 2.2.1: Use algebraic manipulation to show that each of the following funct...
 2.2.2: a. Perform four iterations, if possible, on each of the functions g...
 2.2.3: The following four methods are proposed to compute 211/3. Rank them...
 2.2.4: The following four methods are proposed to compute 71/5. Rank them ...
 2.2.5: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.6: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.7: Use Theorem 2.3 to show that g(x) = + 0.5 sin(x/2) has a unique fix...
 2.2.8: Use Theorem 2.3 to show that g(x) = 2x has a unique fixed point on ...
 2.2.9: Use a fixedpoint iteration method to find an approximation to 3 th...
 2.2.10: Use a fixedpoint iteration method to find an approximation to 3 25...
 2.2.11: For each of the following equations, determine an interval [a, b] o...
 2.2.12: For each of the following equations, use the given interval or dete...
 2.2.13: Find all the zeros of f (x) = x2 +10 cos x by using the fixedpoint...
 2.2.14: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.15: Use a fixedpoint iteration method to determine a solution accurate...
 2.2.16: Let A be a given positive constant and g(x) = 2x Ax2. a. Show that ...
 2.2.17: Find a function g defined on [0, 1] that satisfies none of the hypo...
 2.2.18: a. Show that Theorem 2.2 is true if the inequality g (x) k is rep...
 2.2.19: a. Use Theorem 2.4 to show that the sequence defined by xn = 1 2 xn...
 2.2.20: a. Show that if A is any positive number, then the sequence defined...
 2.2.21: Replace the assumption in Theorem 2.4 that a positive number k < 1 ...
 2.2.22: Suppose that g is continuously differentiable on some interval (c, ...
 2.2.23: An object falling vertically through the air is subjected to viscou...
 2.2.24: Let g C1[a, b] and p be in (a, b) with g( p) = p and g ( p) > 1. ...
Solutions for Chapter 2.2: FixedPoint Iteration
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 2.2: FixedPoint Iteration
Get Full SolutionsThis textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Since 24 problems in chapter 2.2: FixedPoint Iteration have been answered, more than 15792 students have viewed full stepbystep solutions from this chapter. Chapter 2.2: FixedPoint Iteration includes 24 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

Stiffness matrix
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.

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.