MATH 174: NUMERICAL ANALYSIS I LECTURER: JOMAR FAJARDO RABAJANTE 1st Sem AY 2018-2019 IMSP, UPLB

IMPROVING OUR NEWTON-COTES FORMULAS

ROMBERG INTEGRATION Applying Richardson Extrapolation to the composite Trapezoidal Rule. Define the following step sizes:

ROMBERG INTEGRATION Formulas for 1st column: (this is a recursive formula for composite Trapezoidal Rule) 1 panel 2 panels j panels

ROMBERG INTEGRATION O(h2) O(h4) O(h6) O(h8) Trapezoidal Simpson’s Boole’s O(h8)

ROMBERG INTEGRATION O(h2) 1 panel O(h4) 2 panels O(h6) 3 panels O(h8)

ROMBERG INTEGRATION Extrapolation Formula: Example:

AUTOMATIC/ADAPTIVE QUADRATURE Using higher-order quadrature formulas requires higher-order derivatives in getting the error bound. We also know that smaller step sizes (more number of panels) improve accuracy. But how small is small? Some functions also vary wildly over some of their domain and vary slowly through other parts, hence equal step sizes is not appropriate.

AUTOMATIC/ADAPTIVE QUADRATURE “Magmamano-mano ba tayo?” It is very tedious to do trial and error in choosing the appropriate panels, and after evaluation we still need to check if this satisfies our tolerable error (remember it is not easy to compute for the error bound). What if instead of doing trial and error, we start by right away inputting our tolerable error bound, and do a process that would automatically target that goal.

AUTOMATIC/ADAPTIVE QUADRATURE For our example, we use Adaptive Trapezoidal Rule. Recall the basic trapezoidal rule: Old error

AUTOMATIC/ADAPTIVE QUADRATURE We can improve the formula by cutting into half:

AUTOMATIC/ADAPTIVE QUADRATURE Then: Assume

AUTOMATIC/ADAPTIVE QUADRATURE Then: The difference is 3 times the new error New error

AUTOMATIC/ADAPTIVE QUADRATURE Hence, for Adaptive Trapezoidal Rule, the ACCURACY TEST is If this is satisfied then is a good approximate with max error of TOL. Input of the user

AUTOMATIC/ADAPTIVE QUADRATURE If this is NOT satisfied then apply again the method individually to With accuracy tests: Etc…

AUTOMATIC/ADAPTIVE QUADRATURE EXAMPLE: TOL=0.03 Solve this on the blackboard

AUTOMATIC/ADAPTIVE QUADRATURE EXAMPLE: Solve this on the blackboard EXERCISE

AUTOMATIC/ADAPTIVE QUADRATURE EXAMPLE: TOL=0.03 Solve this on the blackboard

AUTOMATIC/ADAPTIVE QUADRATURE For Adaptive Simpson’s Rule, the ACCURACY TEST is Defining