MA3232 Summary 5. d y1 dy1. MATLAB has a number of built-in functions for solving stiff systems of ODEs. There are ode15s, ode23s, ode23t, ode23tb.

Transcription

1 umerical solutions of higher order ODE We can convert a high order ODE into a system of first order ODEs and then apply RK method to solve it. Stiff ODEs Stiffness is a special problem that can arise in the solution of ordinary differential equations. A stiff system is one involving rapidly changing components together with slowly changing ones. In some cases, the rapidly varying components ephemeral transients that die away quickly, after which the solution becomes dominated by the slowly varying components. Although the transient phenomena exist for only a short part of the integration interval, they can dictate the time step for the entire solution. d y dy y A typical example is the van der Pol equation µ ( y ) + = 0, with initial 0 dy 0 = =. When the parameter µ becomes very large, the system is conditions y stiff. MATLAB has a number of built-in functions for solving stiff systems of ODEs. There are ode5s, ode3s, ode3t, ode3tb. Fitting functions to data Two types of fittings: ) Fitting to exact data --- polynomial fitting and cubic spline fitting Practically, polynomial fitting is not a very good tool since the fitting polynomial has large oscillations (Gibbs phenomenon). Cubic spline fitting is a good tool (in Matlab, the built-in function spline does the job for us). ) Fitting to data with noise --- least square fitting (regression model) Given data points: ( x i, y i ), i =,,, y i = Measured value f x i + ε i Exact value oise The least square fitting construct functions of the form g x where g = b g ( x) + b g ( x) + + b p g p ( x) ( x), g ( x),, g p ( x) are given functions and solving G T G b = G T y where T is determined by b = b, b,, b p

2 G = ( g, g,, g p ), g k = g k ( x ), g k ( x ),, g k x y = ( y, y,, ) T. ( T, k =,,, p, Fourier transform If f( t) is periodic with period T. Then = c k exp iπ k t f t k = T T where c k = f ( t)exp iπ k t T T. c k 0 Motivation (applications of Fourier transform): oise removal Data compression Solving PDE Discrete form of the Fourier coefficients: c k = f j exp i π k j Due to aliasing effect, Definition: Sampling rate = Definition: { } are called Fourier coefficients. c k +m = c k for all integer values of m. # of data points time = T yquist frequency = Largest frequency that can be resolved in the sampling = T With sampling rate T, a frequency of k T appears as a frequency of k T m T where T < k T m and m is an integer. T T Sampling rate Discrete Fourier coefficients yquist frequency c k, < k are called discrete Fourier coefficients

3 Discrete Fourier transform DFT (discrete Fourier transform) DFT: F = { f j, j = 0,,, } Y =, k = 0,,, = f j exp i π k j, k = 0,,, ote: DFT is implemented by Matlab built-in function fft. IDFT (inverse discrete Fourier transform) IDFT: f j = { } Y =, k = 0,,, F = exp iπ k j, j = 0,,, k = 0 ote: IDFT is implemented by Matlab built-in function ifft. { } { f j, j = 0,,, } When = p, DFT and IDFT can be calculated very efficiently. Theorem: That is, f j = f j, IDFT DFT = Identity j = 0,,, { } to Relating, k = 0,,, Recall y 0 = c 0 y = c = c k = c + = c + + = c + f j exp i π k j c k, < k = c ( y 0, y,, )= c 0, c,, c, c +,, c, c

4 FFT (Fast Fourier transform) FFT is an efficient way of calculating DFT. Assume = p. DFT: F = { f j, j = 0,,, } Y =, k = 0,,, k j = f j ω, k = 0,,, where ω = exp iπ { } Cost of straightforward calculation = O( ). Let us look at a recursive way of calculating Y = DFT( F). Theorem: Consider DFT: F Y Let F = f 0, f,, f F = f, f 3,, f Y = y 0, y,, =, +,, y Y Then Y = DFT( F )+ V.* DFT F Y = DFT( F ) V.* DFT F where ( ) ( ) V = ω 0, ω,, ω. Let C be the cost of the recursive algorithm on a vector of size. It satisfies C = 0 C ( ) = C + 3 = = C C log ( ) log (a recursive equation)

5 3 C log ( ) = Consider the function g = g g g = 0 g = 0 = C( ) C 3 log 3 log 3 log ( ). We have = 0 = 3 log ( )