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Today’s class
• Spline Interpolation• Quadratic Spline• Cubic Spline
• Fourier Approximation
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
1
Lagrange & Newton Interpolation
• Noticing that the function (black line) has a sharp or sudden change at x = 0.
• Polynomial interpolations work poorly.
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
2
Spline Interpolation
• Spline interpolation applies low-order polynomial to connect two neighboring points and uses it to interpolate between them.
• Typical Spline functions
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
3
Linear Splines
• Use straight lines to connect two neighboring points
Shortcomings: Sharp angle at
connections, or not smooth.
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
4
Linear Splines• Use either Lagrange or Newton interpolations to
determine the equations for the straight lines
• To find y5 at x5, first find which interval x5 is in and then use the linear Spline in that region to calculate y5.
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
5
Quadratic Spline Function• Each two neighboring points are connected
by a 2nd-order (quadratic) polynomial.
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
6
Quadratic Splines
• If number of points is n+1, there are two end points and n-1 interior points. The number of intervals is n.• Since each interval has one quadratic polynomial, there are 3n unknown coefficients (ai, bi & ci ) to be determined.
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
7
Conditions Used to Determine Coefficients• At each interior point, the two neighboring
quadratic polynomials have to pass this point, resulting in 2(n-1) equations
• The first and last quadratics must pass through the end points resulting in 2 more equations.
• At each interior point, the first-order derivatives of the two neighboring polynomials are equal, resulting in (n-1) equations.
• The last equation is obtained by letting the second-order derivative of the first polynomial equal zero (totally arbitrary and may be changed).
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
8
Equations Used to Determine Coefficients
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
9
Quadratic Splines
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
10
Cubic Spline Function• Each two neighboring points are connected or
interpolated by a 3rd-order (Cubic) polynomial.
• If # of points is n+1, then there are two end points and n-1 interior points. # of intervals is n.
• Each interval has a cubic polynomial. There are totally 4n unknown coefficients (ai, bi, ci & di) .
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
11
Conditions Used to Determine Coefficients• At each interior point, the two neighboring cubic
polynomials have to pass this point, resulting in 2(n-1) equations
• Only one cubic polynomial to pass an end point, resulting in 2 equations
• At each interior point, the first-order & second-order derivatives of the two neighboring polynomials are equal, resulting in 2(n-1) equations.
• There are totally 4n-2 equations, two more additional equations are needed by letting the second-order derivatives of the first and last polynomials equal zero.
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
12
Equations Used to Determine Coefficients
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
13
• Second
Cubic Spline Functions• Second derivative is a line • Lagrange interpolating polynomial for
second derivative
• Integrate twice to get fi(x)
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
14
Cubic Spline Functions
• Two constants can be evaluated by applying interval end conditions
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
15
Cubic Spline Functions
• First derivatives at knots must be equal
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
16
at xi
Cubic Spline Functions• Rearranging terms we get the following
relationship
• For all n-1 interior knots, this gives us n-1 equation with n-1 unknowns – the second derivatives
• Once we solve for the second derivatives, we can plug it into the previous equations to solve for the splines
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
17
Cubic Spline Functions
• Example: (3,2.5), (4.5,1), (7,2.5), (9,0.5)• At x=x1=4.5
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
18
Cubic Spline Functions
• Example: (3,2.5), (4.5,1), (7,2.5), (9,0.5)• At x=x2=7
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
19
Cubic Spline Equations
• Solve the system of equations to find the second derivatives
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
20
Cubic Spline Equations
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
21
Cubic Spline Equations
• Substituting for other intervals
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
22
Cubic Splines
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
23
Fourier Approximation
• What if the curve is periodic• Use a sinusoidal function as the least-
squares model
• Select coefficients to minimize least-squares sum
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
24
Least-Squares Approximation of Sinusoidal Functions
• Special case when the data points are spaced at equal intervals of Δt over one period
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
25
Fourier Series• Any periodic function can be represented
by a series of sinusoids of multiples of a common harmonic frequency
[ ]
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
26
Fourier Series
• Example
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
27
Fourier Series
• Example
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
28
Fourier Series
• Example
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
29
Fourier Series
• Example
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
30
Fourier Series
• Example
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
31
Fourier Series
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
32
Next class
• Review
Numerical MethodsLecture 21
Prof. Jinbo BiCSE, UConn
33