Composite Numerical Integration Methods in Mathematics
Learn about composite numerical integration methods like the Trapezoidal Rule and Simpson's Rule to approximate definite integrals more accurately. Understand the principles, applications, and error analysis involved in these techniques.
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Sec:4.4 Composite Numerical Integration
Sec:4.4 Composite Numerical Integration Example The Trapezoidal Rule Use Trapezoidal rule to approximate ? ? ?? ??? ? ? ? ?? =? ???? ?? ?? + ? ?? ? ? ? Absolute Error Apply Trapezoidal rule on [0, 4] ???? ? ? ? 57.6 ? ? Apply Trapezoidal rule on [0, 2] and [2, 4] ???? ???? ? ? ? 16.8 ? ? Apply Trapezoidal rule on [0, 1], [1, 2], [2, 3] and [3, 4] ? ? ? ? ???? ???? ???? ???? ? ? ? ? ? 4.4 ? ? ? ? Remark: The error has been reduced
Sec:4.4 Composite Numerical Integration To generalize this procedure 1) choose an integer n The Trapezoidal Rule ? =? ? ? 2) Subdivide the interval [a, b] into n subintervals ??= ? + ?? ?? ??? ? ? ? ?? =? ? ?? ?? + ? ?? ? = ?,?,?, ,? ? 3) apply Trapezoidal rule on each subintervals [?? 1,??] [?1,?2] [?0,?1] We will apply Trapezoidal rule ? times ?0 ?1 ?? ?2 ?? 1 ? ? ?? ?? + ?(??) ?? ?????? ? ?? ?? ? + ?(??) ?? ?????? ? ? ?? ?? + ?(??) ?? ?????? ? ? ?? = ? ? ??? ??+? ? ?? ? ? ? Appearing twice at interior nodes ? ?? ?????? ? ?? + ? ??+? = ? ? ?? = ? ? ?? ? ?=? ?=? ?=? ? ?? ? ??? ? ? Use Extreme value theorem and Intermediate value theorem =? ?????? ?? ? + ? ? ?? + ?(?) ?=? ?=?
Sec:4.4 Composite Numerical Integration ? ??? Use Intermediate value theorem ? ? ?? ?????? ????? = ????? = ? ? ??? ? ???? ??? ? ?=? ?=? There exists ? ?,? [?,?] Use Extreme value theorem ? ? ? ? ?=? ? [?,?]??? ???? max ? [?,?]??? ???? = ??? m?? ? ? ? ? ? ? ? ? ???? = ???? ? [?,?]??? ???? ? [?,?]??? m?? max ?=? ?=? ?=? ?=? ? ? ???? =? ? ??? ? ? ? ? [?,?]??? ???? ? max ? [?,?]??? ? m?? ?=? ?=? ? ? ????? = ?? ? ? ???? ?? ? [?,?]??? ? ???? max ? [?,?]??? m?? ? ?=? ?=? = ?? ? ? ? ??? ?? ? ? ??? ? ???? ??? ? ?=? ? ? ?? ????? =
Sec:4.4 Composite Numerical Integration The Trapezoidal Rule ? ?? ??? ? ? ? ?? =? ?? ?? + ? ?? ? Theorem Notice that the error term for the Composite Trapezoidal rule is O(h2), whereas it was O(h3) for the standard Trapezoidal rule. However, these rates are not comparable because of different values of h.
Sec:4.4 Composite Numerical Integration Example Simpson s Rule ? ? ? ?? =? Use Simpson s rule to approximate ? ?? ?? + ?? ?? + ?(??) ?? ???? ????? ? ? ? Absolute Error ???? ? Apply Simpson s rule on [0, 4] ? ? ? 3.17143 ? ? Divide into 4 subintervals ???? ???? 0.265 ? ? ? ? ? ? ? Apply 2-Simpson s ? ?/? ?/? Divide into 6 subintervals ???? ???? ???? ?/? ??/? ?/? 0.055 ? ? ?/? ? ?/? ?/? ? Apply 3-Simpson s Remark: we divide into even number of subintervals
Sec:4.4 Composite Numerical Integration To generalize this procedure 1) choose an even integer n Simpson s Rule ? ? =? ? ? ? ?? =? 2) Subdivide the interval [a, b] into n subintervals ??= ? + ?? ?? ?? + ?? ?? + ?(??) ? ? ? = ?,?, ,? ?? ????? 3) apply Simpson s rule on each consecutive pair of subintervals [?? 2,??] [?2,?4] [?0,?2] We will apply Simpson s rule ? times 2 ?0 ?2 ?? ?1 ?4 ?? 2 ?3 ?? 1 ? ? ?? ?? + ?? ?? + ?(??) ?? ?????? ? ?? ?? ? + ?? ?? ? + ?(??) ?? ???? ? ?? ?? + ?? ?? + ?(??) ?? ?????? ? ? ?? = ? ?? ? ?/??? ??? ?/?? ? ?/? ? ??? ?????? ? ??? ? + ?? ??? ? + ? ??? = ?=? ? ? ?? = ? ? ?? Appearing twice ? ?=? ?=? ? ??? ? ?/??? ?/? ? ?/? =? ?????? ?? ? + ? ? ??? + ? ? ??? ? + ?(?) ?=? ?=? ?=?
Sec:4.4 Composite Numerical Integration ?/??? ??? ?/?? ? ?/? ? ??? ?????? ? ??? ? + ?? ??? ? + ? ??? = ?=? ? ? ?? = ? ? ?? Appearing twice ? ?=? ?=? ? ??? ? ?/??? ?/? ? ?/? =? ?????? ?? ? + ? ? ??? + ? ? ??? ? + ?(?) ?=? ?=? ?=? Extreme value theorem and Intermediate value theorem give ?/??? ? ? = ? ? ???????(?) ?????? ? ? = ?=? Theorem Notice that the error term for the Composite Simpson s rule is O(h4), whereas it was O(h5) for the standard Simpson s rule. However, these rates are not comparable because of different values of h. Simpson s Rule ?? ?? + ?? ?? + ?(??) ?? ? ? ? ?? =? ????? ?
Sec:4.4 Composite Numerical Integration Theorem Example ? ? ? ? ???? Use composite Simpson s rule to approximate with n=6 ? ? ?/? ? ??/? ?/? ?/? ? =? ? ? ? ?(? ? ?(?? ? ?(? ? ?(? ? ?(?) ?(?) ?(?) ?) ?) ?) ?) ? ? ?2 ?1 ?4 ?3 ?5 ? ? ? ? ? ? ? 3? ? + 4? ?1 + 2? ?2 + 4? ?3 + 2? ?4 + 4? ?5 + ?(?) ? ? ?? = ?
Sec:4.4 Composite Numerical Integration The subdivision approach can be applied to any of the Newton-Cotes formulas. Such as Midpoint rule The midpoint Rule ? ? ? ?? = ??? ?? +?? ?? ? ? Theorem
Sec:4.4 Composite Numerical Integration Composite Simpson s rule Composite Trapezoidal rule Example (a) 360 (*) Composite Trapezoidal with n=18 requires 19 function evaluations (b) n=18 with 19 function evaluations 1+17+1 1+8+9+1 accurate only to about 5e-3 accurate to within about 1e-5
Sec:4.4 Composite Numerical Integration Remarks: 1) The Newton-Cotes formulas are generally unsuitable for use over large integration intervals. High-degree formulas would be required 2) The values of the coefficients in these formulas are difficult to obtain. 3) The Newton-Cotes formulas are based on interpolatory polynomials that use equally-spaced nodes, a procedure that is inaccurate over large intervals because of the oscillatory nature of high-degree polynomials.
