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WEEK 4
Roots of Equations Multiple Roots
Systems of Nonlinear Equations
2
At the end of this topic, the students will be able:
To solve roots for polynomials systems and
system of non-linear equations
LESSON OUTCOMES
4
Multiple Roots
• Corresponds to a point where a function is tangent to the x axis at that point. For example, following polynomials have:
• Double root
– f(x) = (x–3)(x–1)(x–1) ~ one value of x makes two terms in the equation equal to zero
– Axis is not cross (even roots)
• Triple root
– f(x) = (x–3)(x–1)(x–1)(x–1) ~ one value of x makes three terms in the equation equal to zero
– Axis is cross (odd roots)
375)( 23 xxxxf
310126)( 234 xxxxxf
5
Figure shows (a) through (c) show
several cases of multiple roots. Consequently, multiple roots pose several limitations for the numerical methods. There are two possible difficulties: 1. For a function with even multiple
roots, the function does not change sign over the interval. Thus, it eliminates the use of the reliable bracketing methods.
2. At the multiple roots, both f(x) and f’(x) are zero or approach zero. This poses problems for both Newton-Raphson and secant methods.
To overcome these two problems,
a modified Newton-Raphson method is used.
6
• Introducing the new function u(x) as the ratio of the function and its first derivative as
• Using quotient rule for the derivative,
• The modified Newton-Raphson method in terms of u(x) is then
)(
)()(
i
ii
xf
xfxu
This function has roots
at all the same locations
as the original function
2)(
)('')()(')(')('
i
iiii
xf
xfxfxfxfxu
)('
)(1
i
iii
xu
xuxx
7
• Then, modified Newton-Raphson Method
• The denominator will not be zero even if f’(x) is zero. Therefore, the formula is said to be more stable.
It is preferable for multiple roots, it is somewhat less efficient.
Unless necessary, this method is seldom used in practice due more computational effort involved.
)('')()(
)(')(21
iii
iiii
xfxfxf
xfxfxx
8
375)( 23 xxxxf
Example
Use both standard and modified Newton-Raphson methods to evaluate the multiple root for with an initial guess of x0 = 0.
Refer Example 6.9
9
Exercise
Use the modified Newton-Raphson method to find
the root of the following polynomial function
f(x) = x3 - 7x2 + 8x +16 = 0
until the relative error is less than 0.05%.
Use (i) x = 0 and (ii) x = 2.0 as initial guess.
10
Ite. xi f(xi) f’(xi) f’’(xi) xi+1 εa(%)
1 0.0 16.0 8.0 -14.0 -0.4444 -
2 -0.4444 0.9739 14.8147 -16.6667 -0.8485 47.619
3 -0.8485 3.5618 22.0386 -19.0909 -0.9903 14.316
4 -0.9903 0.2427 24.8053 -19.9415 -0.9999 0.970
5 -0.9999 0.00095 24.9992 -19.9998 -1.0000 0.004
i) x = 0
ii) x = 2
Ite. xi f(xi) f’(xi) f’’(xi) xi+1 εa(%)
1 2.0 12.0 -8.0 -2.0 3.0909 -
2 3.0909 3.3809 -6.6116 4.5455 3.8796 20.327
3 3.8796 0.07083 -1.1613 9.2771 3.9985 2.975
4 3.9985 0.000012 -0.01523 9.9909 3.9999 0.038
Note: f’(x) does not approach zero, so x = 1.0 is not a multiple root.
Note: f’(x) approaches zero, so x = 3.9999 (essentially 4.0) is a multiple root.
11
SYSTEMS OF NONLINEAR EQUATIONS
Linear equations
Which can be solved for
0),,,,(
0),,,,(
0),,,,(
321
3212
3211
nn
n
n
xxxxf
xxxxf
xxxxf
0...)( 2211 bxaxaxaxf nn
12
Non-linear equations
Which can be solved for using expression:
573
10
2
2
xyy
xyx
0573),(
010),(
2
2
xyyyxv
xyxyxu
13
• Solution of the nonlinear equations using open methods
Fixed point iteration
Newton Raphson
14
Example: Fixed point iteration
Determine the roots using fixed-point iteration with initial guesses of x = 1.5 and y = 3.5
0573),(
010),(
2
2
xyyyxv
xyxyxu
Refer Example 6.10
• Taylor series expansion of a function of more than one variable
• The root of the equation occurs at the value of x and y where ui+1 and vi+1 equal to zero.
y
vyy
x
vxxvv
y
uyy
x
uxxuu
iii
iiiii
iii
iiiii
)()(
)()(
111
111
Newton-Raphson method
16
y
vy
x
vxvy
y
vx
x
v
y
uy
x
uxuy
y
ux
x
u
ii
iiii
ii
i
ii
iiii
ii
i
11
11
x
v
y
u
y
v
x
ux
vu
x
uv
yy
x
v
y
u
y
v
x
u
y
uv
y
vu
xx
iiii
ii
ii
ii
iiii
ii
ii
ii
1
1
• Then, the equation can be arranged to
Determinant of
the Jacobian of
the system.
Two equation
version of
Newton-Raphson
method
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Example: Newton Raphson
Use the multiple-equation Newton-Raphson method to determine roots with initial guesses of x = 1.5 and y = 3.5
Refer Example 6.11
0573),(
010),(
2
2
xyyyxv
xyxyxu
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Exercise
Determine the roots of the following
simultaneous equations using
(a) fixed-point iteration
(b) Newton-Raphson method
y = -x2 + x + 0.75
y + 5xy = x2
Employ initial guesses of x=y=1.2 until the
approximate error is less than 0.05%.
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