Computers in Civil Engineering 53:081 Spring 2003
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Computers in Civil Computers in Civil Engineering Engineering 53:081 Spring 2003 53:081 Spring 2003 Lecture #7 Lecture #7 Roots of Equations: Open Roots of Equations: Open Methods Methods
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Computers in Civil Engineering 53:081 Spring 2003. Lecture #7. Roots of Equations: Open Methods. Lecture Outline. Open Methods The Newton-Raphson Algorithm The Secant Algorithm. Newton-Raphson Algorithm Graphical Derivation. . 0. x i+ 1. x i. From figure:. (Newton-Raphson Formula). - PowerPoint PPT Presentation
Transcript of Computers in Civil Engineering 53:081 Spring 2003
Computers in Civil Computers in Civil EngineeringEngineering
53:081 Spring 200353:081 Spring 2003
Lecture #7Lecture #7
Roots of Equations: Open Roots of Equations: Open MethodsMethods
Open Methods– The Newton-Raphson Algorithm– The Secant Algorithm
Lecture OutlineLecture Outline
Newton-Raphson Newton-Raphson AlgorithmAlgorithm
Graphical DerivationGraphical Derivation
xixi+10
)(')tan( ixfβ
)( ixf
)(xf
1
)()(
ii
ii xx
xfx'f
)(
)(1
i
iii x'f
xfxx
From figure:
(Newton-Raphson Formula)
Derivation from Taylor Derivation from Taylor SeriesSeries
))(()(0
0)(that such selecting
))(()()(
...)(!2
)('' ))(()()(
1
11
11
2111
iiii
ii
iiiii
iiiiiii
xxx'fxf
xfx
xxx'fxfxf
xxf
xxx'fxfxf
)(
)(
i
iii x'f
xfxx 1Which can be rearranged as:
(same result as geometrical one)
• Quadratic convergence (Single Roots)• Number of correct decimal places doubles
with each iteration (single root)• Linear convergence (Multiple Roots)• Some problem cases exist
• Slow or no convergence
• Oscillation• Both function and its derivative must be
evaluated:• Inconvenient
• May not be so easy
Newton-Raphson Algorithm Newton-Raphson Algorithm PropertiesProperties
Example: Example: ff((xx) = ) = ee-x-x-x-x
sr
rra x
xx
100%new
oldnew
-1.0
-0.5
0.0
0.5
1.0
0.2 0.4 0.6 0.8 1.0
x
f(x)
Stopping Criteria:
ConvergenceConvergence
Newton-Raphson Bisection
Iteration xr |t|% |t|%
1 0.50000 11.8 11.8
2 0.56631 0.147 32.2 3 0.56714 0.00002 10.2 4 0.56714 < 10-8 0.819
Newton-Raphson Newton-Raphson PitfallsPitfalls
x1 x2x3 x
x
f(x)
f(x)
x1 x2x3 x4
Secant AlgorithmSecant Algorithm
ii
iii xx
xfxfx'f
1
1 )(( ))(
)(
)
1
11
ii
iiiii xfxf
xxxfxx
)(
)((
Motivation: Inconvenient/difficult to evaluate f '(x) analytically in Newton-Raphson algorithm:
Solution: Approximate f '(x) with a backward finite divided difference:
(1) )(
)(
i
iii x'f
xfxx 1
Substituting in (1) yields the secant algorithm:
Secant AlgorithmSecant Algorithm
xixi-10
)( 1ixf
)( ixf
xixi+10
)(')tan( ixfβ
)( ixf
)(xf
Secant Algorithm
•Use approximate f '(x) at xi
•Two initial estimates required
Newton-Raphson Algorithm
Use true f '(x) at xi
xi+1
Multiple RootsMultiple Roots
1 3
Double Root(even # of rootsno sign change)
Single Root(Odd # of roots,sign change)
f x( )
x
Note: = 0f x’( )
1 3
Triple Root(odd # of rootssign change)
Note: = 0f x’( ) Single Root(Odd # of roots,sign change)
f x( )
x
Multiple roots occur where the function is tangent to the axis. In other words, where
0)(')( xfxf
Multiple Roots Multiple Roots (continued)(continued)
At even multiple roots: no sign change => can’t use bracketing methods.
At multiple roots f(x) and f '(x) are zero.
– Newton-Raphson:
– Secant:
)(
)(
i
iii x'f
xfxx 1
)(
)
1
11
ii
iiiii xfxf
xxxfxx
)(
)((
Both formulas contain derivative (or its estimate) in denominator. This could result in division by zero as the solution converges very close to the root.
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EquationsEquations(Read the Textbook!)(Read the Textbook!)
