Circuit Theory/Quadratic Equation Revisited

The Quadratic Equation The roots to the quadratic polynomial are easily derived and many people memorized them in high school: Derivation of the Quadratic Equation To derive this set and complete the square: Solving for gives Taking the square root of both sides and putting everything over a common denominator gives Numerical Instability of the Usual Formulation of the Quadratic Equation Middlebrook has pointed out that this is a poor expression from a numerical point of view for certain values of , , and . [Give an example here] Middlebrook showed how a better expression can be obtained as follows. First, factor out of the expression: Now let Then A More Numerically Stable Formulation of the Negative Root Considering just the negative square root we have Multiplying the numerator and denominator by gives By defining we can write Note that as , . Finding the Positive Root Using the Same Approach Turning now to the positive square root we have Using the two roots and , we can factor the quadratic equation Accuracy for Low For values of the value of is within 10% of 1 and we may neglect it. As noted above, the approximation gets better as . With this approximation the quadratic equation has a very simple factorization: an expression that involves no messy square roots and can be written by inspection. Of course, it is necessary to check the assumption about being small before using the simplification. Without this simplification, needs to be calculated and the roots are slightly more complicated. [Explore the consequences if .]