: Numerical Integration:
: lec1
: Gaussian Elimination
The basic idea behind numerical
differentiation is very simple. If given the set of values
i=0,1,...,n, we determine the interpolating polynomial
through these points. We then differentiate this polynomial
to obtain whose values for any is taken as an
approximation to . Let us very briefly describe this
interpolating polynomial.
Let
be distinct points on an interval
I and let be a real valued function which takes on the
values
, at these n+1 points. To construct
a polynomial of degree not exceeding n which passes through the
n+1 points
we use the method of
Lagrange. We begin by expressing the desired polynomial as
where each
is a polynomial of degree not
exceeding n. The interpolation condition
will be
satisfied if the satisfy:
It is easy to verify that the function defined by
have this property.
The error in approximating by such a polynomial is
given by
where and
and is some
point on the interval containing
Then the error
in the derivative at a tabular point is given by
Where is a point in the interval containing the points
and
If we assume that the interpolating points
are
equally spaced with spacing h, and put
we can
approximate by Newton's forward difference formula given by
Where is the forward difference
operation definition
and
is the binomial coefficient given by
The error is
.
Differentiating (1) w. r. h. x and noting that
We obtain
If we now set
ie we obtain Approximation formulas for for different
values of n. For example, for , we have
with the error of this formula given by
Taking now n=2,
with the error given by
Formulas for approximation higher derivation of can be
obtained in a similar manner. Differentiating (1) twice and
setting , we obtain
with the error given by
Another formula for the derivatives, using central difference, is
given by
where
Using Taylor's expansion about the , we get
This gives
If is continuous on
, we have
as thus the error in the formula is
The simplest formula
based on central difference is
with error
: Numerical Integration:
: lec1
: Gaussian Elimination
root
平成18年1月24日