 0.2000 0.8000
0 0 0 1.0000
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