- 11.5.1E: Use the facts that to find log2(1,000), log2(1,000,000), and log2(1...
- 11.5.2E: Suppose an algorithm requires operations when performed with an inp...
- 11.5.5E: In 5 and 6, trace the action of the binary search algorithm (Algori...
- 11.5.6E: In 5 and 6, trace the action of the binary search algorithm (Algori...
- 11.5.7E: Suppose bot and top are positive integers with bot ? top. Consider ...
- 11.5.8E: Exercises 8–11 refer to the following algorithm segment. For each p...
- 11.5.9E: Exercises 8–11 refer to the following algorithm segment. For each p...
- 11.5.10E: Exercises 8–11 refer to the following algorithm segment. For each p...
- 11.5.11E: Exercises 8–11 refer to the following algorithm segment. For each p...
- 11.5.12E: := n div 3Exercises 12–15 refer to the following algorithm segment....
- 11.5.13E: Exercises 12–15 refer to the following algorithm segment. For each ...
- 11.5.15E: Exercises 12–15 refer to the following algorithm segment. For each ...
- 11.5.20E: In 20 and 21, draw a diagram like Figure 11.5.4 to show how to merg...
- 11.5.21E: In 20 and 21, draw a diagram like Figure 11.5.4 to show how to merg...
- 11.5.22E: In 22 and 23, draw a diagram like Figure 11.5.5 to show how merge s...
- 11.5.23E: In 22 and 23, draw a diagram like Figure 11.5.5 to show how merge s...
- 11.5.24E: Show that given an array a[bot], a[bot + 1], . . . , a[top] of
Solutions for Chapter 11.5: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Constant down each diagonal = time-invariant (shift-invariant) filter.