Numerical Methods,

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Todays class

• Numerical differentiation
• Roots of equation
• Bracketing methods

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Numerical Differentiation

• Finite divided difference
• First forward difference
• First backward difference

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Numerical Differentiation

• Centered difference approximation
• Subtract the two equations

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Numerical Differentiation

• First forward difference

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Numerical Differentiation

• First backward difference

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Numerical Differentiation

• Centered difference

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Error Propagation

• What is the effect of error in one calculation
  propagating to subsequent calculations?
• Example
• Multiplying sin x with cos x
• Single variable functions

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Error Propagation

• Use Taylor series

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Error Propagation
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Error Propagation

• Multivariable functions

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Numerical stability

• Condition of a problem is a measure of its
  sensitivity to changes in input values
• The condition number is defined as the ratio of
  the relative function error to the relative value
  error

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Numerical stability

• Condition number lt 1 indicates a well-conditioned
  function i.e. changes in the input are
  attenuated
• Condition number gt 1 indicates a ill-conditioned
  function i.e. changes in the input are amplified

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Roots of equation

• Given a function f(x), the roots are those values
  of x that satisfy the relation f(x) 0
• Example
• From the quadratic formula, the roots are

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Roots of Equations

• The need to solve for roots show up in many
  engineering problems
• Also, can be used to find solutions to implicit
  variables

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Example

• Find a value of R such that current is 5A at t
  1s

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Example

• It is not possible to isolate R to the left side
  and thus solve for R
• R is know as an implicit variable
• Rewrite the function as a function of R set to 0

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Roots of equations

• Still need a method to solve for this root
• Other examples of difficult to solve roots

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Roots of equations

• Non-computer methods
• Graphical methods

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Graphical methods

• Not exact
• Can give you a rough estimate of the root,
• Can give you insights on the number of roots and
  shape of the curve
• Can use the rough estimate in more precise
  numerical methods

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Graphical methods

• Use to get an initial estimate of the root and
  also to find out how many roots there are

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Graphical methods
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Graphical methods
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Graphical methods
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Roots of equation

• Non-computer/numerical method
• Exhaustive search method
• To find the root in the interval a,b, start at
  xa and check if f(a) 0, then try f(a?),
  f(a2?), and so on, until we get f(x)
  sufficiently close to 0
• If the step value ? is sufficiently small we can
  obtain an accurate result but this could take an
  extremely long time. For example, if the interval
  is 0,10 and the step size is ? 0.001, it will
  take on average 10,000 guesses
• In addition to the inefficiency of this approach,
  if f(x) is a steep function, this approach may
  not produce an accurate results

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Exhaustive search

• Example
• Find the root of the function
• Actual root is at x1.0001
• With an interval of 0.9, 1.1 and a step size of
  ? 0.001. The exhaustive search method will test
  f(1.000) -0.01 and f(1.001) 0.086, neither of
  which are that close to f(x) 0

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Roots of equations

• More systematic methods are required
• Bracketing methods
• Open methods

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Incremental search methods

• Locate an interval where sign changes
• Divide interval into smaller subintervals which
  are then searched for sign changes
• Keep repeating until root is found with
  sufficient confidence

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Bisection method

• Also called
• Binary chopping
• Interval halving
• An incremental search method where the interval
  is cut in half

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Bisection method

• Step 1
• Choose lower xl and upper xu such that the
  function changes sign over that range i.e.
  f(xl) and f(xu) are different signs or f(xl)
  f(xu) lt 0
• Step 2
• Estimate root to be xr(xlxu)/2

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Bisection method

• Step 3
• Determine in which subinterval the root lies
• If f(xr)?0 is within acceptable tolerance, stop
  and root equals xr
• If f(xl) f(xr) lt 0, then root is in lower
  subinterval. Set xu xr, and return to step 2
• If f(xl) f(xr) gt 0, then root is in upper
  subinterval. Set xl xr, and return to step 2

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Bisection method

• Termination criteria
• Use approximate relative error calculation to
  determine when to stop
• In general, ?a is larger than ?t

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Bisection method

• Example
• Use range of 202204
• Root is in upper subinterval

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Bisection method
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Bisection method

• Use range of 203203.5
• Root is in upper subinterval

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Error estimates

• The approximate error is upper bound estimate of
  the true error
• When the root is near one of the ends of the
  interval, the approximate error is fairly close
  to the actual true error
• Error is fairly well-contained

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Error estimates

• You always know that the true root is within ?x/2
  of your estimate

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Bisection method
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Bisection method

• You can calculate an error estimate based just on
  the initial guesses
• You can also make estimates on the error on
  future iterations
• Superscripts indicates the iteration number

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Bisection method

• Each subsequent iteration cuts the approximate
  error in half
• This, allows to determine a priori exactly how
  many iterations are needed to arrive at the
  desired error

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False Position Method

• The false position method works in a similar
  fashion to the bisection method
• Start with an initial interval a,b where f(a)
  and f(b) have opposite signs, which is the same
  as the bisection
• Instead of choosing the initial guess xr as the
  midpoint of the interval, we join the point
  a,f(a) and b,f(b) with a straight line and
  choose xr as the point where that straight line
  crosses the x-axis.

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False Position Method
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False Position Method

• Algorithm is the same as bisection method with
  the same three steps

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False Position Method

• Step 1
• Choose lower xl and upper xu such that the
  function changes sign over that range - i.e.
  f(xl) and f(xu) are different signs - or f(xl)
  f(xu) lt 0
• Step 2
• Estimate new root to be

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False Position Method

• Step 3
• Determine in which subinterval the root lies
• If f(xr) ? 0 is within acceptable tolerance, stop
  and root equals xr
• If f(xl) f(xr) lt 0, then root is in lower
  subinterval. Set xu xr, and return to step 2
• If f(xl) f(xr) gt 0, then root is in upper
  subinterval. Set xl xr, and return to step 2

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Next class

• Roots of equations
• Open methods
• Read chapters 5 and 6
• HW2, due 9/17
• Chapra Canale
• 6th edition 3.5, 3.7, 3.13, 4.5, 4.6, 4.12 (b)
  and (d), and 4.16