CSE 541 Numerical Methods

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CSE 541Numerical Methods

• Integration

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A Very Old Idea

• Archimedes circa 285 - 212 B.C.
• Area of a circle
• Approximate with a triangle
• We know the area of a triangle
• So, keep subdividing

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Definition Quadrature

• Quadrature Rules
• 1. The process of making something square. 2.
  Mathematics The process of constructing a square
  equal in area to a given surface. 3. Astronomy A
  configuration in which the position of one
  celestial body is 90 from another celestial
  body, as measured from a third.
• The American Heritage Dictionary Fourth
  Edition.  2000

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Calculus

• Formal definition
• Basic definition of a definite integral

f (x)
where
sum of height ? width
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Calculate the Integral

• Let be the
  antiderivative of a continuous function f (x),
  where
• Fundamental Theorem of Calculus
• There is a relationship between integration and
  differentiation
• Solve the definite integral with the
  antiderivative

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Why Numerical Integration?

• Some functions do not have antiderivatives
• An analytical solution is complicated
• Integrand is not precisely defined by an
  equation, i.e., we are given a set of data (xi,
  yi), i 1, 2, 3, ,n
• Remember, the function f (x) is a black box

Yuck! Quick, someone give me a symbolic solver
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Reimann Integral Theorem

• All numerical approximations can be represented
  by
• where wi are the weights, xi are the sampling
  points, and Et is the truncation error
• Integration is a summing process
• The fcn is continuous on the closed and bounded
  interval of integration

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Partitioning the Integral

• The most common numerical integration formula is
  based on equally spaced data points
• Divide x0 , xn into n intervals (n ? 1)

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Riemann Integration

• Take a continuous function and partition into
  intervals
• What height should we choose?

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Upper Sums

• WLOG, assume that f (x) gt 0 everywhere
• If within each interval, we could determine the
  maximum value of the function, then we have
• where

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Upper Sums
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Lower Sums

• If within each interval, we could determine the
  minimum value of the function, then we have
• where

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Lower Sums
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Finer Partitions

• Define a bound on the integral for some partition
  (x0 , , xn)
• Riemann-integrable functions
• As n ? ?, the sum of the upper bounds and the sum
  of the lower bounds approach each other

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Bounding the Integral
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Bounding the Integral
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Bounding the Integral
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Monotonic Functions

• Special case Monotonic functions
• If a function is monotonically increasing (or
  decreasing), then
• Lower sum corresponds to the left partition
  values
• Upper sum corresponds to the right partition
  values

Choose?