- 5.10.1: To prove Theorem 5.20, part (i), show that the hypotheses imply tha...
- 5.10.2: For the Adams-Bashforth and Adams-Moulton methods of order four, a....
- 5.10.3: Use the results of Exercise 32 in Section 5.4 to show that the Rung...
- 5.10.4: Consider the differential equation y = f(t, y), ab. Part (a) sugges...
- 5.10.5: Given the multistep method 3 I Wf+i = --wi + 3w,_i - Wi2 + 3/j/(ti,...
- 5.10.6: Obtain an approximate solution to the differential equation y' = -y...
- 5.10.7: Investigate stability for the difference method vv',+i = -4wi + 5wi...
- 5.10.8: Consider the problem y' = 0, 0 < r < 10, >'(0) = 0, which has the s...
Solutions for Chapter 5.10: Stability
Full solutions for Numerical Analysis | 10th Edition
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > 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 .
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Outer product uv T
= column times row = rank one matrix.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
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.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).