 11.4.1: Use the Nonlinear FiniteDifference method with h = 0.5 to approxim...
 11.4.2: Use the Nonlinear FiniteDifference method with h = 0.25 to approxi...
 11.4.3: Use the Nonlinear FiniteDifference Algorithm with TOL = 104 to app...
 11.4.4: Use the Nonlinear FiniteDifference Algorithm with TOL = 104 to app...
 11.4.5: Repeat Exercise 4(a) and 4(b) using extrapolation.
 11.4.6: In Exercise 7 of Section 11.3, the deflection of a beam with suppor...
 11.4.7: Show that the hypotheses listed at the beginning of the section ens...
Solutions for Chapter 11.4: FiniteDifference Methods for Nonlinear Problems
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 11.4: FiniteDifference Methods for Nonlinear Problems
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9780538733519. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Chapter 11.4: FiniteDifference Methods for Nonlinear Problems includes 7 full stepbystep solutions. Since 7 problems in chapter 11.4: FiniteDifference Methods for Nonlinear Problems have been answered, more than 15971 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

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.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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).

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

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.