 5.9.1: Use the RungeKutta method for systems to approximate the solutions...
 5.9.2: Use the RungeKutta method for systems to approximate the solutions...
 5.9.3: Use the RungeKutta for Systems Algorithm to approximate the soluti...
 5.9.4: Use the RungeKutta for Systems Algorithm to approximate the soluti...
 5.9.5: Change the Adams FourthOrder PredictorCorrector Algorithm to obta...
 5.9.6: Repeat Exercise 2 using the algorithm developed in Exercise 5.
 5.9.7: Repeat Exercise 1 using the algorithm developed in Exercise 5.
 5.9.8: Suppose the swinging pendulum described in the lead example of this...
 5.9.9: The study of mathematical models for predicting the population dyna...
 5.9.10: In Exercise 9 we considered the problem of predicting the populatio...
Solutions for Chapter 5.9: HigherOrder Equations and Systems of Differential Equations
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 5.9: HigherOrder Equations and Systems of Differential Equations
Get Full SolutionsSince 10 problems in chapter 5.9: HigherOrder Equations and Systems of Differential Equations have been answered, more than 13879 students have viewed full stepbystep solutions from this chapter. Chapter 5.9: HigherOrder Equations and Systems of Differential Equations includes 10 full stepbystep solutions. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9.

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).

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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

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!).

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

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·