 5.3.1: Use Taylors method of order two to approximate the solutions for ea...
 5.3.2: Use Taylors method of order two to approximate the solutions for ea...
 5.3.3: Repeat Exercise 1 using Taylors method of order four.
 5.3.4: Repeat Exercise 2 using Taylors method of order four.
 5.3.5: Use Taylors method of order two to approximate the solution for eac...
 5.3.6: Use Taylors method of order two to approximate the solution for eac...
 5.3.7: Repeat Exercise 5 using Taylors method of order four.
 5.3.8: Repeat Exercise 6 using Taylors method of order four.
 5.3.9: Given the initialvalue problem y = 2 t y + t 2 et , 1 t 2, y(1) = ...
 5.3.10: Given the initialvalue problem y = 1 t2 y t y2 , 1 t 2, y(1) = 1, ...
 5.3.11: A projectile of mass m = 0.11 kg shot vertically upward with initia...
 5.3.12: Use the Taylor method of order two with h = 0.1 to approximate the ...
Solutions for Chapter 5.3: HigherOrder Taylor Methods
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 5.3: HigherOrder Taylor Methods
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CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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.