- 5.8.1: Use the Extrapolation Algorithm with tolerance TOL I0-4 , hmax 0.25...
- 5.8.2: Use the Extrapolation Algorithm with TOL 10~4 to approximate the so...
- 5.8.3: Use the Extrapolation Algorithm with tolerance TOL 10-6 , hmax 0.5,...
- 5.8.4: Use the Extrapolation Algorithm with tolerance TOL = 10~6 , hmax = ...
- 5.8.5: Suppose a wolf is chasing a rabbit. The path ofthe wolftoward the r...
- 5.8.6: The Gompertz population model was described in Exercise 26 ofSectio...
Solutions for Chapter 5.8: Extrapolation Methods
Full solutions for Numerical Analysis | 10th Edition
ISBN: 9781305253667
Solutions for Chapter 5.8
23
5
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
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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).
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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.)
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Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
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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.
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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.
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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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Iterative method.
A sequence of steps intended to approach the desired solution.
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Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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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).
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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 ().
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Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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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·