 21.2.21.1.21: Carry out and show the details of the calculations leading to (4 H7...
 21.2.21.1.22: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.23: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.24: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.25: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.26: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.27: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.28: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.29: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.30: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.31: Solve the initial value problems by AdamsMoulton. 10 steps with I ...
 21.2.21.1.32: Show that by applying the method in the text to a polynomial of sec...
 21.2.21.1.33: Use Prob. 12 to solve y' = 2x.\". y(O) = I (10 steps, It = 0.1, RK ...
 21.2.21.1.34: How much can you reduce the error in Prob. 13 by halving II (20 ste...
 21.2.21.1.35: CAS PROJECT. AdamsMoulton. (a) Accurate starting is important in (...
Solutions for Chapter 21.2: Multistep Methods
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 21.2: Multistep Methods
Get Full SolutionsAdvanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Since 15 problems in chapter 21.2: Multistep Methods have been answered, more than 49128 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 21.2: Multistep Methods includes 15 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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).

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].

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

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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].

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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·

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

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

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