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ChE 374Computational Methods in Engineering

Solution of Non-Linear Equations

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Roots of Equations• Many engineering problems require the

determination of the values of the variable xwhich satisfies the non-linear equation

( ) 0f x =

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Non-Linear Eqn.

• Bracketing methods– Bisection– Regula falsi (False position)

– Graphical methods to provide insight– Error formulation

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Non-Linear Eqn.• Open methods

– Systematic trial and error– Computationally efficient, but do not always

work• Newton Raphson• Secant• One-point iteration

• Roots of Polynomials– Muller’s– Bairstow’s

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Example

• Using the graphical approach to determine the value of the drag coefficient ( c ) needed for a parachutist of mass m=68.1 kg to have a velocity of 40 m/s after free falling time of 10s.

• NOTE g = 9.8 m/s2.

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Drag coefficient to attain a given velocity at a set time period

( )( )/1 e c m tgmvc

−= −

( )( )/( ) 1 e c m tgmf c vc

−= − −

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c f (c )

4

8

12

16

20

34.115

17.653

6.067

-2.269

-8.401

f (c )

-15

-10

-5

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25

f (c )

( )( )/( ) 1 e c m tgmf c vc

−= − − g= 9.8m/s2, m=68.1 kg, t=10s, v = 40 m/s

The curve crosses the x axis between 12 and 16. A rough estimate of the root is 14.8. The graphical method is not very accurate, but provides a good estimate. Also helps to understand the properties of the function.

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Plot of tan (x) showing singularitiestanx

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 1 2 3 4 5 6 7

tanx

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Root search algorithm%Incremental search for a root of f(x)%USAGE: [x1,x2] = rootsearch(func, a,b, dx)%INPUT:% func = handle of function that returns f(x)% a,b = limits of search% dx = search increment% OUTPUT% x1, x2 = bounds of function that returns f(x)% set to NaN if no root is found%function [x1,x2] = rootsearch(func, a,b, dx)

x1 = a;f1 = feval(func,x1);x2 = a + dx;f2 = feval(func,x2);

while f1*f2 > 0.0if x1 >= b

x1=NaN; x2 = NaN; returnendx1 = x2;f1 = f2;x2 = x1+dx;f2 = feval(func,x2);

end

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Bisection AlgorithmStep 1 Choose lower xl and upper xu guesses for the root

such that the function changes sign over the interval. This can be checked by ensuring that f(xl)f(xu) <0

Step 2 An estimate of the root xr is determined byxr = (xl + xu)/2

Step 3 Make the following evaluations to determine in which subinterval the root lies

(a) If f(xl)f(xr) <0, root lies in lower subinterval, set xu=xl and go to step 2

(b) If f(xl)f(xr) >0, root lies in upper interval, set xl = xrand return to step 2

(c) If f(xl)f(xr) =0, the root equals xr; end computation

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Number of bisections

1

(0)

Bisection is repeated until the original intervalhas been reduced to a small value, so that

If is the original intervalthe number of bisections required to attaina prescribed absolute

i ix x

x

εε−− ≤

Δ

error is

(0)

ln

ln 2

x

nε

⎛ ⎞Δ⎜ ⎟⎝ ⎠=

Slow but sure !!!

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Termination criteria and error estimatesDefine approximate percent error as

100%

root for the present iteration

root from the previous iteration

new oldr r

a newr

newroldr

x xx

x

x

ε −= ×

≡

≡

(using absolute values because we are concerned about the magnitude,Not the sign)

When the error becomes less than specified stopping criterion, the computation stopped.

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Regula Falsi (False Position) Method

2x

1x

1( )f x

2( )f x

( )f x

3x 4x

( )11

1

( )( ) ( )

i i ii i

i i

f x x xx x

f x f x−

+−

−= −

−

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Newton Raphson Method

1( )f x′

( )f x

( )f x

3x4x

1( ) 1,2,3....( )

ii i

i

f xx x if x+ = − =′

x

2( )f x′

3( )f x′

2x 1x

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Newton Method: Properties

• Requires derivative of the function f’ = df/dx• Very fast convergence in most cases

(Quadratic convergence) !!• May not converge if the estimate is far off• May not converge if the derivative (slope) at

an estimate is close to zero• Can be used to find complex roots

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Newton : Non-convergent

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Convergence criteria for stopping iterative procedure

1

1

3 6

0

( )

a small number 10 10

i i

i ii

i

i

x x

x x xx

f x

or

ε

ε

ε

ε

−

−

− −

− ≤

−≤ ≠

≤

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Secant method• Similar to Newton’s method but different in

that the derivative f’ is approximated using two consecutive iterates

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1

( ) ( )( ) i ii

i i

f x f xf xx x

−

−

−′ =−

( ) 1

1

( )( )( ) ( ) ( )

i i ii i i i

i i i

f x x xf x x x f xf x f x f x

−

−

⎡ ⎤−= − = − ⎢ ⎥′ −⎣ ⎦

2,3, 4,i = …• Needs two initial guesses

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Convergence of the Secant Method

1( )f x

( )f x

( )f x

3x4x

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1

( ) 2,3,4....( ) ( )

i ii i i

i i

x xx x f x if x f x

−+

−

⎡ ⎤−= − =⎢ ⎥−⎣ ⎦

Exact root

x

2( )f x

3( )f x

Convergence of Secant method

2x 1x

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Secant : non-convergence

1x2x

( )f x

x

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Fixed Point Iteration

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Roots of Polynomials

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Roots of Polynomials2

0 1 2( ) nnf x a ax ax ax= + + + +

n = order of polynomiala’s = constant coefficients

For real coefficients

• An nth order equation will have n real or complex roots• If n is odd, at least one root is real• If complex roots exist, they exist in conjugate pairs, i.e

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Polynomial Deflation

• If a root of a polynomial has been found, it is desirable to factor the polynomial

( ) 1( ) ( )n nP x x r P x−= −

• Deflation or synthetic division

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Roots of Polynomials

• Polynomials• Polynomial deflation• Muller’s Method• Bairstow’s Method