ChE 250 Numeric Methods

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ChE 250 Numeric Methods

• Lecture 4, Chapra Chapter 4
• 20070124

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Outline

• Chapter 4
• Truncation Error and The Taylor Series
• The Taylor Series
• Numeric Differentiation
• Error Propagation
• Stability and Condition
• Controlling Errors

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Truncation Errors and The Taylor Series

• The Taylor Series
• Any smooth function can be represented by a
  polynomial (at least in the neighborhood)
• Conceptually, the first term is a constant and
  gives a good guess of the value of the function.
• Including the second term gives us a linear (or
  first order) equation

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Truncation Errors and The Taylor Series

• The Taylor Series
• Example 4.1, p. 76
• Replace (xi1-xi) with h like last chapter

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Truncation Errors and The Taylor Series

• The Taylor Series
• The Taylor series will eventually model a
  polynomial perfectly if high enough order
• Exponential and trigonometric functions require
  much more terms, but when staying in the
  neighborhood, and h is small, they can be used
  effectively

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Truncation Errors and The Taylor Series

• The Taylor Series
• Truncation error is the Rn term that is left
  over after a finite number of terms have been
  summed
• To characterize the the truncation error we can
  examine functions with known results and examine
  the relationship between Rn and the parameters n
  and h

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Truncation Errors and The Taylor Series

• The Taylor Series
• Although there is a breakeven point when reducing
  step size, often double precision computing
  reduces the round off error
• Round off error increases with more machine
  computations and is inversely proportional to h

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• The Taylor series leads directly to the finite
  difference method we used in Chapter 1, and we
  can now elaborate on that method
• O(h) is the truncation error which is log
  proportional to h, Figure 4.5

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• forward finite difference
• Backward finite difference
• Centered finite difference

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• Error Propagation
• First expand the function into a Taylor series of
  all the variables

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• Error Propagation
• Total error can be estimated in a function of any
  number of variables by taking a first order
  Taylor expansion

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• Error Propagation
• Total error can be estimated in a function of any
  number of variables by taking a first order
  Taylor expansion
• And defining the error in terms of the difference
  between the measurement (approximation) and the
  true value.

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• Error Propagation
• Stability and Condition
• The condition of a problem is a measure of its
  sensitivity to changes in its input values
• The condition is derived from the true error and
  the relative error
• Low condition number (lt1) is good ?
• High number (gt1) is bad ?

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Truncation Errors and The Taylor Series

• The Taylor Series
• Numeric Differentiation
• Error Propagation
• Stability and Condition
• Controlling Errors
• Numeric Error
• No systematic approaches?
• Avoid subtracting similar numbers
• Rearrange formulas to preserve significant
  figures
• Use extended precision if necessary
• Calculate condition number and other error
  estimations
• Compare solution with those derived for simpler
  models
• Modeling Errors
• Mistakes!
• Watch Fundamentals! Always check for
  conservation laws
• Formulation
• Using the right simplification
• Sometimes different equation are used over ranges
  of the variable

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Preparation for Jan 26th

• Reading
• Chapra
• Part II Orientation
• Chapter 5 Bracketing Methods
• Homework
• Homework set 2 due on Wednesday 31Jan.
• 4.2, 4.8, 4.12, 4.19
• 5.1, 5.3, 5.5, 5.9, 5.13, 5.17