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Rounding errors in the computation of would mean a nonzero (though small) value of so that eventually the parasitic solution () would dominate. Therefore, this method is not zero-stable. Therefore, this method is not zero-stable.
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u {\displaystyle u} , the exact solution to a differential equation in an appropriate normed space ( V , | | | | ) {\displaystyle (V,||\ ||)} .
Computing the square root of 2 (which is roughly 1.41421) is a well-posed problem.Many algorithms solve this problem by starting with an initial approximation x 0 to , for instance x 0 = 1.4, and then computing improved guesses x 1, x 2, etc.
A famous example is the recurrence for the Fibonacci numbers, = + where the order is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on .
To compute the terms of a recurrence through according to Miller's algorithm, one first chooses a value much larger than and computes a trial solution taking initial condition to an arbitrary non-zero value (such as 1) and taking + and later terms to be zero. Then the recurrence relation is used to successively compute trial values for , down ...
For linear multistep methods, an additional concept called zero-stability is needed to explain the relation between local and global truncation errors. Linear multistep methods that satisfy the condition of zero-stability have the same relation between local and global errors as one-step methods.
The previous example involved an indicial polynomial with a repeated root, which gives only one solution to the given differential equation. In general, the Frobenius method gives two independent solutions provided that the indicial equation's roots are not separated by an integer (including zero).
The order of the sequence is the smallest positive integer such that the sequence satisfies a recurrence of order d, or = for the everywhere-zero sequence. [ citation needed ] The definition above allows eventually- periodic sequences such as 1 , 0 , 0 , 0 , … {\displaystyle 1,0,0,0,\ldots } and 0 , 1 , 0 , 0 , … {\displaystyle 0,1,0,0 ...