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Solutions for Chapter 5.8: Extrapolation Methods

Numerical Analysis | 10th Edition | ISBN: 9781305253667 | Authors: Richard L. Burden J. Douglas Faires, Annette M. Burden

Full solutions for Numerical Analysis | 10th Edition

ISBN: 9781305253667

Numerical Analysis | 10th Edition | ISBN: 9781305253667 | Authors: Richard L. Burden J. Douglas Faires, Annette M. Burden

Solutions for Chapter 5.8: Extrapolation Methods

Numerical Analysis was written by and is associated to the ISBN: 9781305253667. This expansive textbook survival guide covers the following chapters and their solutions. Since 6 problems in chapter 5.8: Extrapolation Methods have been answered, more than 58055 students have viewed full step-by-step solutions from this chapter. Chapter 5.8: Extrapolation Methods includes 6 full step-by-step solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Change of basis matrix M.

    The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

  • Characteristic equation det(A - AI) = O.

    The n roots are the eigenvalues of A.

  • Cholesky factorization

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

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Graph G.

    Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = O.

  • Partial pivoting.

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

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • 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).

  • Rotation matrix

    R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

  • Standard basis for Rn.

    Columns of n by n identity matrix (written i ,j ,k in R3).

  • Symmetric matrix A.

    The transpose is AT = A, and aU = a ji. A-I is also symmetric.

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • 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·