Gauri nanda clocky

MATH466/462 Project 4. Due in class on Wed, Mar 18, 2020 Instruction: your project report should include necessary mathematical justification, description, and details of your algorithms/conclusions. Your MATLAB codes and generated outputs may be attached in the end of the report. Make sure you addressed all the questions in each problem. Both the report and codes will be graded. Please submit a printed hard-copy. Problem A (20 pts): Circulant Preconditioners for Toeplitz Systems Toeplitz matrix arises in many different applications. Its matrix-vector product can be computed efficiently via fast Fourier transform (FFT). Many circulant preconditioners have been proposed for solving Toeplitz systems. In this project, we will compare several circulant preconditioners for solving various Toeplitz systems. Task 1: Study Toeplitz matrix, Circulant matrix, and the use of FFT (read our Textbook Chap 10.1). Read Chapters 1 and 3: https://ee.stanford.edu/~gray/toeplitz.pdf Understand more on FFT: https://arxiv.org/pdf/1805.05533v2.pdf Task 2: play and understand the following codes: For any Circulant matrix 𝐶 , the matrix-vector products 𝑥 = 𝐶−1𝑏 can also be computed via FFT:

1 n=10; C=gallerygallerygallery('circul',(1:n));%Construct a Circulant matrix using (1:n) as 1st row 2 b=ones(n,1); x2=C\b; %Direct solve: O(n^3) operations 3 ev=fftfftfft(C(:,1));%the eigenvalues of C by FFT of its first column 4 x1= ifftifftifft(fftfftfft(b)./ev); %Solve C\b using FFT: O(n log n) operations 5 normnormnorm(x1-x2,inf) %should be zero

By embedding a Toeplitz matrix 𝑇 into a Circulant matrix, the product 𝑇𝑣 can also be computed via FFT:

1 n=10;t=(n:-1:1); T=toeplitztoeplitztoeplitz(t,t'); %construct a full symmetric Toeplitz matrix: T'=T 2 v=randrandrand(n,1);y2=T*v;%compute y2=T*v using direct multiplication: O(n^2) operations 3 gev = fftfftfft([t 0 t(n:-1:2)].');%the eigenvalues of the embeding larger Circulant matrix 4 y = ifftifftifft(fftfftfft([v;zeroszeroszeros(n,1)]).*gev);%compute y1=T*v using FFT: only O(n log n) operations 5 y1 = y(1:n); %take the first half of the long vector 6 normnormnorm(y1-y2,inf) %should be close to zero

Task 3: understand the construction of 3 circulant preconditioners. Let 𝑇𝑛 be an 𝑛-by-𝑛 Toeplitz matrix with 𝑇𝑛 (𝑖, 𝑗) = 𝑡𝑖−𝑗 , where {𝑡𝑘 }𝑛−1𝑘=1−𝑛 are given diagonals. We can define at least 3 different circulant preconditioners as follows:

1. Strang’s Preconditioner: Strang’s preconditioner 𝑆𝑛 with 𝑆𝑛 (𝑖, 𝑗) = 𝑠𝑖−𝑗 is defined to be the circulant ma- trix obtained by copying the central diagonals of 𝑇𝑛 and bringing them around to complete the circulant requirement. Assume 𝑛 = 2𝑚 is even, the diagonals 𝑠𝑘 of 𝑆𝑛 are given by

𝑠𝑘 =

 𝑡𝑘 if 0 ≤ 𝑘 ≤ 𝑚 − 1 0 if 𝑘 =𝑚 𝑡𝑘−𝑛 if𝑚 < 𝑘 ≤ 𝑛 − 1 𝑠−𝑘