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Solutions for Chapter 8.4: Multistep Methods

Full solutions for Elementary Differential Equations and Boundary Value Problems | 9th Edition

ISBN: 9780470383346

Solutions for Chapter 8.4: Multistep Methods

This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. Chapter 8.4: Multistep Methods includes 16 full step-by-step solutions. Since 16 problems in chapter 8.4: Multistep Methods have been answered, more than 12779 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • 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!

  • Circulant matrix C.

    Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

  • Eigenvalue A and eigenvector x.

    Ax = AX with x#-O so det(A - AI) = o.

  • 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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • 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.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • 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.

  • Nullspace N (A)

    = All solutions to Ax = O. Dimension n - r = (# columns) - rank.

  • 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.

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Schwarz inequality

    Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

  • Subspace S of V.

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

  • Trace of A

    = sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

  • Transpose matrix AT.

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

  • Wavelets Wjk(t).

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

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