Region of the coefficients of a quadratic equation that cause the roots of it to be in the unit diskFrom Simon Haykin's Adaptive Filter Theory: consider the characteristic equation is 1+a1z−1+a2z−2=0, then for the roots to be inside the unit circle (or in the unit disk), the coefficients of the quadratic must satisfy the following conditions: −1≤a1+a2,−1≤a2−a1, and −1≤a2≤1.I could figure this out some of the way. For complex roots it follows that the coefficients lie in the region defined by the intersection of 4a2&gt;a12 and a2&lt;1. For real roots, I could prove that |a2|&lt;1, and obviously 4a2≤a12. But I cannot figure out how the line boundaries of the region can be derived.

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Region of the coefficients of a quadratic equation that cause the roots of it to be in the unit diskFrom Simon Haykin&#039;s Adaptive Filter Theory: consider the characteristic equation is 1+a1z−1+a2z−2=0, then for the roots to be inside the unit circle (or in the unit disk), the coefficients of the quadratic must satisfy the following conditions: −1≤a1+a2,−1≤a2−a1, and −1≤a2≤1.I could figure this out some of the way. For complex roots it follows that the coefficients lie in the region defined by the intersection of 4a2&amp;gt;a12 and a2&amp;lt;1. For real roots, I could prove that |a2|&amp;lt;1, and obviously 4a2≤a12. But I cannot figure out how the line boundaries of the region can be derived.

Jared Kemp · Accepted Answer

From the quadratic equationz=d−a12±(da12)2−a2For the greater real root and the upper boundary on the real line1≥−a12+(a12)2−a2&amp;nbsp;&amp;nbsp;1+a12≥+(a12)2−a2&amp;nbsp;&amp;nbsp;1+a1+(a12)2≥(a12)2−a2&amp;nbsp;&amp;nbsp;a1+a2≥−1For the lesser real root and the lower boundary on the real line−1≤−a12−(a12)2−a2&amp;nbsp;&amp;nbsp;−1+a12≤−(a12)2−a2&amp;nbsp;&amp;nbsp;−1+a1+(a12)2≤(a12)2−a2&amp;nbsp;&amp;nbsp;−1≤a2−a1For the complex roots being inside the unit circle1≥|−a12±ia2−(a12)2|&amp;nbsp;&amp;nbsp;1≥(a12)2+a2−(a12)2&amp;nbsp;&amp;nbsp;1≥a2&amp;nbsp;&amp;nbsp;12≥(a2)2&amp;nbsp;&amp;nbsp;1≥|a2|&amp;nbsp;&amp;nbsp;−1≤a2≤1

Region of the coefficients of a quadratic equation

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