1.1 4.375 = 100.0011 = 1.00011 x 2^2
    exp is 2+127 = 129
    rep is: 0 10000001 0001100

1.2 comparison is made simpler because the numbers are in the
    same order as the integers.

1.3 last mantissa bit represents 2^-7.  Exponent is 2, so
    eps = 2^-5 = 1/32.

1.4 2^-130 is a denormalized number so its exponent is 0, or
    2^-126.  The value of the last bit is 2^-7 * 2^-126 = 2^-133.

1.5 when the exponent is 7, eps = 2^7 * 2^-7 = 1.  The first
    number with exponent 7 is 1 * 2^7 = 128.

1.6 The "half" format is better when you care about precision and
    the range is sufficient. Our alternative format is better if
    you need to represent very large or very small numbers and
    can tolerate lower precision.


2A.1. only LU requires pivoting.

2A.2. LU, QR, SVD.

2B.1. QR: x = R \ Q^T b
        1. m-v multiply
        2. backsolve
      SVD: x = V S^-1 U^T b
        1. m-v multiply
        2. vector scale
        3. m-v multiply

2B.2. QR: x = R \ Q^T b
        1. m-v multiply (n x m)
        2. backsolve    (n x n)
      SVD: x = V S^-1 U^T b
        1. m-v multiply (n x m)
	2. vector scale (n)
	3. m-v multiply (n x n)

2B.3. we need to multiply a Householder matrix by a submatrix of
      size (m-k) x (n-k).  This is A' - (beta u) (u^T A') (where
      A' is the submatrix).
        1. v-m product (m-k) x (n-k)
        2. outer product (m-k) x (n-k)
	3. matrix subtraction (m-k) x (n-k)

2B.4. only SVD can do this one.  Requires x = V_1 S_1^-1 U_1^T b
        1. m-v mult (r x m)
	2. vector scale (r)
	3. m-v mult (n x r)


3.1 7 nonzero sigmas -> rank 7

3.2 double precision can easily compute 10^-9 as a fraction of
    10, so still rank 7.

3.3 in single precision 10^-9 is on the order of rounding errors
  for this matrix, so rank 5.

3.4 in a matrix with 140 entries, the F-norm of an error of 0.02
    in every entry exceeds 0.2.  The F-norm error of truncating
    the singular values after the first 2 is about the same order
    (1.7) so the 0.1s are explained by noise and we have to
    consider the "true" rank to be 2.


4.
	      bis  newt   sec
      deriv    N     Y     N
  guarantee    Y     N     N
    fastest    N     Y     N
    slowest    Y     N     N