Trapezoidal, simpsons rule, and higher order approximations

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SUMMARY

The discussion centers on the derivation of the trapezoidal rule and Simpson's rule using matrix methods to determine coefficients for linear and quadratic polynomials. The user highlights the complexity of algebraic manipulations involved in proving these rules and expresses interest in extending the approximation to cubic equations. They reference the Newton-Cotes formulas and Gaussian quadrature as established numerical techniques for further exploration. The conversation emphasizes the simplification of terms in polynomial approximations as a key outcome of their work.

PREREQUISITES
  • Matrix algebra for coefficient determination
  • Understanding of polynomial functions, specifically linear and quadratic
  • Familiarity with numerical integration techniques
  • Knowledge of Newton-Cotes formulas and Gaussian quadrature
NEXT STEPS
  • Research cubic polynomial approximations in numerical integration
  • Explore the derivation and application of Newton-Cotes formulas
  • Study Gaussian quadrature methods for improved accuracy
  • Investigate the simplification of higher-order polynomial terms in numerical methods
USEFUL FOR

Mathematicians, numerical analysts, and students interested in advanced numerical integration techniques and polynomial approximation methods.

okkvlt
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hi. i was able to prove the trapezoidal rule and simpsons rule. (basically i used matrices to determine the coefficients m and b for mx+b when proving the trapezoidal rule and a,b,c for ax^2+bx+c such that the points coincide, then i integrated the approximating polynomial) the amount of number-crunching and expanding products of linear terms was probably the most work I've ever done, but i was amazed to see terms cancel out to yeild vastly simplified formulas. but i was wondering, suppose i approximated each step with suppose, a cubic equation. i know how to do this, but after setting up the 4x4 matrix i quickly realized that this would be extremely mundane due to all the algebraic manipulations. so before i try deriving a numerical method that uses cubics to approximate the curve pieces, i want to know whether it will simplify. will it simplify?
 
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