- 6.9.1: Biology A strain of E-coli Beu 397-recA441 is placed into a nutrien...
- 6.9.2: Biology A strain of E-coli SC18del-recA718 is placed into a nutrien...
- 6.9.3: Chemistry A chemist has a 100-gram sample of a radioactive material...
- 6.9.4: Cigarette Exports The following data represent the number of cigare...
- 6.9.5: Economics and Marketing The following data represent the price and ...
- 6.9.6: Economics and Marketing The following data represent the price and ...
- 6.9.7: Population Model The following data represent the population of the...
- 6.9.8: Population Model The following data represent the world population....
- 6.9.9: Cable Subscribers The following data represent the number of basic ...
- 6.9.10: Cell Phone Users Refer to the data in Table 9. (a) Using a graphing...
- 6.9.11: Age versus Total Cholesterol The following data represent the age a...
- 6.9.12: Income versus Crime Rate The following data represent crime rate ag...
- 6.9.13: Depreciation of a Chevrolet Impala The following data represent the...
Solutions for Chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from Data
Full solutions for College Algebra | 9th Edition
Solutions for Chapter 6.9: Building Exponential, Logarithmic, and Logistic Models from DataGet Full Solutions
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.
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.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
= Xl (column 1) + ... + xn(column n) = combination of columns.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Outer product uv T
= column times row = rank one matrix.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.