 10.2.1: Use Newton's method with x (0* = 0 to compute x <2) for each ofthe ...
 10.2.2: Use Newton's method with x 101 = 0 to compute x'2 ' for each of the...
 10.2.3: Use the graphing facilities of your CAS or calculator to approximat...
 10.2.4: Use the graphing facilities of your CAS or calculator to approximat...
 10.2.5: Use the answers obtained in Exercise 3 as initial approximations to...
 10.2.6: Use the answers obtained in Exercise 4 as initial approximations to...
 10.2.7: Use Newton's method to find a solution to the following nonlinearsy...
 10.2.8: The nonlinear system 4X] X2 + X3 = X1X4, X] + 3X2 2X3 = X2X4, X 2X...
 10.2.9: The nonlinear system 1 3xi  cos(x2X3) _ 2 = 0 ' x?  625x2   = 0...
 10.2.10: The amount of pressure required to sink a large, heavy object in a ...
 10.2.11: In calculating the shape of a gravityflow discharge chute that wil...
 10.2.12: An interesting biological experiment (see [Schr2]) concerns the det...
 10.2.13: Show that when n = \, Newton's method given by Eq. {10.9) reducesto...
 10.2.14: What does Newton's method reduce to for the linear system Ax = b gi...
Solutions for Chapter 10.2: Newton's Method
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 10.2: Newton's Method
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9781305253667. Chapter 10.2: Newton's Method includes 14 full stepbystep solutions. Since 14 problems in chapter 10.2: Newton's Method have been answered, more than 14946 students have viewed full stepbystep solutions from this chapter. 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.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Norm
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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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).

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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