Math128B: Numerical Analysis Programming Project, Due April 30, 2018

Transcription

1 Math128B: Numerical Analysis Programming Project, Due April 30, 2018 Choose any one of the following 4 projects, details of which are in Week 11 Lecture Notes. Turn in a.m file Projectxyyyy.m, where x is the project number and yyyy is your student ID. your.m file to your GSI by 11:59PM, April 30, 2018.

2 Math128B: Numerical Analysis, Programming Project, Due April 30, Proj. 1 Solving linear equations with Gaussian Elimination Implement GE, GEPP, GECP, with and without iterative refinement. Compare with matlab \ function in terms of accuracy (residual norm) and execution time. function [GEPP,GECP] = Project1yyyy(A,b,tau) where GEPP is an object with the following fields: GEPP.P GEPP.L GEPP.U GEPP.x GEPP.xref GEPP.rnorm GEPP.time GEPP.matlab.rnorm GECP.matlab.time Row permutation L matrix U matrix solution refined solution 2-norm of residual 2-norm of residual in matlab solution of matlab solution GECP is an object with the following fields: GECP.Pr GECP.Pc GECP.L GECP.U GECP.x GECP.xref GECP.rnorm GECP.time GECP.matlab.rnorm GECP.matlab.time Row permutation Column permutation L matrix U matrix solution refined solution 2-norm of residual 2-norm of residual in matlab solution of matlab solution

3 Math128B: Numerical Analysis, Programming Project, Due April 30, Proj. 2 QR Algorithm for Symmetric Eigenproblem Householder reduction to tridiagonal Implicit bulge Chasing scheme for tridiagonal QR Algorithm. Recover eigenvectors. function EIG = Project2yyyy(A,tau) where EIG is an object with the following fields: EIG.H EIG.T EIG.QT EIG.Q EIG.D EIG.res norm EIG.orth norm EIG.time EIG.matlab.time Product of all Householder reflections tridiagonal matrix Eigenvector matrix of T Eigenvector matrix of A Vector of all eigenvalues of A 2-norm of A (EIG.Q) (EIG.Q) diag (EIG.D) 2-norm of (EIG.Q) T (EIG.Q) I of matlab function eig

4 Math128B: Numerical Analysis, Programming Project, Due April 30, Proj. 3 QR Algorithm for SVD Householder reduction to bidiagonal Implicit bulge Chasing scheme for bidiagonal QR Algorithm. Recover singular vectors. function SVD = Project3yyyy(A,tau) where SVD is an object with the following fields: SVD.G SVD.H SVD.B SVD.U SVD.V SVD.S SVD.res norm SVD.U orth norm SVD.V orth norm SVD.time SVD.matlab.time Product of all left Householder reflections Product of all right Householder reflections bidiagonal matrix Left singular vector matrix of A Right singular vector matrix of A Vector of all singular values of A 2-norm of A (SVD.V) (SVD.U) diag (SVD.S) 2-norm of (SVD.U) T (SVD.U) I 2-norm of (SVD.V) T (SVD.V) I of matlab function svd

5 Math128B: Numerical Analysis, Programming Project, Due April 30, Proj. 4 Quasi-Newton Methods (BFGS) for min x,c g (x, A, y, c, λ) Implement BFGS and L-BFGS Run BFGS and L-BFGS with logistic objective function g (x, A, y, c, λ) = 1 n log ( φ ( ( y i a T n i x + c ))) + λ i=1 2 x 2 2, where x R p, c R, A = a T 1. a T n Rn p, y = y 1. y n Rn, and φ (t) = (1 + exp ( t)) 1. For numerical stability, however, the function log (φ ( )) should be computed as follows: log (φ (t)) = min (0, t) log (1 + exp ( t )). Checkout more details about logistic function at function [BFGS,LBFGS] = Project4yyyy(A,y, lambda, m, IT) where IT is the number of BFGS and L-BFGS iterations. BFGS is an object with the following fields: BFGS.flg BFGS.x BFGS.c BFGS.obj 0 if BFGS succeeds, and 1 otherwise vector x computed by BFGS scalar computed by BFGS computed optimal objective value. LBFGS is an object with the following fields: LBFGS.flg LBFGS.x LBFGS.c LBFGS.obj 0 if LBFGS succeeds, and 1 otherwise vector x computed by LBFGS scalar computed by LBFGS computed optimal objective value. You can set B 1 0 to be a small multiple of the identity matrix.