- 11.2.1: Use the Nonlinear Shooting Algorithm with h = 0.5 to approximate th...
- 11.2.2: Use the Nonlinear Shooting Algorithm with h = 0.25 to approximate t...
- 11.2.3: Use the Nonlinear Shooting method with TOL = 104 to approximate the...
- 11.2.4: Use the Nonlinear Shooting method with TOL = 104 to approximate the...
- 11.2.5: a. Change Algorithm 11.2 to incorporate the Secant method instead o...
- 11.2.6: The Van der Pol equation, y (y2 1)y + y = 0, > 0, governs the flow ...
Solutions for Chapter 11.2: The Shooting Method for Nonlinear Problems
Full solutions for Numerical Analysis | 9th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Every v in V is orthogonal to every w in W.
Outer product uv T
= column times row = rank one matrix.
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
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.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.