1. Computers in Civil Engineering 53:081 Spring 2003
Lecture #7
Roots of Equations: Open Methods

2. Lecture Outline Open Methods The Newton-Raphson Algorithm
The Secant Algorithm

3. Newton-Raphson Algorithm Graphical Derivation
xi+1 xi
From figure:
(Newton-Raphson Formula)

4. Derivation from Taylor Series
Which can be rearranged as:
(same result as geometrical one)

5. Newton-Raphson Algorithm Properties
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

6. Example: f(x) = e-x-x f(x) x Stopping Criteria: 1.0 0.5 0.0 0.2 0.4
0.6
0.8
1.0
-0.5
-1.0 x
Stopping Criteria:

7. Convergence Newton-Raphson Bisection Iteration xr |t|% |t|%
<

8. Newton-Raphson Pitfalls
f(x) x3 x1 x2 x
f(x) x1 x3 x4 x2 x

9. Secant Algorithm
Motivation: Inconvenient/difficult to evaluate f '(x) analytically in Newton-Raphson algorithm:
Solution: Approximate f '(x) with a backward finite divided difference:
Substituting in (1) yields the secant algorithm:

10. Secant Algorithm xi+1 xi-1 xi xi+1 xi  Secant Algorithm
Use approximate f '(x) at xi
Two initial estimates required
xi+1
xi-1 xi
Newton-Raphson Algorithm
Use true f '(x) at xi 
xi+1 xi

11. Multiple Roots
Multiple roots occur where the function is tangent to the axis. In other words, where

12. Multiple Roots (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:
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.

13. Next: Systems of Nonlinear Equations (Read the Textbook!)