PHYS100 Lecture 16 Numerical Differentiation 110309

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PHYS100Lecture 16Numerical Differentiation11/03
/09
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• 5) Numerical Differentiation (Kuisalaas,
  p.182)
• Given a function, y f(x), one must determine
  derivative, df(x)/dx, at x xk
• Given means
• a) Have algorithm for computing f(x)
• Used for checking algorithm
• b) Possess set of data points xi, yi i
  1n
• Primary use

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• Numerical differentiation related to
  interpolation
• Subject to trade off/balance between round off
  error,
• created by 16-digit machine precision, and
  approximation error
• Usually error of numerical derivative
  significantly greater than eps
• Primary tool ? Taylor series approximation

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• 5.2) Finite Difference Approximation

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• Continuing
• Consider grid of function values given n points
  x1 .. xn
• Central differences use function values on
  both sides of any x
• Thus central differences cannot be employed at
  x1 and xn

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• Eq. a ?
• Eq. b ?
• Truncation error is O(h) -

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• Eq. c minus 4 times Eq. a ?
• Additional algebra produces

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• Example 2nd Order df/dx
• Uniform grid x1 to xn
• forward
• df_dx(1)(-f(x(3)) 4f(x(2)) -3f(x(1)) )/(2h)
• central
• for k 2n 1
• df_dx(k) (f(x(k 1)) - f(x(k -
  1)))/(2h)
• end
• backward
• df_dx(n)(f(x(n-2))-4f(x(n-1))
  3f(x(n)))/(2h)

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• gtgt h pi/200
• gtgt x (0hpi)'
• gtgt df_dx first_derivative(_at_sin, x)
• gtgt plot(x,abs(df_dx -cos(x)),'r')

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• Investigate trade off between truncation and
  round off
• FD1 Minimum error of 1.378e-010 at h
  8.315e-009

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• CD2 Minimum error of 5.126e-013 at h
  1.170e-006

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• CD4 Minimum error of 1.188e-014 at h
  9.682e-004

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• CD6 Minimum error of 3.331e-016 at h
  8.869e-003

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• CD8 Minimum error of 1.110e-016 at h
  1.748e-003

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• 5.3) Richardson Extrapolation (Kuisalaas
  p. 188)
• Simple, elegant method for improving
  finite-difference accuracies
• exact value represented by G
• step-size dependent approximation represented
  by g(h)
• step-size dependent error by E(h) and E(h)
  form is
• E(h) chp with c and p constant
• Thus
• G g(h) E(h)
• Start with h h1

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• Central difference error -
• E(h) O(h2) c1h2 c2h4 c3h6
• Eliminate dominant error with p 2
• Example of Richardson extrapolation
• Approximate 1st derivative of e-x at x 1
• 2nd Order central difference

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• gtgt df_dxc _at_(f, x, h) (f(x h) - f(x -
  h))./(2h)
• gtgt f _at_(x) exp(-x)
• gtgt h1 0.1 h2 h1/2
• gtgt x 1.0
• gtgt gh1 df_dxc(g, x, h1)
• gtgt gh2 df_dxc(g, x, h2)
• gtgt G (4gh2 - gh1)/3
• G -3.6788e-001
• gtgt exact -f(x)
• exact -3.6788e-001
• gtgt error G - exact
• error 7.6664e-008
• gtgt error1 gh1 - exact
• error1 -6.1344e-004
• gtgt error2 gh2 - exact
• error2 -1.5330e-004