Mathematical Modeling

1. Mathematical Modeling Team 1 & 7 Presentation : JiHoon Lee Special Linear Systems -Tridiagonal system

2. Contents Preliminary - Tridiagonal system solver - The Economical Storage scheme Problem & Result - Compute x for Ax = b by using Tridiagonal system solver - Find an economical storage scheme Summary Moreover Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M

3. Preliminary -Tridiagonal system solver Let T be a tridiagonal matrix When the LU factorization of T exists (T=LU) L and U have very simple structures. Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M • : Algorithm for the LU factorization of T • and L, U

4. Preliminary After we find L and U with just 2n-2 flops, we can solve Tx = b by solving two special bidiagonal systems Ly = b and Ux = y It needs only 4n-4 flops to compute x This is called Tridiagonal system solver Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M

5. Preliminary -The Economical Storage scheme Let T be a n by n tridiagonal matrix. Since T is a n by n matrix, we need space to store it. However, if LU factorization of T exists, by storing l2~ln, u1~unand b1~bn-1, We can need just 3n storage space Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M

6. Problem A is a tridiagonal matrix (from gallery) Choose b so that Ax=b has solution vector x with all elements are 1 For n=50, 100, compute x (use tridiagonal system solver w/o pivoting) 4. Find economical storage scheme (Do not store or operate on entries outside the bands) Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M

7. Codes Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M Tridiagonal system solver function [x e d_b u l]=tridsol(n) A=gallery('tridiag',n); d_p=diag(A(2:n,1:n-1)); d_m=diag(A); d_b=diag(A,1); u(1)=d_m(1); % d_p : vector of subdiagonal entries of A % d_m : vector of main diagonal entries of A % d_b : vector of superdiagonal entries of A fori=2:n l(i-1)=(d_p(i-1))/u(i-1); u(i)=d_m(i)-l(i-1)*d_b(i-1); if (u(i)==0) ifi==n break; elsedisp('Itcan not be solved by the tridiagonal system solver.') pause end; end; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% b(1)=d_m(1)+d_b(1); b(n)=d_p(n-1)+d_m(n); for j=2:n-1 b(j)=d_p(j-1)+d_m(j)+d_b(j); end % b=A*x where x=ones(n,1) y(1)=b(1); for k=2:n y(k)=b(k)-l(k-1)*y(k-1); end % L*y=b x(n)=y(n)/u(n); for f=1:n-1 h=n-f; x(h)=(y(h)-d_b(h)*x(h+1))/u(h); end % U*x=y e=norm(ones(n,1)-x'); % error

8. Codes Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M Original Matrix Loading each entry function T=Ori(d_b,u,l) n=length(u); T = zeros(n); T(1,1)=u(1); fori = 2:n; T(i,i-1) = l(i-1) * u(i-1); T(i,i) = u(i) + l(i-1)*d_b(i-1); T(i-1,i) = d_b(i-1); End function a = sto(d_b,u,l,x,y) if x ==1 if x==y a = u(1); elseif y==x+1 a = d_b(1); else a= 0; end; end; if x>=2 if x==y, a= u(x) + l(x-1)*d_b(x-1); elseif x==y+1 a= l(x-1)*u(x-1); elseif y==x+1 a = d_b(x); else a=0; end; end;

9. Result (n=50) Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M x = [ 1 1 1 1 1 1 ÿ 1 1 1 1 1 1 ] : 1x50 e = 3.9886e-014 l = [-0.5 -0.6667 -0.75 ÿ -0.9792 -0.9796 -0.98] : 1x49 u = [2 1.5 1.3333 1.25 ÿ 1.0208 1.0204 1.02] : 1x50 d_b = [-1 -1 -1 -1 -1 ÿ -1 -1 -1 -1 -1 -1 -1]: 1x49

10. Result (n=100) Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M x = [ 1 1 1 1 1 1 ÿ 1 1 1 1 1 1 ] : 1x100 e = 7.7300e-014 l = [-0.5 -0.6667 -0.75 ÿ -0.9898 -0.9899 -0.9900] : 1x99 u = [2 1.5 1.3333 1.25 ÿ 1.0102 1.0101 1.0100] : 1x100 d_b = [-1 -1 -1 -1 -1 ÿ -1 -1 -1 -1 -1 -1 -1]: 1x99

11. Economical Storage Scheme Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M For a tridiagonal matrix T(n by n matrix), By stroing only d_b, l, u We can store the whole information of T. So we can save about n^2 – 3n storage space. We can load T from d_b, l and u by using a function T=Ori(d_b,u,l) And also T(x,y) by using a function a=sto(d_b,u,l,x,y)

12. Economical Storage Scheme Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M When n = 50 we need 2500 space for storing T. However by storing d_b, u and l, we need only 148 space. So we can save 2500-148 = 2352 storage space. When n = 100 we need 10,000 space for storing T. However by storing d_b, u and l, we need only 298 space. So we can save 10,000-298 = 9702 storage space. ∴It is economical

13. Summary Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M Let T is a tridiagonal matrix, and there exists LU factorization. Then we can use tridiagonal system solver to solve Tx = b with 4n-4 flops Furthermore, we can save a lot space of storage -> 3n by storing l2~ln, u1~unand b1~bn-1, instead of T. For a tridiagonal system, tridiagonal system solver is so useful.

14. Summary Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M Plot of T (n=50) Plot of T2 (n=100) Storing T(or T2) is a waste of storage space

15. Moreover - 1 Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M However, we can’t use Tridiagonal system solver for all tridiagonal matrices. Counter Example : B is a 6x6 tridiagonal matrix Since u(2) = 0, l(2) = inf => error occurs It can’t be solved with tridiagonal system solver.

16. Moreover - 2 Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M Condition number of T (n= 50) Condition number of T2 (n=100) Tridiagonal matrix in gallery is insensitive. How about Hillbert matrix?

17. Thank you Mathematical Modeling : Special Linear Systems – Tridiagonal system M.M