Linear Polynomial, Quadratic Polynomial

<body onLoad="MM_checkPlugin('Shockwave Flash','','http://www.riskglossary.com/default_nofl.htm',false);return document.MM_returnValue" stylesrc="../_private/template.htm" bgcolor="#FFFFFF" link="#669933" vlink="#000000" alink="#000000"> <div align="left"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table1"> <tr> <td width="740"> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" height="37" id="table2"> <tr> <td height="35" width="100%"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" id="table3"> <tr> <td width="100%" align="left" valign="top"> <p class="header_gold">Linear Polynomial, Quadratic Polynomial</td> <td width="45" align="right" valign="bottom"> <map name="FPMap6"> <area href="http://www.contingencyanalysis.com" shape="rect" coords="1, 0, 44, 11" target="_top"> </map> <img border="0" src="../images/home.gif" usemap="#FPMap6" width="45" height="12"></td> </tr> </table> </td> </tr> <tr> <td width="100%" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td width="0%" valign="top" bgcolor="#FFFFCC"> <table border="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="0" cellspacing="1" id="table4"> <tr> <td width="0%" align="left" valign="top"> <p class="header2_black">Explained:<br> <img border="0" src="../images/blank_1x100.gif" width="100" height="1"></td> <td width="0%" align="left" valign="top"><p class="word_list"> <a href="#linear">linear polynomial</a><p class="word_list"> <a href="#quadratic">quadratic polynomial</a><br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> <td width="100%" align="left" valign="top"><p class="word_list"> <br> <img border="0" src="../images/blank_1x160.gif" width="160" height="1"></td> </tr> </table> </td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="165" align="left" id="table5"> <tr> <td width="160">  </td> <td width="11"> </td> </tr> <tr> <td width="160"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script> </td> <td width="11"> </td> </tr> </table> <p>A polynomial from <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16"> to <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16"> is <b> <a name="linear">linear</a></b> if it has form </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table6"> <tr> <td width="100%" align="center"> <i>p</i>(<i>x</i>) = <i>bx</i> + <i>a</i></td> <td width="5%" align="right">[1]</td> </tr> </table> <p>with <i>a</i> and <i>b</i> scalars. It is <b><a name="quadratic"> quadratic</a></b> if it has form: </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table7"> <tr> <td width="100%" align="center"> <i>p</i>(<i>x</i>) = <i>cx</i><sup>2</sup> + <i>bx</i> + <i>a</i></td> <td width="5%" align="right">[2]</td> </tr> </table> <p>with <i>a</i>, <i>b</i> and <i>c</i> scalars. These notions generalize to higher dimensions. A polynomial from <img border="0" src="../symbols/real_numers_n_dimensional.gif" align="absbottom" width="18" height="16"> to <img border="0" src="../symbols/real_numbers.gif" align="absbottom" width="11" height="16"> is linear if it has form </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table8"> <tr> <td width="100%" align="center"> <i>p</i>(<b><i>x</i></b>) = <i><b>bx</b></i> + <i>a</i></td> <td width="5%" align="right">[3]</td> </tr> </table> <p>with <i><b>b</b></i> an <i>n</i>-dimensional row vector and <i>a</i> a scalar. It is quadratic if it has <a name="[4]">form </a></p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table9"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/polynomial_[4].gif" width="136" height="22"></td> <td width="5%" align="right">[4]</td> </tr> </table> <p>with <i><b>c</b></i> an <i>n</i><img border="0" src="../symbols/cross.gif" align="absbottom" width="11" height="13"><i>n</i> matrix, <i><b>b</b></i> an <i>n</i>-dimensional row vector and <i>a</i> a scalar. Without loss of generality, we may assume <i><b>c</b></i> is symmetric. For example, the quadratic polynomial </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table10"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/polynomial_[5].gif" width="279" height="28"></td> <td width="5%" align="right">[5]</td> </tr> </table> <p>can be expressed in form [<a href="#[4]">4</a>] with </p> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" cellpadding="0" height="36" id="table11"> <tr> <td width="100%" align="center"> <img border="0" src="../formulas/polynomial_[6].gif" width="124" height="76"></td> <td width="5%" align="right">[6]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-top: 24; margin-bottom:24"><i><b>b</b></i> = ( 0  1  0 )</td> <td width="5%" align="right">[7]</td> </tr> <tr> <td width="100%" align="center"> <p style="margin-bottom: 24"><i>a</i> = 17</td> <td width="5%" align="right">[8]</td> </tr> </table> <table border="0" cellpadding="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" id="table12"> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Related Internal Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table13"> <tr> <td width="100%"> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/imaginary_numbers.htm">complex number</a> </strong>A number of the form <i>a</i> + <i>bi</i>, where <i>a</i> and <i>b</i> are real, and <i>i</i> is the imaginary square root of –1.</p> <p align="left" class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/interpolation.htm">interpolation</a> Any procedure for fitting a function to a set of points in such a manner that the function intercepts each of the points.</p> <p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><strong style="font-weight: 400"><a href="../articles/least_squares.htm">method of least squares</a> Any of several techniques for fitting a curve to data so as to minimize the sum of squared differences between the curve and data points.</strong><p class="explanatory_text"> <img border="0" src="../images/bullet_link_glossary.gif" width="26" height="15"><a href="../articles/taylor_series.htm">Taylor series expansion</a> In calculus, a power series obtained as a limit of Taylor polynomials that may approximate or equal the function from which it is constructed.</td> </tr> </table> </td> </tr> <tr> <td height="30" valign="bottom" colspan="2"> <p class="header2_black">Sponsored Links</td> </tr> <tr> <td colspan="2" bgcolor="#CC9966" height="2"> <img border="0" src="../images/blank_2x2.gif" width="2" height="2"></td> </tr> <tr> <td bgcolor="#FFFFCC" width="75"> <img border="0" src="../images/ad_side_banner_1.jpg" width="103" height="285"></td> <td bgcolor="#FFFFCC" width="99%"> <script type="text/javascript"></script> <script type="text/javascript" src="http://pagead2.googlesyndication.com/pagead/show_ads.js"> </script>  </td> </tr> <tr> <td bgcolor="#FFFFCC" colspan="2"> <table border="0" cellspacing="0" style="border-collapse: collapse" bordercolor="#111111" width="100%" cellpadding="6" id="table14"> <tr> <td width="100%" bgcolor="#FFFFFF"> <p align="center" style="margin-top: 18"><map name="FPMap365"> <area target="_top" href="http://www.riskglossary.com" shape="rect" coords="0, 0, 164, 102"> </map><img border="0" src="../images/logo.jpg" usemap="#FPMap365" width="165" height="103"></p> <p align="center"><span class="verdana10"> <a href="http://www.riskglossary.com/main/disclaimer.htm" target="_self"> Disclaimer</a></span></p> <p align="center">website: <span class="verdana10"> <a target="_top" href="http://www.contingencyanalysis.com"> http://www.contingencyanalysis.com</a></span><br> glossary direct link: <span class="verdana10"> <a target="_top" href="http://www.riskglossary.com"> http://www.riskglossary.com</a></span><br> copyright © Contingency Analysis, 2006<p align="center"><map name="FPMap1"> <area target="_top" href="http://www.contingencyanalysis.com" shape="rect" coords="0, 0, 240, 88"> </map><img border="0" src="../images/logo_glossary_white.gif" usemap="#FPMap1" width="241" height="89"></td> </tr> </table> </td> </tr> </table> </td> <td>  </td> </tr> </table> </div> </body>