- 7.1.1: Find the three-digit decimal floating-point representation of each ...
- 7.1.2: Find the absolute error and the relative error when each of the rea...
- 7.1.3: Represent each of the following as five-digit base 2 floating-point...
- 7.1.4: Use four-digit decimal floating-point arithmetic to do each of the ...
- 7.1.5: Let x1 = 94,210, x2 = 8631, x3 = 1440, x4 = 133, and x5 = 34. Calcu...
- 7.1.6: What would the machine epsilon be for a computer that uses 16-digit...
- 7.1.7: What would the machine epsilon be for a computer that uses 36-digit...
- 7.1.8: How many floating-point numbers are there in the system if t = 2, L...
Solutions for Chapter 7.1: Floating-Point Numbers
Full solutions for Linear Algebra with Applications | 8th Edition
peA) = det(A - AI) has peA) = zero matrix.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.