 11.1.1: The boundaryvalue problem y" = A{yx), 0
 11.1.2: The boundaryvalue problem y" = y' + 2y + Cosx, 0 < x < f, y(0) =0...
 11.1.3: Use the linearshooting method to approximate the solution to the fo...
 11.1.4: Although q(x) <0 in the following boundaryvalue problems, unique s...
 11.1.5: Use the Linear Shooting Algorithm to approximate the solution y = e...
 11.1.6: Let u represent the electrostatic potential between two concentric ...
 11.1.7: Write the secondorder initialvalue problems (11.3) and (11.4) as ...
 11.1.8: Show that, under the hypothesis ofCorollary 11.2, if y2 is the solu...
 11.1.9: Consider the boundaryvalue problem /+ y = 0, 0
 11.1.10: Attempt to apply Exercise 9 to the boundaryvalue problem y" y = 0, 0
Solutions for Chapter 11.1: The Linear Shooting Method
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 11.1: The Linear Shooting Method
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. Chapter 11.1: The Linear Shooting Method includes 10 full stepbystep solutions. Since 10 problems in chapter 11.1: The Linear Shooting Method have been answered, more than 13019 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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).

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Iterative method.
A sequence of steps intended to approach the desired solution.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

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.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Outer product uv T
= column times row = rank one matrix.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.