 9.1.1: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.2: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.3: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.4: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.5: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.6: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.7: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.8: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.9: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.10: In 110 use the improved Eulers method to obtain a fourdecimal appr...
 9.1.11: Consider the initialvalue problem y (x y 1)2 , y(0) 2. Use the imp...
 9.1.12: Although it might not be obvious from the differential equation, it...
 9.1.13: Consider the initialvalue problem y 2y, y(0) 1. The analytic solut...
 9.1.14: Repeat using the improved Eulers method. Its global truncation erro...
 9.1.15: Repeat using the initialvalue problem y x 2y, y(0) 1. The analytic...
 9.1.16: Repeat using the improved Eulers method. Its global truncation erro...
 9.1.17: Consider the initialvalue problem y 2x 3y 1, y(1) 5. The analytic ...
 9.1.18: Repeat using the improved Eulers method, which has a global truncat...
 9.1.19: Repeat for the initialvalue problem y ey , y(0) 0. The analytic so...
 9.1.20: Repeat using the improved Eulers method, which has global truncatio...
 9.1.21: Answer the question Why not? that follows the three sentences after...
Solutions for Chapter 9.1: EULER METHODS AND ERROR ANALYSIS
Full solutions for A First Course in Differential Equations with Modeling Applications  10th Edition
ISBN: 9781111827052
Solutions for Chapter 9.1: EULER METHODS AND ERROR ANALYSIS
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. A First Course in Differential Equations with Modeling Applications was written by Sieva Kozinsky and is associated to the ISBN: 9781111827052. Since 21 problems in chapter 9.1: EULER METHODS AND ERROR ANALYSIS have been answered, more than 20405 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: A First Course in Differential Equations with Modeling Applications, edition: 10th. Chapter 9.1: EULER METHODS AND ERROR ANALYSIS 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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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).

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Outer product uv T
= column times row = rank one matrix.

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 ].

Solvable system Ax = b.
The right side b is in the column space of A.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.