- 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 three-di...
- 1.2.5: Use three-digit rounding arithmetic to perform the following calcul...
- 1.2.6: Repeat Exercise 5 using four-digit rounding arithmetic.
- 1.2.7: Repeat Exercise 5 using three-digit chopping arithmetic.
- 1.2.8: Repeat Exercise 5 using four-digit 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 three-digit r...
- 1.2.13: Use four-digit rounding arithmetic and the formulas (1.1), (1.2), a...
- 1.2.14: Repeat Exercise 13 using four-digit chopping arithmetic.
- 1.2.15: Use the 64-bit 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 two-by-two linear system ax + by = e, cx + dy = f , where a, b,...
- 1.2.20: Repeat Exercise 19 using four-digit 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 k-digit 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 floating-point number x to t ...
- 1.2.28: The opening example to this chapter described a physical experiment...
Solutions for Chapter 1.2: Round-off Errors and Computer Arithmetic
Full solutions for Numerical Analysis | 9th Edition
A = CTC = (L.J]))(L.J]))T for positive definite A.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + 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, off-diagonal entries = 0.
A sequence of steps intended to approach the desired solution.
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.
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.
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 = Q-1 = 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 II·
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.