- 3.R.1RQ: a) Define the term algorithm.________________b)What are the differe...
- 3.R.2RQ: a) Describe, using English, an algorithm for finding the largest in...
- 3.R.3RQ: a) State the definition of the fact that f(n) is O(g(n)), where f(n...
- 3.R.4RQ: List these functions so that each function is big- O of the next fu...
- 3.R.5RQ: a) How can you produce a big-O estimate for a function that is the ...
- 3.R.6RQ: a) Define what the worst-case time complexity, average case time co...
- 3.R.7RQ: a) Describe the linear search and binary search algorithm for findi...
- 3.R.8RQ: a) Describe the bubble sort algorithm.________________b) Use the bu...
- 3.R.9RQ: a) Describe the insertion sort algorithm.________________b)Use the ...
- 3.R.10RQ: a) Explain the concept of a greedy algorithm.________________b) Pro...
- 3.R.11RQ: Define what it means for a problem to be tractable and what it mean...
Solutions for Chapter 3.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Column space C (A) =
space of all combinations of the columns of 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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.