 2.2.1: Use algebraic manipulation to show that each ofthe following functi...
 2.2.2: a. Perform four iterations, ifpossible, on each ofthe functions g d...
 2.2.3: Let f(x) = x 3 2x F 1. To solve fix) 0, the following four fixedp...
 2.2.4: Let fix) = x 4  3x2 2. To solve /(x) = 0, the following four fix...
 2.2.5: The following four methods are proposed to compute 21V3 . Rank them...
 2.2.6: The following four methods are proposed tocompute 7 1,5 . Rank them...
 2.2.7: Use a fixedpointiteration method to determine a solution accurate ...
 2.2.8: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.9: Use Theorem 2.3 to show that g(x) n + 0.5 sin(x/2) has a unique fix...
 2.2.10: Use Theorem 2.3 to show that g(x) = 2 _A has a unique fixed point o...
 2.2.11: Use a fixedpoint iteration method to find an approximation to ^3 t...
 2.2.12: Use a fixedpoint iteration method to find an approximation to \/25...
 2.2.13: For each ofthe following equations, determine an interval [a, b] on...
 2.2.14: For each of the following equations, use the given interval or dete...
 2.2.15: Find all the zeros off(x) = x 2 +10 cosx by using the fixedpoint i...
 2.2.16: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.17: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.18: An object falling vertically through the air is subjected to viscou...
 2.2.19: Let g e C'fa, fej and p be in (a, b) with g(p) = p and g'(p) > 1....
 2.2.20: Let A be a given positive constant and g{x) = 2x Ax2 . a. Show that...
 2.2.21: Find a function g defined on [0, 1J that satisfies none ofthe hypot...
 2.2.22: a. Show that Theorem 2.3 is true if the inequality g'(x) < k is r...
 2.2.23: a. Use Theorem 2.4 to show that the sequence defined by 1 I xn =x_...
 2.2.24: a. Show that if A is any positive number, then the sequence defined...
 2.2.25: Replace the assumption in Theorem 2.4 that "a positive number k < I...
 2.2.26: Suppose that g is continuously differentiable on some interval (c, ...
Solutions for Chapter 2.2: FixedPoint Iteration
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 2.2: FixedPoint Iteration
Get Full SolutionsSince 26 problems in chapter 2.2: FixedPoint Iteration have been answered, more than 12890 students have viewed full stepbystep solutions from this chapter. Chapter 2.2: FixedPoint Iteration includes 26 full stepbystep 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. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Solvable system Ax = b.
The right side b is in the column space of A.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.