Is there a rule that characterizes when Euler's method over-estimates or under-estimates?For example, "if f ( x ) is increasing, then Euler's method underestimates," or something similar?

Question

Is there a rule that characterizes when Euler&#039;s method over-estimates or under-estimates?For example, &quot;if   f  (  x  ) is increasing, then Euler&#039;s method underestimates,&quot; or something similar?

Odin Jacobson · Accepted Answer

Generally speaking, Euler&#039;s method will overestimate when the second derivative of   f is negative. This comes from the Taylor&#039;s series of the function:  f  (  x  )  =  f  (      x    0    )  +  (  x  −      x    0    )      f    ′    (  x  )  +      1    2    (  x  −      x    0        )    2        f    ″    (  x  )  +  …Euler&#039;s method accounts for the       f    ′   term but no more. This is not absolute. For a given step size, the third derivative could be huge and swamp the effect of the second. Higher order methods account for more terms of the Taylor series.