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Truncation Errors and the Taylor Series Chapter 4

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Truncation Errors and the Taylor Series Chapter 4. How does a CPU compute the following functions for a specific x value? cos (x) sin(x) e x log(x) etc.

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How does a CPU compute the following functions for a specific x value?

cos(x) sin(x) ex log(x)etc.

  • Non-elementary functions such as trigonometric, exponential, and others are expressed in an approximate fashion using Taylor series when their values, derivatives, and integrals are computed.
  • Taylor series provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point.
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Taylor Series (nth order approximation):

The Reminder term, Rn, accounts for all terms from (n+1) to infinity.

Define the step size as h=(xi+1- xi), the series becomes:

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Any smooth function can be approximated as a polynomial.

Take x = xi+1 Then f(x) ≈ f(xi) zero orderapproximation

first orderapproximation

Second orderapproximation:

nth orderapproximation:

  • Each additional term will contribute some improvement to the approximation. Only if an infinite number of terms are added will the series yield an exact result.
  • In most cases, only a few terms will result in an approximation that is close enough to the true value for practical purposes
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ExampleApproximate the function f(x) = 1.2 - 0.25x- 0.5x2 - 0.15x3 - 0.1x4 from xi = 0 with h = 1 and predictf(x) at xi+1 = 1.

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Example:

computing f(x) = ex using Taylor Series expansion

Choose x = xi+1and xi = 0 Then f(xi+1) = f(x) and(xi+1 – xi)= x

Since First Derivative of ex is also ex :

(2.) (ex )” = ex (3.)(ex)”’ = ex, … (nth.)(ex)(n) = ex

As a result we get:

Looks familiar?

Maclaurin series for ex

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Yet another example:

computing f(x) = cos(x) using Taylor Series expansion

Choose x=xi+1 and xi=0 Then f(xi+1) = f(x) and (xi+1 – xi) = x

Derivatives of cos(x):

(1.) (cos(x))’ = -sin(x)(2.) (cos(x) )” = -cos(x),

(3.) (cos(x) )”’ = sin(x) (4.) (cos(x) )”” = cos(x),

……

As a result we get:

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Notes on Taylor expansion:
  • Each additional term will contribute some improvement to the approximation. Only if an infinite number of terms are added will the series yield an exact result.
  • In most cases, several terms will result in an approximation that is close enough to the true value for practical purposes
  • Reminder value R represents the truncation error
  • The order of truncation error is hn+1 R=O(hn+1),

If R=O(h), halving the step size will halve the error.

If R=O(h2), halving the step size will quarter the error.

error propagation
Error Propagation
  • Let xfl refer to the floating point representation of the real number x.
  • Since computer has fixed word length, there is a difference between x and xfl (round-off error)

and we would like to estimate the error in the calculation of f(x) :

  • Both x and f(x) are unknown.
  • If xfl is close to x, then we can use first order Taylor expansion and compute:

Result: If f’(xfl) and Dx are known, then we can estimate the error using this formula

Solve from Example 4.5 p.95