Download Contents 1. Vector Spaces

BACKGROUND NOTES FOR APPROXIMATION THEORY
MATH 441, FALL 2009
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This is simply the area of the trapezoid obtained by connecting the points
(a, 0), (a,
f (a)), (b, f (b)) and (b, 0).
3
ror is O (b − a)
by assuming the f
Taylor's formula on
F (x) =
b
x
It isn't too dicult to verify that the erhas a continuous second derivative and using
f (t) dt.
Next, break the interval up into equal sized subintervals using step size
(b − a) /N
h =
and partition points
xj = a + (j − 1) h, j = 0, 1, . . . , N.
Apply the simple trapezoidal rule to each subinterval and add up the results to
obtain
b
(Composite) Trapezoidal Rule:
a
b−a
f (x) dx ≈
2N
N
−1
X
(
One can show that the error of the composite rule
)
f (x0 ) + 2
f (xk ) + f (xN ) .
k=1
2
is O h
by summing the
errors of each subinterval.
Exercises
Exercise 6.1.
Determine the order of
f (h) = sin h2
as
h→0
in terms of
h.
References
[1] M.J.D. Powell,
1981.
[2] T. Shores,
Approximation Theory and Methods
, Cambridge University Press, Cambridge,
Applied Linear Algebra and Matrix Analysis
, Springer, New York, 2007.