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Round-O Errors
How big is the round-o errors in a given oating-point number system?
Consider the mantissa only. The rounding results in an absolute
error bounded by half of the last digit, i.e.,
1
| | t .
2
For any number x that is within the range of
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Newtons Interpolation Formula
Newtons interpolation formula is mathematically equivalent to the Lagranges formula, but is much more ecient.
One of the most important features of Newtons formula is that
one can gradually increase the support data withou
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Examples
Consider the case n = 1, i.e., two points (2, 2.5) and (3, 4) are to be
interpolated.
The two Lagrange polynomials are easy to construct.
t3
23
t2
.
1 (t) =
32
0 (t)
=
Their geometry is sketched below.
Figure 1: First degree Lagrange polynomia
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Conditioning
A mathematical problem is said to be well-conditioned if small changes
in the data of a problem result in small changes in the solution.
Conditioning is a qualitative description of the sensitivity dependence of the solution to the change
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Stability
An algorithm may be unstable. A problem may be ill-conditioned.
Why is stability an important issue in the design of an algorithm?
Since a computer can only represent nitely many numbers, there
is a good chance that most real numbers will ha
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Polynomial Evaluation
Given a polynomial in the natural form
p(t) = an tn + an1 tn1 + . . . + a1 t + a0 ,
the evaluation of p = p(t) can be done stably by an algorithm called
synthetic division:
p = a[n]
for i = n-1:-1:0
p = p*t + a_[i]
end
Synthetic d
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Implementation of Newton Interpolant
Let dij denote the (i, j)-entry in the following table wherethe indexing
begins with d00 = f [x0 ].
f [x0 ] = f0
f [x1 ] = f1 f [x0 , x1 ]
f [x2 ] = f2 f [x1 , x2 ] f [x0 , x1 , x2 ]
f [x3 ] = f3 f [x2 , x3 ] f [x1
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Error in Interpolation
Suppose the polynomial p(t) interpolates a function f (t) at nodes t =
x0 , x1 , . . . , xn , i.e., suppose p(xi ) = fi = f (xi ) for all i = 0, 1, . . . , n.
Dene
e(t) = f (t) p(t)
as the error function. What can be said about t
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Lagrange Polynomials
Can we construct n polynomials j (t) for j = 0, 1, . . . , n, each of which
has degree n and does the following interpolation?
j (xi )
0,
1,
=
if i = j,
if i = j.
The Lagrange polynomials dened by
n
j (t) :=
i=0,i=j
t xi
, j = 0, 1
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Polynomial Interpolation
Two distinct points can uniquely determine a straight line. What can
three points in a plane that are not collinear determine?
Given cfw_(xi , fi )2 , determine a quadratic polynomial
i=0
p(t) a0 + a1 t + a2 t2
such that
p(xi )
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Floating-Point Number System
Two kinds of computers:
Analog Computer: Numbers are represented by some physical
quantities, such as the length of a bar or the intensity of a voltage.
Digital Computer: Numbers are represented by a sequence of digits wher