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1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
2<html>
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4<head>
5<meta http-equiv="Content-Language" content="en-us">
6<title>YALMIP Example : Polynomial expressions</title>
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18<table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
19  <tr>
20    <td width="100%" align="left" height="100%" valign="top">
21        <h2>Polynomial expressions</h2>
22    <hr noShade SIZE="1">
23    <p>Starting from YALMIP 3, polynomial expressions are supported. These
24    nonlinear expressions can be used for, e.g., SDPs with BMI constraints,
25    quadratic programming, or
26    to solve sum-of-squares problems.</p>
27    <p>Nonlinear expressions are built using
28    <a href="reference.htm#sdpvar">
29    sdpvar</a> objects, and are manipulated in same way</p>
30    <table cellPadding="10" width="100%">
31      <tr>
32        <td class="xmpcode">
33        <pre>x = sdpvar(1,1);
34y = sdpvar(1,1);
35p = 1+x*y+x^2+y^3;
36Y = sdpvar(3,3);
37Z = Y*Y+Y.*Y;</pre>
38        </td>
39      </tr>
40    </table>
41    <p>A convenient command is
42    <a href="reference.htm#sdisplay">
43    sdisplay</a> (symbolic display)</p>
44    <table cellPadding="10" width="100%">
45      <tr>
46        <td class="xmpcode">
47        <pre>sdisplay(p)
48<font color="#000000"> ans =
49&nbsp;&nbsp; '1+xy+x^2+y^3'</font></pre>
50        </td>
51      </tr>
52    </table>
53    <p>Some simple operators for polynomials have been implemented, such as
54    differentiation.</p>
55    <table cellPadding="10" width="100%">
56      <tr>
57        <td class="xmpcode">
58        <pre>dp = jacobian(p);
59d2p = jacobian(jacobian(p)');</pre>
60        </td>
61      </tr>
62    </table>
63    <p>Checking the degree is easily done</p>
64    <table cellPadding="10" width="100%">
65      <tr>
66        <td class="xmpcode">
67        <pre>degree(p)
68<font color="#000000"> ans =
69  3
70</font>
71degree(p,x)
72<font color="#000000"> ans =
73  2</font></pre>
74        </td>
75      </tr>
76    </table>
77    <p>Of course, all standard operators applies to the nonlinear objects.</p>
78    <table cellPadding="10" width="100%">
79      <tr>
80        <td class="xmpcode">
81        <pre>x = sdpvar(3,1);
82p = 5*trace(x*x') + jacobian(sum(x.^4))</pre>
83        </td>
84      </tr>
85    </table>
86        <p><img border="0" src="demoicon.gif" width="16" height="16"> Clear the internals of YALMIP on a regular basis with
87    the command <code>yalmip('clear')</code> when working with polynomial
88    expressions. The reason is that every time a nonlinear variable is defined,
89    a description on how it is created is saved inside YALMIP (all monomials
90        generate new variables). With many
91    nonlinear terms this list grows fast, making YALMIP slower and slower since
92    the list has to be searched in when polynomial expressions are manipulated.<br>
93    <br>
94        <img border="0" src="demoicon.gif" width="16" height="16"> The current implementation of the
95    polynomial objects is inefficient for large problems. Multiplying two
96        matrices of dimension, say 20, takes several seconds. But if you have a
97        problem with this type of non-linearity, the solver will probably be the
98        bottle-neck anyway...
99    </td>
100  </tr>
101</table>
102
103</div>
104
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106
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