TRAPEZOIDAL METHOD: ERROR FORMULA
TRAPEZOIDAL METHOD: ERROR FORMULATheoremAssumef(x) twice continuously differentiable on theinterval [a,b]. ThenETn(f) := baf(x)dx Tn(f) = h2(b a)12f (cn)for somecnin the interval [a,b].Later we will say something about the proof of this result, as itleads to some other useful formulas for the above FORMULA says that the ERROR decreases in a manner thatis roughly proportional toh2. Thus doublingn(and halvingh)should cause the ERROR to decrease by a factor of approximately is what we observed with some past examples from thepreceding evaluatingI= 20dx1 +x2using the TRAPEZOIDAL methodTn(f). Let us bound the errorETn(f) = h2(b a)12f (cn)Here,b a= 2. We bound|f (cn)|by max0 x 2|f (x)|.Calculate the derivatives:f (x) = 2x(1 +x2)2,f (x) = 2 + 6x2(1 +x2)3,f (x) =24x(1 x2)(1 +x2)4Forx (0,2),f (x) = 0 only whenx= 1. Somax0 x 2 f (x) = max{ f (0) , f (1) , f (2) }= 2Therefore, ETn(f) h2(2)12 2 =h23 ETn(f) h23How large shouldnbe chosen in order to ensure that ETn(f) 5 10 6(1)To ensure this, we choosehso small thath23 5 10 6This is equivalent to choosinghandnto satisfyh.
The corrected trapezoidal rule In general, I(f) T n(f) ˇ h2 12 f0(b) f0(a) I(f) ˇCT n(f) := T n(f) h2 12 f0(b) f0(a) This is the corrected trapezoidal rule. It is easy to obtain from the trapezoidal rule, and in most cases, it converges more rapidly than the trapezoidal rule.
Download TRAPEZOIDAL METHOD: ERROR FORMULA
Information
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document: