NUMERICAL DIFFERENTIATION

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NUMERICAL DIFFERENTIATION
The derivative of f (x) at x0 is
An approximation to this is
for small values of h.
Forward Difference Formula
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Find an approximate value for
The exact value of
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Assume that a function goes through three points
Lagrange Interpolating Polynomial
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If the points are equally spaced, i.e.,
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Three-point formula
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If the points are equally spaced with x0 in the
middle
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Another Three-point formula
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Alternate approach (Error estimate)
Take Taylor series expansion of f(xh) about x
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2 X Eqn. (1) Eqn. (2)
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The Second Three-point Formula
Take Taylor series expansion of f(xh) about x
Take Taylor series expansion of f(x-h) about x
Subtract one expression from another
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Summary of Errors
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Summary of Errors continued
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Example
Find the approximate value of
with
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Using the Forward Difference formula
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Using the 1st Three-point formula
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Using the 2nd Three-point formula
The exact value of
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Comparison of the results with h 0.1
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Second-order Derivative
Add these two equations.
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NUMERICAL INTEGRATION
area under the curve f(x) between
In many cases a mathematical expression for f(x)
is unknown and in some cases even if f(x) is
known its complex form makes it difficult to
perform the integration.
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Area of the trapezoid
The length of the two parallel sides of the
trapezoid are f(a) and f(b) The height is b-a
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Simpsons Rule
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Composite Numerical Integration
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Riemann Sum
The area under the curve is subdivided into n
subintervals. Each subinterval is treated as a
rectangle. The area of all subintervals are
added to determine the area under the curve.
There are several variations of Riemann sum as
applied to composite integration.
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In Left Riemann sum, the left-side sample of the
function is used as the height of the individual
rectangle.
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In Right Riemann sum, the right-side sample of
the function is used as the height of the
individual rectangle.
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In the Midpoint Rule, the sample at the middle of
the subinterval is used as the height of the
individual rectangle.
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Composite Trapezoidal Rule Divide the interval
into n subintervals and apply Trapezoidal Rule in
each subinterval.
where
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Find
by dividing the interval into 20 subintervals.
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Composite Simpsons Rule Divide the interval
into n subintervals and apply Simpsons Rule on
each consecutive pair of subinterval. Note that
n must be even.
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where
Find
by dividing the interval into 20 subintervals.
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