# Zero stability

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'''Zero-stability''', also known as D-stability in honor of [Germund Dahlquist](/source/Germund_Dahlquist),<ref>{{cite journal |last1=Dahlquist |first1=Germund |title=Convergence and stability in the numerical integration of ordinary differential equations |journal=Mathematica Scandinavica |date=1956 |volume=4 |issue=4 |pages=33–53 |doi=10.7146/math.scand.a-10454 |jstor=24490010 |doi-access=free }}</ref> refers to the stability of a numerical scheme applied to the simple initial value problem <math> y'(x) = 0</math>.

A [linear multistep method](/source/linear_multistep_method) is ''zero-stable'' if all roots of the characteristic equation that arises on applying the method to <math>y'(x) = 0</math> have magnitude less than or equal to unity, and that all roots with unit magnitude are simple.<ref name=HNW>{{cite book |last1=Hairer |first1=Ernst |last2=Nørsett |first2=Syvert |last3=Wanner |first3=Gerhard |title=Solving Ordinary Differential Equations I |date=1987 |publisher=Springer-Verlag |location=Berlin |pages=326–328}}</ref>  This is called the ''root condition''<ref>{{cite book |last1=Butcher |first1=John C |title=The Numerical Analysis of Ordinary Differential Equations |date=1987 |publisher=Wiley |page=11}}</ref> and means that the parasitic solutions of the [recurrence relation](/source/recurrence_relation) will not grow exponentially.  

==Example==
The following third-order method has the highest order possible for any explicit two-step method<ref name=HNW /> for solving <math>y'(x) = f(x)</math>:
<math display="block">y_{n+2} + 4 y_{n+1} - 5y_n = h(4f_{n+1} + 2 f_n).</math>
If <math>f(x)=0</math> identically, this gives a linear recurrence relation with characteristic equation 
<math display="block">r^2 + 4r - 5=(r-1)(r+5) = 0.</math>
The roots of this equation are <math>r=1</math> and <math>r=-5</math> and so the general solution to the recurrence relation is <math>y_n = c_1\cdot 1^n + c_2 (-5)^n</math>. Rounding errors in the computation of <math>y_1</math> would mean a nonzero (though small) value of <math>c_2</math> so that eventually the parasitic solution <math>(-5)^n</math> would dominate.  Therefore, this method is not zero-stable.

==References==
{{Reflist}}

Category:Numerical differential equations

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