 5.7.1: Use the Adams Variable StepSize PredictorCorrector Algorithm with...
 5.7.2: Use the Adams Variable StepSize PredictorCorrector Algorithm with...
 5.7.3: Use the Adams Variable StepSize PredictorCorrector Algorithm with...
 5.7.4: Construct an Adams Variable StepSize PredictorCorrector Algorithm...
 5.7.5: An electrical circuit consists of a capacitor of constant capacitan...
Solutions for Chapter 5.7: Variable StepSize Multistep Methods
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 5.7: Variable StepSize Multistep Methods
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.7: Variable StepSize Multistep Methods includes 5 full stepbystep solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Since 5 problems in chapter 5.7: Variable StepSize Multistep Methods have been answered, more than 12595 students have viewed full stepbystep solutions from this chapter. Numerical Analysis was written by and is associated to the ISBN: 9780538733519.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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).

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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 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)·(b  Ax) = o.

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.

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

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

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