Numerical Methods for Eng [ENGR 391]
[Lyes KADEM 2007]
IX. Multiple roots
A multiple root corresponds to a point where a function is tangent to the x-axis:
Double root
Triple root
f(x)=(x-2) (x-2) (x-3)
f(x)= (x-2) (x-2) (x2) (x-3)
In general, odd multiple roots cross the x-axis whereas even ones do not.
The presence of a multiple root is very problematic for the methods defined previously because:
-
-
There is no change in the sign of the function, so the bracketing methods cannot be
applied.
f x  but also f ' x  goes to zero, and Newton-Raphson and secant method contain
f ' x  in the denominator. However, it can be demonstrated theoretically that f x  → 0
'
before f x  → 0. Therefore, a zero check for f x  must be incorporated into the
'
computer program and the computation can be terminated before f x  reaches zero.
For multiple roots, Newton-Raphson and the secant methods are linearly rather than
quadratically convergent.
For open methods, a new approach has to be used to overcome the limitations introduced by
multiple roots.
The new formula can be written under the following form:
xi1  xi  m
f xi 
f '  xi 
Where (m) is the multiplicity of the root (m=2 for double; m=3 for triple, …). The problem here is
that you have to know the multiplicity of the root.
Another approach is to define a new function U(x) as the ratio of the function to its derivative:
U x  
f x 
f ' x 
This function has the same root as f x  , it is then substituted into the original formula of NewtonRaphson:
xi1  xi 
U  xi 
U '  xi 
with
U ' x  
f ' x  f ' x   f x  f '' x 
 f x
'
Therefore,
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2
Numerical Methods for Eng [ENGR 391]
[Lyes KADEM 2007]
xi1  xi 
f xi  f ' xi 
 f x   f x  f x 
2
'
''
i
i
i
However, the price to pay for this approach is more computational effort.
For the secant method the formula will be:
xi1  xi 
U xi xi1  xi 
U xi1   U xi 
X. Systems of non-linear equations
Usually to solve a system of non-linear equations, we will use an extension of open methods.
An example of a system of non-linear equations is:
f1 x1 , x2   x12  x22  8 x1  4 x2  11  0
f 2 x1 , x2   x12  x22  20 x1  75  0
X.1. Fixed point iteration for systems of non-linear equations
One of the most important drawbacks of the fixed iteration method is that the convergence of the
method is dependent on how the equations are formulated.
It can be shown that sufficient convergence criteria for two equations are:
f1
f
 1 1
x1 x2
and
f 2 f 2

1
x1 x2
This represents a very restrictive criteria and that’s why fixed point iteration method is not used to
solve systems of non-linear equations.
X.2. Newton-Raphson for systems of non-linear equations
The Newton-Raphson formula is the following:
xi1  xi 
f  xi 
f '  xi 
This formula can be obtained using Taylor series expansion [assignment II]. We can do the same
approach for a system of equations, but considering a Taylor series that account for the presence
of both variables:
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Numerical Methods for Eng [ENGR 391]
f1i1  f1i   x1i1  x1i  
f1i 
[Lyes KADEM 2007]
 x2i1  x2i  
x1
f1i 
x2
 ...
and
f 2i1  f 2i   x1i1  x1i  
f 2i 
x1
 x2i1  x2i  
f 2i 
x2
 ...
For the root estimate f1i 1 and f 2 i 1 must be equal zero.
Therefore:
f1i 
f
f
f
x1i1  1i  x2i1   f1i   x1i  1i   x2i  1i 
x1
x2
x1
x2
and
f 2i 
f
f
f
x1i1  2i  x2i1   f 2i   x1i  2i   x2i  2i 
x1
x2
x1
x2
Finally;
x1i1  x1i  
f1i 
f 2i 
x2
f1i  f 2i 
x1 x2
x2i1  x2i  
f 2 i 
f1i 
 f 2 i 

f 2i 
x1
f 2i 
x1 x2
x2
f1i  f 2i 
x2 x1
 f1i 

f1i 
f1i 
x1
f1i  f 2i 
x2 x1
The denominator is called the Jacobian.
This is the two equation version for Newton-Raphson method. This method is also highly
dependent upon your initial guess. For the one equation problem, plotting the function was a
good reflex to choose the initial guess. However, this is not possible with the system of equations.
Therefore, we will mostly use a trial an error method or consider the physical system being
modeled.
Note also that the Newton-Raphson method can be generalized to solve N simultaneous
equations.
To solve this system, we have to write it under matrix form:
J  x   f 
Where x1,2,3,… = x1,2,3,…(i+1) - x1,2,3,…(i)
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Numerical Methods for Eng [ENGR 391]
[Lyes KADEM 2007]
Example:
Solve the following system using Newton-Raphson method:
f1 x1 , x2   x12  x22  8 x1  4 x2  11  0
f 2 x1 , x2   x12  x22  20 x1  75  0
By tacking a starting point as (x1=2; x2=4) and   10 5 .
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