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**Introduction to Numerical Analysis I**

MATH/CMPSC 455 Introduction to Numerical Analysis I Numerical Integration

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**Numerical Integration**

Mathematical Problem: Example: Example:

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**By calculus, find that , then use**

Numerical Integration: replace by another function that approximates well and is easily integral, then we have

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**Newton-Cotes Formulas**

Idea: use polynomial interpolation to find the approximation function Step 1: Select nodes in [a,b] Step 2: Use Lagrange form of polynomial interpolation to find the approximation function Step 3:

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Trapezoid rule Use two nodes: and

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Simpson’s rule Use three nodes:

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**Example: Apply the Trapezoid Rule and Simpson’s Rule to approximate**

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**Error of the trapezoid rule:**

The trapezoid rule is exact for all polynomial of degree less than or equal to 1.

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**Error of the Simpson’s rule:**

The Simpson’s rule is exact for all polynomial of degree less than or equal to 3.

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**The Composite Trapezoid Rule**

Why? ? The high order polynomial interpolations are unbounded! Step 1: Partition the interval into n subintervals by introducing points Step 2: Use the trapezoid rule on each subinterval Step 3: Sum over all subintervals

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**The Composite Simpson’s Rule**

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**Error of Composite Rules**

Error of the composite trapezoid rule: Error of the composite Simpson’s rule:

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**Example: Apply the composite Trapezoid Rule and Simpson’s Rule ( 4 subintervals ) to approximate**

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