 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 Sieva Kozinsky 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 18803 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.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column space C (A) =
space of all combinations of the columns of 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.

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

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose 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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.