 4.8.1: Find the value of z when each of the algorithm segments in 1 and 2 ...
 4.8.2: Find the value of z when each of the algorithm segments in 1 and 2 ...
 4.8.3: Consider the following algorithm segment: if x y > 0 then do y := 3...
 4.8.4: Find the values of a and e after execution of the loops in 4 and 5:
 4.8.5: Find the values of a and e after execution of the loops in 4 and 5:
 4.8.6: Make a trace table to trace the action of Algorithm 4.8.1 for the i...
 4.8.7: Make a trace table to trace the action of Algorithm 4.8.1 for the i...
 4.8.8: The following algorithm segment makes change; given an amount of mo...
 4.8.9: Find the greatest common divisor of each of the pairs of integers i...
 4.8.10: Find the greatest common divisor of each of the pairs of integers i...
 4.8.11: Find the greatest common divisor of each of the pairs of integers i...
 4.8.12: Find the greatest common divisor of each of the pairs of integers i...
 4.8.13: Use the Euclidean algorithm to handcalculate the greatest common d...
 4.8.14: Use the Euclidean algorithm to handcalculate the greatest common d...
 4.8.15: Use the Euclidean algorithm to handcalculate the greatest common d...
 4.8.16: Use the Euclidean algorithm to handcalculate the greatest common d...
 4.8.17: Make a trace table to trace the action of Algorithm 4.8.2 for the i...
 4.8.18: Make a trace table to trace the action of Algorithm 4.8.2 for the i...
 4.8.19: Prove that for all positive integers a and b, a  b if, and only if...
 4.8.20: a. Prove that if a and b are integers, not both zero, and d = gcd(a...
 4.8.21: Complete the proof of Lemma 4.8.2 by proving the following: If a an...
 4.8.22: a. Prove: If a and d are positive integers and q and r are integers...
 4.8.23: a. Prove that if a, d, q, and r are integers such that a = dq + r a...
 4.8.24: An alternative to the Euclidean algorithm uses subtraction rather t...
 4.8.25: Exercises 2529 refer to the following definition
 4.8.26: Exercises 2529 refer to the following definition
 4.8.27: Exercises 2529 refer to the following definition
 4.8.28: Exercises 2529 refer to the following definition
 4.8.29: Exercises 2529 refer to the following definition
Solutions for Chapter 4.8: Application: Algorithms
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 4.8: Application: Algorithms
Get Full SolutionsSince 29 problems in chapter 4.8: Application: Algorithms have been answered, more than 23995 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. Chapter 4.8: Application: Algorithms includes 29 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th.

Block matrix.
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.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Inverse matrix AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = 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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
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