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# Solutions for Chapter 1.2: Round-off Errors and Computer Arithmetic

## Full solutions for Numerical Analysis | 9th Edition

ISBN: 9780538733519

Solutions for Chapter 1.2: Round-off Errors and Computer Arithmetic

Solutions for Chapter 1.2
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##### ISBN: 9780538733519

Chapter 1.2: Round-off Errors and Computer Arithmetic includes 28 full step-by-step 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: Round-off Errors and Computer Arithmetic have been answered, more than 15507 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9.

Key Math Terms and definitions covered in this textbook
• Cholesky factorization

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.

• Diagonalization

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.

• Fibonacci numbers

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.

• 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.

• 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 = 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 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.

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