- 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
A = CTC = (L.J]))(L.J]))T for positive definite A.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).