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

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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

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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

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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

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Quadratic Spline Function• Each two neighboring points are connected

by a 2nd-order (quadratic) polynomial.

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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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

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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

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Equations Used to Determine Coefficients

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Quadratic Splines

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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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

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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

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Equations Used to Determine Coefficients

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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• 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

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Cubic Spline Functions

• Two constants can be evaluated by applying interval end conditions

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Cubic Spline Functions

• First derivatives at knots must be equal

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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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

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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

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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

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Cubic Spline Equations

• Solve the system of equations to find the second derivatives

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Cubic Spline Equations

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Cubic Spline Equations

• Substituting for other intervals

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Cubic Splines

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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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

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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

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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

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Fourier Series

• Example

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Fourier Series

• Example

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Fourier Series

• Example

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Fourier Series

• Example

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Fourier Series

• Example

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Fourier Series

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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Next class

• Review

Numerical MethodsLecture 21

Prof. Jinbo BiCSE, UConn

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