Mathematics 1
Applied Informatics
Štefan BEREŽNÝ
3rd lecture
Contents
• Approximate Solution of a Nonlinear
Equation
• Separation of a Root
• Darboux Theorem
• Bisection’s method
• Newton’s method
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
3
Approximate Solution of
a Nonlinear Equation
Definition:
Let f(x) be a function. Every point c  D(f) such
that f(c) = 0 is called the root of the equation
f(x) = 0.
• initial approximation c0
• iterative sequence c1, c2, c3, … etc
• Methods based on the construction of an iterative
sequence are called iterative methods
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Approximate Solution of
a Nonlinear Equation
If iterative sequence converges to the root c
of the equation f(x) = 0 then
lim
cn  c
n
Error estimates:
cn – c n, where n  0 for n  
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
5
Separation of a Root
By the separation of a root we understand
the specification of an interval a, b such
that the equation f(x) = 0 has a unique root
c in a, b.
Intervals (–, b0  a1, b1  a2, b2  ... 
 an–1, bn–1  an, bn  an+1, ) = D(f)
separate the roots of the equation f(x) = 0, if
each of the intervals includes at most one
root.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Darboux theorem
If function f is continuous on an interval
I = a, b and x1, x2 are any two points from
interval
I
then
to
any
given
number
 between f(x1) and f(x2) there exists a point
 between x1 and x2 such that f() = .
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Darboux theorem
Corollary:
If function f is continuous on an interval
I = a, b and f (a)  f (b)  0 then exists a
point c in (a, b) such that f(c) = 0 (the root of
the equation f(x) = 0).
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
8
Bisection’s method
Suppose that function f is continuous and
strictly monotonic in the interval I = a, b and
f (a)  f (b)  0.
These assumptions guarantee the existence
of a unique root c of the equation f(x) = 0 in
interval I = a, b.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Bisection’s method
Choice of the initial approximation:
Put c0 = (a + b)/2.
Calculation of the further approximations:
If f (c0)  f (b)  0 then c  (c0, b.
Therefore we change a and we put a = c0.
If f (c0)  f (b)  0 then c  a, c0.
We change b and we put b = c0.
Further, we put c1 = (a + b)/2.
Similarly, we obtain c2, c3, … etc.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Bisection’s method
The error estimate:
Denote by d the length of the interval I = a, b at
the beginning of the calculation. Since c  a, b,
c0 – c d/2. The length of the “variable” interval
I = a, b (where the root c is separated) decreases
by one half at each step. Hence c0 – c d/2n+1.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Newton’s method
Suppose that:
- function f has a second derivative f ′′(x)
at each point x  a, b and f ′′(x) does
not change its sign in I = a, b.
- f ′(x) ≠ 0 for all x  a, b,
- f (a)  f (b)  0.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Newton’s method
Choice of the initial approximation:
The initial approximation c0 can be chosen to
be equal an arbitrary point of the interval
a, b such that f (c0)  f ′′(c0)  0.
(Among others, this inequality is satisfied by
one of the points a and b.)
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Newton’s method
Calculation of the further approximations:
To approximate the curve y = f (x) in the
neighborhood of the point [c0, f (c0)], we use
a tangent line to the graph of f at this point.
The point where this line crosses the x-axis
is called c1. Similarly, the point where the
tangent line to the graph of f at point
[c1, f (c1)] crosses the x-axis is the next
approximation c2, etc.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Newton’s method
This procedure can easily be expressed
computatively. Suppose that you already
know the approximation cn and you wish to
find the next approximation cn+1. The
equation for the tangent line to the graph of f
at the point [cn, f (cn)] is:
y  f (cn )  f (cn )  ( x  cn )
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Newton’s method
y = 0 corresponds to x = xn+1. So we get the
equation , which yields:
f (cn )
cn1  cn 
f (cn )
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
16
Newton’s method
The error estimate:
It follows from the Mean Value Theorem,
applied on the interval with end points cn and
c, that exists  between cn and c such that:
f (cn )  f (c)  f ( )  (cn  c)
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
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Newton’s method
f (cn )
cn  c 
f ( )
f (cn )
cn  c 
m
m  min f ( )
  a,b
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MATHEMATICS 1
Applied Informatics
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Thank you for your attention.
Štefan BEREŽNÝ
MATHEMATICS 1
Applied Informatics
19