 2.3.1: Let f(x) = x 2 6 and po = 1. Use Newton's method to find pi
 2.3.2: Let fix) = x 3 cosx and po = l  Use Newton's method to find pi Co...
 2.3.3: Let /(x) = x 2 6. With po = 3 and p\ = 2, find p^. a. Use the Secan...
 2.3.4: Let f(x) = x 3 cosx. With po = I and p\ = 0, find p^. a. Use the Se...
 2.3.5: Use Newton's method to find solutions accurate to within 104 for t...
 2.3.6: Use Newton's method to find solutions accurate to within 105 for t...
 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: The fourthdegree polynomial fix) = 230x4 + 18x3 + 9x2  221x  9 h...
 2.3.14: The function /(x) = lanjrx 6 has a zero at (1/tt) arctan6 % 0.44743...
 2.3.15: The equation 4x2 e x e~x 0 has two positive solutions xi and X2. Us...
 2.3.16: The equation x 210cosx = 0 hastwo solutions, 1.3793646. Use Newton'...
 2.3.17: The function described by /(x) = ln(x2 + I) e {l4x costtx has an in...
 2.3.18: Use Newton's method to solve the equation II, 1 n 0 = x xsinx cos2...
 2.3.19: Use Newton's method to approximate, to within 104 , the value of x...
 2.3.20: Use Newton's method to approximate, to within 104 , the value of x...
 2.3.21: The sum oftwo numbers is 20. If each number is added to its square ...
 2.3.22: Find an approximation for accurate to within 10~4 , for the populat...
 2.3.23: involving the amount of money required to pay off a mortgage over a...
 2.3.24: The accumulated value of a savings account based on regular periodi...
 2.3.25: The logistic population growth model is described by an equation of...
 2.3.26: The Gompertz population growth model is described by Pit)  PLece ...
 2.3.27: Player A will shut out (win by a score of 210) player B in a game ...
 2.3.28: A drug administered to a patient produces a concentration in the bl...
 2.3.29: In the design ofallterrain vehicles, itis necessary to considerthe...
 2.3.30: The iteration equation for the Secant method can be written in the ...
 2.3.31: The following describes Newton's method graphically; Suppose that f...
 2.3.32: Derive the error formula for Newton's method M 2 \P  Pn+\ < ..fl( ...
Solutions for Chapter 2.3: Newton's Method and Its Extensions
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 2.3: Newton's Method and Its Extensions
Get Full SolutionsChapter 2.3: Newton's Method and Its Extensions includes 32 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. Since 32 problems in chapter 2.3: Newton's Method and Its Extensions have been answered, more than 13809 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10.

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).

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

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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 •

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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).

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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·

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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.