- 8.5.1: In each of 1 through 5, determine approximate values of the solutio...
- 8.5.2: In each of 1 through 5, determine approximate values of the solutio...
- 8.5.3: In each of 1 through 5, determine approximate values of the solutio...
- 8.5.4: In each of 1 through 5, determine approximate values of the solutio...
- 8.5.5: In each of 1 through 5, determine approximate values of the solutio...
- 8.5.6: Consider the example problem x_ = x 4y, y_ = x + y with the initial...
- 8.5.7: Consider the initial value problem x__ + t2x_ + 3x = t, x(0) = 1, x...
- 8.5.8: Consider the general initial value problem x_ = f (t, x, y) and y_ ...
Solutions for Chapter 8.5: Systems of First-Order Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
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.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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.
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.
lA-II = 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.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
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.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
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.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.