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MATH 175: Numerical Analysis II

MATH 175: Numerical Analysis II. Lecturer: Jomar Fajardo Rabajante IMSP, UPLB 2 nd Sem AY 2012-2013. 3 rd Method: Regula Falsi /False Position/Inverse Linear Interpolation (also a bracketing method). We will still use the Intermediate Zero Theorem The function should be continuous

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MATH 175: Numerical Analysis II

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  1. MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante IMSP, UPLB 2ndSem AY 2012-2013

  2. 3rd Method: RegulaFalsi/False Position/Inverse Linear Interpolation (also a bracketing method) We will still use the Intermediate Zero Theorem • The function should be continuous • The function values of the two endpoints of the line should be of opposite signs

  3. 3rd Method: RegulaFalsi y-axis x-axis Approximate root

  4. 3rd Method: RegulaFalsi We will use inverse linear interpolation (in slope-intercept form): To get x3, set f(x3)=0 (from the zero of the line):

  5. 3rd Method: RegulaFalsi y-axis Set new x1 = old x3 New Approximate root x-axis Old Approximate root

  6. 3rd Method: RegulaFalsi y-axis Old Approximate root x-axis Set new x2 = old x3 New Approximate root

  7. 3rd Method: RegulaFalsi y-axis 2nd 3rd x-axis 1st

  8. 3rd Method: RegulaFalsi Try this at home: create a flowchart of the RegulaFalsi method Hint: use bisection algorithm, however change the formula for x3 to

  9. 3rd Method: RegulaFalsi • Like bisection method, RegulaFalsi is guaranteed to converge to the root (assuming IZT is met). • The rate at which this method converges will depend on how nearly linear f(x) is near its zero. If f(x) is sufficiently differentiable then it is well approximated by a straight line over small intervals. • RegulaFalsi is often (not always) faster than bisection method, but still the order of convergence is linear.

  10. 3rd Method: RegulaFalsi • We can use abs(x3,k-x3,k-1)<tol as our stopping criterion but not (b-a)/2k<tol (where [a,b] is the initial bracket) since the width of the bracket may not converge to zero like in bisection. • But a better stopping criterion is • (Let x3,k be the approximate root at iteration k, and λbe the asymptotic error constant. Invoking the error evolution equation: ) where

  11. y-axis There’s an improved RegulaFalsi method called the Modified RegulaFalsi. 1/2 3rd 2nd x-axis 1st

  12. Bisection, RegulaFalsi and Modified RegulaFalsi methods are called Interval/Bracketing Methods. Next topics: Iterative methods, such as secant method, Newton’s method, fixed point iteration, Mϋller’s method, Bairstow’s method Usually, faster methods require more assumptions and offer fewer guarantees.

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