 2.3.1: Let f (x) = x2 6 and p0 = 1. Use Newtons method to find p2.
 2.3.2: Let f (x) = x3 cos x and p0 = 1. Use Newtons method to find p2. Cou...
 2.3.3: Let f (x) = x2 6. With p0 = 3 and p1 = 2, find p3. a. Use the Secan...
 2.3.4: Let f (x) = x3 cos x. With p0 = 1 and p1 = 0, find p3. a. Use the S...
 2.3.5: Use Newtons method to find solutions accurate to within 104 for the...
 2.3.6: Use Newtons method to find solutions accurate to within 105 for the...
 2.3.7: Repeat Exercise 5 using the Secant method.
 2.3.8: Repeat Exercise 6 using the Secant method.
 2.3.9: Repeat Exercise 5 using the method of False Position.
 2.3.10: Repeat Exercise 6 using the method of False Position.
 2.3.11: Use all three methods in this Section to find solutions to within 1...
 2.3.12: Use all three methods in this Section to find solutions to within 1...
 2.3.13: Use Newtons method to approximate, to within 104, the value of x th...
 2.3.14: Use Newtons method to approximate, to within 104, the value of x th...
 2.3.15: The following describes Newtons method graphically: Suppose that f ...
 2.3.16: Use Newtons method to solve the equation 0 = 1 2 + 1 4 x2 x sin x 1...
 2.3.17: The fourthdegree polynomial f (x) = 230x4 + 18x3 + 9x2 221x 9 has ...
 2.3.18: The function f (x) = tan x 6 has a zero at (1/ ) arctan 6 0.4474315...
 2.3.19: The iteration equation for the Secant method can be written in the ...
 2.3.20: The equation x210 cos x = 0 has two solutions, 1.3793646. Use Newto...
 2.3.21: The equation 4x2 ex ex = 0 has two positive solutions x1 and x2. Us...
 2.3.22: Use Maple to determine how many iterations of Newtons method with p...
 2.3.23: The function described by f (x) = ln(x2 + 1) e0.4x cos x has an inf...
 2.3.24: Find an approximation for , accurate to within 104, for the populat...
 2.3.25: The sum of two numbers is 20. If each number is added to its square...
 2.3.26: The accumulated value of a savings account based on regular periodi...
 2.3.27: involving the amount of money required to pay off a mortgage over a...
 2.3.28: A drug administered to a patient produces a concentration in the bl...
 2.3.29: Let f (x) = 33x+1 7 52x . a. Use the Maple commands solve and fsolv...
 2.3.30: Repeat Exercise 29 using f (x) = 2x2 3 7x+1.
 2.3.31: The logistic population growth model is described by an equation of...
 2.3.32: The Gompertz population growth model is described by P(t) = PLecekt...
 2.3.33: Player A will shut out (win by a score of 210) player B in a game o...
 2.3.34: In the design of allterrain vehicles, it is necessary to consider ...
Solutions for Chapter 2.3: Newton's Method and Its Extensions
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 2.3: Newton's Method and Its Extensions
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9780538733519. Since 34 problems in chapter 2.3: Newton's Method and Its Extensions have been answered, more than 16035 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Chapter 2.3: Newton's Method and Its Extensions includes 34 full stepbystep solutions.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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 ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.