 5.5.1: In each of 1 through 6, draw a direction field for the differential...
 5.5.7: In each of 1 through 6, generate approximate numerical values of th...
 5.5.13: In each of 1 through 6, use the secondorder Taylor method and the ...
 5.5.2: In each of 1 through 6, draw a direction field for the differential...
 5.5.8: In each of 1 through 6, generate approximate numerical values of th...
 5.5.14: In each of 1 through 6, use the secondorder Taylor method and the ...
 5.5.3: In each of 1 through 6, draw a direction field for the differential...
 5.5.9: In each of 1 through 6, generate approximate numerical values of th...
 5.5.15: In each of 1 through 6, use the secondorder Taylor method and the ...
 5.5.4: In each of 1 through 6, draw a direction field for the differential...
 5.5.10: In each of 1 through 6, generate approximate numerical values of th...
 5.5.16: In each of 1 through 6, use the secondorder Taylor method and the ...
 5.5.5: In each of 1 through 6, draw a direction field for the differential...
 5.5.11: In each of 1 through 6, generate approximate numerical values of th...
 5.5.17: In each of 1 through 6, use the secondorder Taylor method and the ...
 5.5.6: .In each of 1 through 6, draw a direction field for the differentia...
 5.5.12: In each of 1 through 6, generate approximate numerical values of th...
 5.5.18: In each of 1 through 6, use the secondorder Taylor method and the ...
Solutions for Chapter 5: Approximation of Solutions
Full solutions for Advanced Engineering Mathematics  7th Edition
ISBN: 9781111427412
Solutions for Chapter 5: Approximation of Solutions
Get Full SolutionsSince 18 problems in chapter 5: Approximation of Solutions have been answered, more than 36861 students have viewed full stepbystep solutions from this chapter. Chapter 5: Approximation of Solutions includes 18 full stepbystep solutions. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9781111427412. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Outer product uv T
= column times row = rank one matrix.

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).

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.