 2.1.1: Use the Bisection method to find p3 for f (x) = x cos x on [0, 1].
 2.1.2: Let f (x) = 3(x + 1)(x 1 2 )(x 1). Use the Bisection method on the ...
 2.1.3: Use the Bisection method to find solutions accurate to within 102 f...
 2.1.4: Use the Bisection method to find solutions accurate to within 102 f...
 2.1.5: Use the Bisection method to find solutions accurate to within 105 f...
 2.1.6: Use the Bisection method to find solutions, accurate to within 105 ...
 2.1.7: a. Sketch the graphs of y = x and y = 2 sin x. b. Use the Bisection...
 2.1.8: a. Sketch the graphs of y = x and y = tan x. b. Use the Bisection m...
 2.1.9: a. Sketch the graphs of y = ex 2 and y = cos(ex 2). b. Use the Bise...
 2.1.10: Let f (x) = (x + 2)(x + 1)2x(x 1)3(x 2). To which zero of f does th...
 2.1.11: Let f (x) = (x + 2)(x + 1)x(x 1)3(x 2). To which zero of f does the...
 2.1.12: Find an approximation to 3 correct to within 104 using the Bisectio...
 2.1.13: Find an approximation to 3 25 correct to within 104 using the Bisec...
 2.1.14: Use Theorem 2.1 to find a bound for the number of iterations needed...
 2.1.15: Use Theorem 2.1 to find a bound for the number of iterations needed...
 2.1.16: Let f (x) = (x 1)10, p = 1, and pn = 1 + 1/n. Show that f ( pn) <...
 2.1.17: Let{ pn} be the sequence defined by pn = n k=1 1 k . Show that{ pn}...
 2.1.18: The function defined by f (x) = sin x has zeros at every integer. S...
 2.1.19: A trough of length L has a cross section in the shape of a semicirc...
 2.1.20: A particle starts at rest on a smooth inclined plane whose angle is...
Solutions for Chapter 2.1: The Bisection Method
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 2.1: The Bisection Method
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9780538733519. This expansive textbook survival guide covers the following chapters and their solutions. Since 20 problems in chapter 2.1: The Bisection Method have been answered, more than 15879 students have viewed full stepbystep solutions from this chapter. Chapter 2.1: The Bisection Method includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Column space C (A) =
space of all combinations of the columns of A.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Iterative method.
A sequence of steps intended to approach the desired solution.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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(DÂ» O.

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

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.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

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

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