 1.2.1: Compute the absolute error and relative error in approximations of ...
 1.2.2: Find the largest interval in which p must lie to approximate p with...
 1.2.3: Suppose p must approximate p with relative error at most 103. Find ...
 1.2.4: Perform the following computations (i) exactly, (ii) using threedi...
 1.2.5: Use threedigit rounding arithmetic to perform the following calcul...
 1.2.6: Repeat Exercise 5 using fourdigit rounding arithmetic.
 1.2.7: Repeat Exercise 5 using threedigit chopping arithmetic.
 1.2.8: Repeat Exercise 5 using fourdigit chopping arithmetic.
 1.2.9: The first three nonzero terms of the Maclaurin series for the arcta...
 1.2.10: The number e can be defined by e = n=0(1/n!), where n! = n(n 1) 2 1...
 1.2.11: Let f (x) = x cos x sin x x sin x . a. Find limx0 f (x). b. Use fou...
 1.2.12: Let f (x) = ex ex x . a. Find limx0(ex ex )/x. b. Use threedigit r...
 1.2.13: Use fourdigit rounding arithmetic and the formulas (1.1), (1.2), a...
 1.2.14: Repeat Exercise 13 using fourdigit chopping arithmetic.
 1.2.15: Use the 64bit long real format to find the decimal equivalent of t...
 1.2.16: Find the next largest and smallest machine numbers in decimal form ...
 1.2.17: Suppose two points (x0, y0) and (x1, y1) are on a straight line wit...
 1.2.18: The Taylor polynomial of degree n for f (x) = ex is n i=0(xi /i!). ...
 1.2.19: The twobytwo linear system ax + by = e, cx + dy = f , where a, b,...
 1.2.20: Repeat Exercise 19 using fourdigit chopping arithmetic.
 1.2.21: a. Show that the polynomial nesting technique described in Example ...
 1.2.22: A rectangular parallelepiped has sides of length 3 cm, 4 cm, and 5 ...
 1.2.23: Let Pn(x) be the Maclaurin polynomial of degree n for the arctangen...
 1.2.24: Suppose that f l(y) is a kdigit rounding approximation to y. Show ...
 1.2.25: The binomial coefficient m k = m! k!(m k)! describes the number of ...
 1.2.26: Let f C[a, b] be a function whose derivative exists on (a, b). Supp...
 1.2.27: The following Maple procedure chops a floatingpoint number x to t ...
 1.2.28: The opening example to this chapter described a physical experiment...
Solutions for Chapter 1.2: Roundoff Errors and Computer Arithmetic
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 1.2: Roundoff Errors and Computer Arithmetic
Get Full SolutionsChapter 1.2: Roundoff Errors and Computer Arithmetic includes 28 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: 9780538733519. Since 28 problems in chapter 1.2: Roundoff Errors and Computer Arithmetic have been answered, more than 15507 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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).

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.

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

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

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.