Is there a rule that characterizes when Euler's method over-estimates or under-estimates?For example, "if f...

Santino Bautista

Santino Bautista

Answered

2022-06-25

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?

Answer & Explanation

Odin Jacobson

Odin Jacobson

Expert

2022-06-26Added 17 answers

Generally speaking, Euler's method will overestimate when the second derivative of f is negative. This comes from the Taylor's series of the function:
f ( x ) = f ( x 0 ) + ( x x 0 ) f ( x ) + 1 2 ( x x 0 ) 2 f ( x ) +
Euler'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.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?