Math 6800 Problem Set 3: Computing Reduced QR Factorization and Solving Linear Systems - P | Assignments Linear Algebra

Math 6800

D. Schwendeman Problem Set 3

Due:

Thursday, 9/30/10

1. (a) Write a computer subroutine (or Matlab function) that computes the reduced QR-

factorization of an m×nreal matrix Ausing the modified Gram-Schmidt algorithm. Your

code may assume that m≥nand should check for rank deficiency of Ain a sensible way

(and return an error message if such a case occurs).

(b) Test your code using the m×nVandermonde matrix







1t1t2

1· · · tn−1

1t2t2

2· · · tn−1

1tmt2

m· · · tn−1







where the nodes ti,i= 1,2, . . . , m, are distinct (so that Ahas full rank). Consider the case

when m= 21, n= 6 and ti= (i−11)/10 which are equally spaced nodes on [−1,1]. Check

that kˆ

Qˆ

R−Akand kˆ

QTˆ

Q−Ikare both small, and print out these values.

Qand αis a scalar constant. Choose α

so that the last element pis 1. Plot the data (ti,pi), i= 1, . . . , m, for each column of ˆ

on the same graph. Observe that these curves are approximations of the first six Legendre

polynomials. Explain this observation. (Hint: consider a discrete approximation of the

integral orthogonality for the Legendre polynomials.)

(d) Redo parts (b) and (c) using the Chebyshev nodes ti= cos((2i−1)π/(2m)). For this

case, the curves are no longer approximations of Legendre polynomials. What orthogonal

functions do you think they approximate. Explain.

2. (a) Consider the real m×nsystem Ax=b, where m≥n. Write a subroutine that

computes the QR-factorization of Ausing Householder triangulation. Output the results

of the algorithm compactly in a matrix of size Aas discussed in class. (You might need an

extra row for this compact storage.) Your code should check for rank deficiency and return

an error message as before. Write a second subroutine that computes the solution of the

linear system given the QR-factorization of Aand the right-hand-side vector b. (For the

case m > n, the solution is in a least squares sense.)

(b) Test your code by considering the following curve fitting problem. Find the best fit

polynomial

P(t) = a0+a1t+a2t2+· · · +an−1tn−1

to the data (ti, yi), i= 1,2, . . . , m. The problem is equivalent to solving Ax=b, where

Ais the Vandermonde matrix above, bi=yi, and xj=aj−1. Run your codes for the case

when ti,i= 1,2, . . . , m, are equally spaced nodes on [−5,5] and yi= 1/(1 + t2

i). Compute

and plot the polynomial interpolants for m= 21 and n= 3, 7, 15, and 21. (Use many more

nodes than the 21 data points to plot the smooth polynomial interpolants.) Comment on

the accuracy and behavior of the curves.

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Math 6800

D. Schwendeman Problem Set 3

Due: Thursday, 9/30/

(a) Write a computer subroutine (or Matlab function) that computes the reduced QR- factorization of an m × n real matrix A using the modified Gram-Schmidt algorithm. Your code may assume that m ≥ n and should check for rank deficiency of A in a sensible way (and return an error message if such a case occurs).

(b) Test your code using the m × n Vandermonde matrix

A =

   

1 t 1 t^21 · · · tn 1 −^1 1 t 2 t^22 · · · tn 2 −^1 .. .

1 tm t^2 m · · · tn m−^1

   

where the nodes ti, i = 1, 2 ,... , m, are distinct (so that A has full rank). Consider the case when m = 21, n = 6 and ti = (i − 11)/10 which are equally spaced nodes on [− 1 , 1]. Check that ‖ Qˆ Rˆ − A‖ and ‖ QˆT^ Qˆ − I‖ are both small, and print out these values.

(c) Let p = αqj , where qj is the jth^ column of Qˆ and α is a scalar constant. Choose α so that the last element p is 1. Plot the data (ti, pi), i = 1,... , m, for each column of Qˆ on the same graph. Observe that these curves are approximations of the first six Legendre polynomials. Explain this observation. (Hint: consider a discrete approximation of the integral orthogonality for the Legendre polynomials.)

(d) Redo parts (b) and (c) using the Chebyshev nodes ti = cos((2i − 1)π/(2m)). For this case, the curves are no longer approximations of Legendre polynomials. What orthogonal functions do you think they approximate. Explain.

(a) Consider the real m × n system Ax = b, where m ≥ n. Write a subroutine that computes the QR-factorization of A using Householder triangulation. Output the results of the algorithm compactly in a matrix of size A as discussed in class. (You might need an extra row for this compact storage.) Your code should check for rank deficiency and return an error message as before. Write a second subroutine that computes the solution of the linear system given the QR-factorization of A and the right-hand-side vector b. (For the case m > n, the solution is in a least squares sense.)

(b) Test your code by considering the following curve fitting problem. Find the best fit polynomial P (t) = a 0 + a 1 t + a 2 t^2 + · · · + an− 1 tn−^1

to the data (ti, yi), i = 1, 2 ,... , m. The problem is equivalent to solving Ax = b, where A is the Vandermonde matrix above, bi = yi, and xj = aj− 1. Run your codes for the case when ti, i = 1, 2 ,... , m, are equally spaced nodes on [− 5 , 5] and yi = 1/(1 + t^2 i ). Compute and plot the polynomial interpolants for m = 21 and n = 3, 7, 15, and 21. (Use many more nodes than the 21 data points to plot the smooth polynomial interpolants.) Comment on the accuracy and behavior of the curves.

Math 6800 Problem Set 3: Computing Reduced QR Factorization and Solving Linear Systems - P, Assignments of Linear Algebra

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