 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 24181 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.

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!

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

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.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

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

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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·