- 11.4.1: Use the Nonlinear Finite-Difference method with h = 0.5 to approxim...
- 11.4.2: Use the Nonlinear Finite-Difference method with h = 0.25 to approxi...
- 11.4.3: Use the Nonlinear Finite-Difference Algorithm with TOL = 104 to app...
- 11.4.4: Use the Nonlinear Finite-Difference 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: Finite-Difference Methods for Nonlinear Problems
Full solutions for Numerical Analysis | 9th Edition
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 S-I AS = A = eigenvalue matrix.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Invert A by row operations on [A I] to reach [I A-I].
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal 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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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+ (Moore-Penrose 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.
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.
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.