Does the Gaussian elimination without pivoting always produce an LU factorization for any given matrix B?

SKU: assim9 Category:

Determine the largest positive number and the smallest positive number in F

 

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1.Consider the finite floating point number system F with nonzero numbers of the form

x = ±0.d1d2d3 × 10e ,        1 ≤ d1 ≤ 9,              0 ≤ d2, d3 ≤ 9,                −2 ≤ e ≤ 3

 

(a) Determine the total number of floating point numbers in F (including 0)

 

(b) Determine the largest positive number and the smallest positive number in F.

 

(c) Determine the machine precision (unit roundoff) µ of the system F.

 

(d) Let axb with a = 40, b = 30 ∈ F. Suppose we use the formula a2bto compute ||x||2 on a machine with the number system F. What is the computed result? How to compute ||x||2 more accurately?

 

 

2.You are not allowed to use computer programs for solving this problem. You need to provide exact solutions.

12

 

(a) Use the Gaussian elimination without pivoting to determine a unit lower triangular matrix L and an upper triangular matrix U such that A = LU.

 

(b) Use the LU factorization computed in (a) to solve the system of linear equations Ax = b.

 

(c) Use the Gaussian elimination with partial pivoting to determine a permutation matrix P, a unit lower triangular matrix L, and an upper triangular matrix U such that P A = LU. Determine the growth factor ρ.

 

(d) Use the factorization P A = LU computed in (c) to solve the system of linear equations Ax = b.

 

(e) Does the Gaussian elimination without pivoting always produce an LU factorization for any given matrix B? Give a brief explanation or a counterexample.

 

(f) Does the Gaussian elimination with partial pivoting always produce a factorization P B = LU for any given matrix B? Give a brief explanation or a counterexample.

 

 

3.3

(a) Use Householder matrices or Givens rotations to compute a full QR factorization of A. That is, compute A = QR, where Q is 3 × 3 and R is 3 × 2.

 

(b) Use the QR factorization computed in (a) to compute PN , the orthogonal projection onto the null space of AT . Then compute bN = PNb.

 

(c) Use the QR factorization computed in (a) to solve the least squares problem minx∈R2 ||b − Ax||2 .

 

(d) Compute the residual r = b − Ax, where x is the least squares solution from (c). Check whether r is the same as bN obtained in (b), and give a brief explanation about what you have observed.

 

 

4.You are not allowed to use computer programs for solving this problem. You need to provide exact solutions.

leta

(a) Compute the upper triangular matrix H1 for the Cholesky factorization A = h1.  Then compute the matrix a11

 

(b) Compute the upper triangular matrix H2 for the Cholesky factorization A1=h2. Then compute the matrix

h12

 

 

(c)Compute q1.Verify that Q is a real orthogonal matrix.

 

(d) Compute R = H2H1. Verify A = QR and A2 = RQ.

 

 

5.blat

 

(a) Apply one power iteration to A and x(º) to compute x(¹) .

 

(b) Apply one Rayleigh quotient iteration to A and x(º) to compute x(¹)

 

(c) Perform one QR iteration on A with the Rayleigh quotient shift σ:

A − σI = QR, A1 = RQ + σI,

to compute A1.

 

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