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Published byMagda Fábián Modified over 5 years ago
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Splines There are cases where polynomial interpolation is bad overshoot oscillations
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Example Interpolation at -4,-3,-2,-1,0,1,2,3,4
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Idea behind splines use lower order polynomials to connect subsets of data points make connections between adjacent splines smooth because lower order polynomials, avoid oscillations and overshoots
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We will cover linear splines quadratic splines cubic splines - most common Autocad uses splines extensively
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Linear splines - connect each two points with straight line functions connecting each pair of points are m is slope between points
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Linear splines are exactly the same as linear interpolation!
Example:
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Problem with linear splines - not smooth at data points (or knots)
first derivative is not continuous use higher order splines to get continuous derivatives - equate derivatives at neighboring splines
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Quadratic splines - continuous first derivatives
can have discontinuous second and higher derivatives Derive second order polynomial between each pair of points For n+1 points (i=0,1,…,n) - n intervals - 3n unknown parameters (a’s, b’s, and c’s) Need 3n equations
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Equations for interior knots (data points), equations on either side must equal knot at knot n-1 interior knots 2n-2 interior equations
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the first and last functions must pass through the end points
2 more equations: total is 2n so far
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first derivatives at interior nodes must be equal
Another n-1 equations for a total of 3n-1 equations We need 3n equations. What is the last equation?
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Last equation is arbitrary
We use assume second derivative is 0 at first point Connect first two points by a straight line
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Example: n=3 9 unknowns Set up interior equations (4 of them)
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Set up end-point equations (2 of them)
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Set up first derivative equations at interior knots (2 of them)
Final arbitrary equation
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The 9 equations are
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Rearrange, and use a1=0 to eliminate
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Put in matrix form
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Solve by any matrix method
Get
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Result
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Cubic splines avoid the straight line and the over-swing
Can develop method like we did for quadratic - 4n unknowns - 4n equations interior knot equality end point fixed interior knot first derivative equality assume derivative value if needed
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Can do a set of algrebraic manipulations (which book shows) to reduce 4n equations to n-1 equations
Unknowns - second derivatives
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Second derivatives are evaluated using
with end conditions Becomes a tridiagonal system
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Method outline Set up equations for second derivatives get second dertivatives and plug into spline equation for each section
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Example: Note that n is 3 - we will not have a true tridiagonal setup Get second derivatives with end conditions
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Write the equations as
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Now plug second derivative values into cubic spline equation
This gets messy, so look at Matlab code
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