- 9.4.1: A certain computer can execute 10 gigaflops per second. Use Formula...
- 9.4.2: A certain computer can execute 100 gigaflops per second. Use Formul...
- 9.4.3: Today's personal computers can execute 70 gigaflops per second. Use...
- 9.4.4: The IBM Roadrunner computer can operate at speeds in excess of 1 pe...
- 9.4.5: (a) Approximate the time required to execute the forward phase of G...
- 9.4.6: About how many teraflops per second must a computer be able to exec...
- 9.4.7: In Exercises 710, A and B are matrices and c is a real number. How ...
- 9.4.8: In Exercises 710, A and B are matrices and c is a real number.How m...
- 9.4.9: In Exercises 710, A and B are matrices and c is a real number. How ...
- 9.4.10: In Exercises 710, A and B are matrices and c is a real number. If A...
- 9.4.e: In parts (a)(g) determine whether the statement is true or false, a...
- 9.4.f: In parts (a)(g) determine whether the statement is true or false, a...
- 9.4.g: In parts (a)(g) determine whether the statement is true or false, a...
Solutions for Chapter 9.4: Comparison of Procedures for Solving Linear Systems
Full solutions for Elementary Linear Algebra: Applications Version | 10th Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Upper triangular systems are solved in reverse order Xn to Xl.
Column space C (A) =
space of all combinations of the columns of A.
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.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
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.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
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.
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.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.