- 10.4.1: Use the method of Steepest Descent with TOL 0.05 to approximate the...
- 10.4.2: Use the method of Steepest Descent with TOL 0.05 to approximate the...
- 10.4.3: Use the results in Exercise I and Newton's method to approximate th...
- 10.4.4: Use the results of Exercise 2 and Newton's method to approximate th...
- 10.4.5: Use the method of Steepest Descent to approximate minima to within ...
- 10.4.6: Exercise 12 in Section 10.2 concerns a biological experiment to det...
- 10.4.7: As people grow older, they tend to wonder if they will outlive thei...
- 10.4.8: a. Show thatthe quadratic polynomial P(a) g\ + h\a + ftjafa - ai) i...
Solutions for Chapter 10.4: Steepest Descent Techniques
Full solutions for Numerical Analysis | 10th Edition
Tv = Av + Vo = linear transformation plus shift.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det 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].
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
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.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Solvable system Ax = b.
The right side b is in the column space of A.
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.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.