 1.15.1: Using the improved Euler method with h = 0.1, determine an approxim...
 1.15.2: Using the improved Euler method with h = 0.1, determine an approxim...
 1.15.3: Using the improved Euler method with h = 0.1, determine an approxim...
 1.15.4: Using the improved Euler method with h = 0.1, determine an approxim...
 1.15.5: Using the improved Euler method with h = 0.1, determine an approxim...
 1.15.6: Using the improved Euler method with h = n/40, determine an approxi...
Solutions for Chapter 1.15: An improved Euler method
Full solutions for Differential Equations and Their Applications: An Introduction to Applied Mathematics  3rd Edition
ISBN: 9780387908069
Solutions for Chapter 1.15: An improved Euler method
Get Full SolutionsDifferential Equations and Their Applications: An Introduction to Applied Mathematics was written by and is associated to the ISBN: 9780387908069. Since 6 problems in chapter 1.15: An improved Euler method have been answered, more than 5898 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Their Applications: An Introduction to Applied Mathematics, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.15: An improved Euler method includes 6 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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.

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).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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.

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.