 5.6.1: Use all the AdamsBashforth methods to approximate the solutions to...
 5.6.2: Use all the AdamsBashforth methods to approximate the solutions to...
 5.6.3: Use each ofthe AdamsBashforth methods to approximate the solutions...
 5.6.4: Use each ofthe AdamsBashforth methods to approximate the solutions...
 5.6.5: Use all the AdamsMoulton methods to approximate the solutions to t...
 5.6.6: Use all the AdamsMoulton methods to approximate the solutions to t...
 5.6.7: Use Algorithm 5.4 to approximate the solutions to the initialvalue...
 5.6.8: Use Algorithm 5.4 to approximate the solutions to the initialvalue...
 5.6.9: Use Algorithm 5.4 to approximate the solutions to the initialvalue...
 5.6.10: Use Algorithm 5.4 to approximate the solutions to the initialvalue...
 5.6.11: Use the MilneSimpson predictorcorrector method to approximate the...
 5.6.12: Use the MilneSimpson predictorcorrector method to approximate the...
 5.6.13: The initialvalue problem y' = e y , 0 < t < 0.20, >>(0) = 1, has s...
 5.6.14: The Gompertz differential equation N'it) = a\n^Nit) serves as a mo...
 5.6.15: Change Algorithm 5.4 so that the corrector can be iterated for a gi...
 5.6.16: a. Derive the AdamsBashforth twostep method by using the Lagrange...
 5.6.17: Derive the AdamsBashforth threestep method by the following metho...
 5.6.18: Derive the AdamsMoulton twostep method and its local truncation e...
 5.6.19: Derive Simpson's method by applying Simpson's rule to the integral ...
 5.6.20: Derive Milne's method by applying the open NewtonCotes formula (4....
 5.6.21: Verify the entries in Table 5.12 on page 305
Solutions for Chapter 5.6: Multistep Method
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 5.6: Multistep Method
Get Full SolutionsThis textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. 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 21 problems in chapter 5.6: Multistep Method have been answered, more than 15013 students have viewed full stepbystep solutions from this chapter. Chapter 5.6: Multistep Method includes 21 full stepbystep solutions.

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).

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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

Column space C (A) =
space of all combinations of the columns of A.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).