- 8.2.1: F(x) = x2 2x
- 8.2.2: F(x) = x5
- 8.2.3: F(x) = sin x
- 8.2.4: F(x) = x3 x
- 8.2.5: F(x) = arctan x
- 8.2.6: F(x) = 3x(1 x)
- 8.2.7: F(x) = (/2)sin x
- 8.2.8: F(x) = x2 3
- 8.2.9: F(x) = 1/x
- 8.2.10: F(x) = 1/x2
- 8.2.11: F(x) = ex
- 8.2.12: F(x) = x5 + 1
- 8.2.13: F(x) = 1 x2
- 8.2.14: F(x) = (/2) cos x
- 8.2.15: F(x) = x + 1, if x < 3.5; 2x 8, if x 3.5.
- 8.2.16: F(x) = 1 x3
- 8.2.17: F(x) = |x 2| 1
- 8.2.18: F(x) = x x2
- 8.2.19: F(x) = 1/x
- 8.2.20: F(x) = sin x
- 8.2.21: F(x) = tan x
- 8.2.22: F(x) = x + x3
- 8.2.23: F(x) = x x3
- 8.2.24: F(x) = ex1 (fixed point is 1)
- 8.2.25: F(x) = e ex (fixed point is 1)
- 8.2.26: Find all fixed points for Fc(x) = x2 + c for all values of c. Deter...
- 8.2.27: How many fixed points does F(x) = tan x have? Are they attracting, ...
- 8.2.28: What can you say about fixed points for Fc(x) = cex with c > 0? Wha...
- 8.2.29: Consider the function T (x) = 4x, x < 1/2; 4 4x, x 1/2. Does T have...
- 8.2.30: Recall from calculus that Newtons method is an iterative procedure ...
Solutions for Chapter 8.2: FIXED POINTS AND PERIODIC POINTS
Full solutions for Differential Equations 00 | 4th Edition
peA) = det(A - AI) has peA) = zero matrix.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
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.
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.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
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.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).