**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 with a = 40, b = 30 ∈ F. Suppose we use the formula to 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.

**(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.**

**(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.

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

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

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

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

**5.**

**(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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