Topic:Numerical Analysis/stability of Multistep methods
For multistep methods, the problems involved with consistence, convergence and stability are complicated because of the number of approximations involved at each step. In the one step method, the approximation depends only on the previous approximation , whereas the multistep methods use at least two of the previous approximations, and the usual method that are employed involve more.
Contents
Stable Multistep Method[edit]
Characteristic polynominals[edit]
For the multistep method:
 ,
 ,
the polynominal
 ,
is called the characteristic polynominal of the above mutistep method.
The stability of a multistep method with respect to roundoff error is dictated by the magnitudes of the zeros of the characteristic polynominal.
Root condition[edit]
Let denote the (not necessarily distinct) roots of the characteristic equation
 ,
associated with the multistep difference method
 ,
If the absolute value , for each i=1,2,...,m, and all roots with abosolute value 1 are the simple roots, then the difference method is said to satisfy the root condition.
Stability Definitions[edit]
 Methods that satisfy the root condition and have as the only root of the characteristic equation of the magnitude one are called strongly stable.
 Methods that satisfy the root condition and have more than one distinct root with magnitude one are called weakly stable.
 Methods that do not satisfy the root condition are called unstable.
Stable Multistep methods[edit]
A multistep method of the form
 ,
 ,
is stable if and only if it satisfies the root condition. Moreover, if the difference method is consistent with the differential equation, then the method is stable if and only if it is convergent.
Examples[edit]
Example1[edit]
The fourthorder AdamsBashfoth method is
where The characteristic equation is
so we find the characteristic roots to be
Thus the 4thorder Adams Method satisfies the root condition and is strongly stable.
Example 2[edit]
Milne's Method is
where The characteristic equation is
so the characteristic roots are
Thus the 4thorder Milne's Method satisfies the root condition, but it is only weakly stable.
Absolute Stability & Region of Absolute stability[edit]
Definition[edit]
Apply a numerical method to initialvalue problem
Suppose that a round off error is introduced in the initial condition for this method. At the nth step the round off error is . Provided the step size h is chosen to satisfy , the numerical method is said to be absolutely stable for the step size h. And is said to be region of absolute stability for numerical method.
Region of absolute stability for onestep method[edit]
In general, when a onestep method is applied to the test equation, a function exists with the property that the difference method gives
 .
The initial roundoff error and the roundoff error at the ith step will satisfy which is .
The inequality will hold if . Thus, region R of absolute stability for a onestep method is
Region of absolute stability for multistep methods[edit]
Consider the kstep method
Applying it to
we get
The characteristic polynominal of the method is
The region R of absolute stability for a multistep method is
 .
Example[edit]
Determine the region of absolute stability for two step method .
Applying this method to: , gives
The characteristic polynominal of above method is
All the zeros of the characteristic polynominal are
Thus the region of absolute stability for this method is .
Exercises[edit]
Exercise 1[edit]
Find the region of the absolute stability of Euler's method:
Solution:
Applying Euler's method to test equation: , gives
The region R of absolute stability for Euler's method is
That is .
Exercise 2[edit]
Find the region R of the absolute stability of Trapezoidal method
Solution:
Applying Trapezoidal method method to test equation: , gives
which simplifies to
where , so .
The inequality will hold if . The region R of absolute stability is
Exercise 3[edit]
Make a table of the interval of absolute stability for four onestep methods: Euler's method, Backward Euler, MEuler and Trapezoidal method.
Solution:

Methods the interval Euler BackwardEuler MEuler The trapezoidal method
Exercise 4[edit]
Find the Region of absolute stability of AdamsBashforth explict Fourstep Method
Solution:
Applying AdamsBashforth explict Fourstep Method to the test equation gives
The characteristic polynominal of above method is
The region R of absolute stability is
Quiz[edit]
The following is a quiz covering information presented on the associated multistep method.