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Solutions for Chapter 19.3: Laurent Series

Advanced Engineering Mathematics | 6th Edition | ISBN: 9781284105902 | Authors: Dennis G. Zill

Full solutions for Advanced Engineering Mathematics | 6th Edition

ISBN: 9781284105902

Advanced Engineering Mathematics | 6th Edition | ISBN: 9781284105902 | Authors: Dennis G. Zill

Solutions for Chapter 19.3: Laurent Series

Solutions for Chapter 19.3
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Textbook: Advanced Engineering Mathematics
Edition: 6
Author: Dennis G. Zill
ISBN: 9781284105902

Chapter 19.3: Laurent Series includes 28 full step-by-step solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781284105902. Since 28 problems in chapter 19.3: Laurent Series have been answered, more than 36448 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics , edition: 6. This expansive textbook survival guide covers the following chapters and their solutions.

Key Math Terms and definitions covered in this textbook
  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Column picture of Ax = b.

    The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

  • Diagonalizable matrix A.

    Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

  • Elimination.

    A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

  • Fibonacci numbers

    0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Iterative method.

    A sequence of steps intended to approach the desired solution.

  • Kronecker product (tensor product) A ® B.

    Blocks aij B, eigenvalues Ap(A)Aq(B).

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. 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.

  • 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).

  • Partial pivoting.

    In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

  • 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).

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Schur complement S, D - C A -} B.

    Appears in block elimination on [~ g ].

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Stiffness matrix

    If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

  • Vandermonde matrix V.

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

  • Volume of box.

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

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