- 8.3.1: F(x) = x + x2 + , = 0
- 8.3.2: F(x) = x, = 1
- 8.3.3: F(x) = sin x, = 1
- 8.3.4: F(x) = sin x, = 1
- 8.3.5: F(x) = x2, = 1/4
- 8.3.6: F(x) = arctan x, = 1
- 8.3.7: F(x) = (x + x2), = 0
- 8.3.8: Consider the family of functions F(x) = x3 + . Find all -values for...
- 8.3.9: Consider the family of functions given by T(x) = x, if x < 1/2; (1 ...
- 8.3.10: For the logistic family Fk (x) = kx(1 x), show explicitly that ther...
- 8.3.11: Consider the family of functions Fc(x) = x2 + c. Sketch the graphs ...
Solutions for Chapter 8.3: BIFURCATIONS
Full solutions for Differential Equations 00 | 4th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
A = CTC = (L.J]))(L.J]))T for positive definite A.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
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.
Gram-Schmidt 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.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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 .
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
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.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.