Computers in Civil Engineering 53:081 Spring 2003

Computers in Civil Engineering53:081 Spring 2003 Lecture #7 Roots of Equations: Open Methods

Lecture Outline • Open Methods • The Newton-Raphson Algorithm • The Secant Algorithm

Newton-Raphson AlgorithmGraphical Derivation  0 xi+1 xi From figure: (Newton-Raphson Formula)

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

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

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

Convergence 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 Pitfalls f(x) x3 x1 x2 x f(x) x1 x3 x4 x2 x

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

Secant Algorithm • Secant Algorithm • Use approximate f '(x) atxi • Two initial estimates required xi+1 xi-1 0 xi Newton-Raphson Algorithm Use true f '(x) atxi  0 xi+1 xi

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

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.

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

Computers in Civil Engineering 53:081 Spring 2003