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Solutions for Chapter 2.4: Numerical Approximation: Eulers Method

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition | ISBN: 9780321796981 | Authors: C. Henry Edwards, David E. Penney, David T. Calvis

Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition

ISBN: 9780321796981

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition | ISBN: 9780321796981 | Authors: C. Henry Edwards, David E. Penney, David T. Calvis

Solutions for Chapter 2.4: Numerical Approximation: Eulers Method

Solutions for Chapter 2.4
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Textbook: Differential Equations and Boundary Value Problems: Computing and Modeling
Edition: 5
Author: C. Henry Edwards, David E. Penney, David T. Calvis
ISBN: 9780321796981

Since 31 problems in chapter 2.4: Numerical Approximation: Eulers Method have been answered, more than 15581 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. Chapter 2.4: Numerical Approximation: Eulers Method includes 31 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Differential Equations and Boundary Value Problems: Computing and Modeling was written by and is associated to the ISBN: 9780321796981.

Key Math Terms and definitions covered in this textbook
  • 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).

  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • 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.

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • Multiplication Ax

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

  • Nullspace matrix N.

    The columns of N are the n - r special solutions to As = 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).

  • Permutation matrix P.

    There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

  • Polar decomposition A = Q H.

    Orthogonal Q times positive (semi)definite H.

  • Random matrix rand(n) or randn(n).

    MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • 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 ().

  • Similar matrices A and B.

    Every B = M-I AM has the same eigenvalues as A.

  • Toeplitz matrix.

    Constant down each diagonal = time-invariant (shift-invariant) filter.

  • Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

    T- 1 has rank 1 above and below diagonal.

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

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