- 4.8.1: Use Algorithm 4.4 with n = m = 4 to approximate the following doubl...
- 4.8.2: Find the smallest values for = m so that Algorithm 4.4 can be used ...
- 4.8.3: Use Algorithm 4.5 with n m 2to approximate the integrals in Exercis...
- 4.8.4: Find the smallest values of n m so that Algorithm 4.5 can be used t...
- 4.8.5: Use Algorithm 4.4 with (i) n 4, m 8, (ii) n 8, m 4, and (iii) n m 6...
- 4.8.6: Find the smallest values for n = m so that Algorithm 4.4 can be use...
- 4.8.7: Use Algorithm 4.5 with (i) n = m = 3, (ii) n = 3, m = 4, (iii) m = ...
- 4.8.8: Use Algorithm 4.5 with n = m = 5 to approximate the integrals in Ex...
- 4.8.9: Use Algorithm 4.4 with n m \4 and Algorithm 4.5 with n m 4 to appro...
- 4.8.10: Use Algorithm 4.4 to approximate Jxy + y 2 dA, R where R is the reg...
- 4.8.11: Use Algorithm 4.6 with n m p 2 to approximate the following triple ...
- 4.8.12: Repeat Exercise 11 using n = m = p = 3
- 4.8.13: Repeat Exercise 11 using n in p 4.
- 4.8.14: Repeat Exercise 11 using n = m = p = 5.
- 4.8.15: Use Algorithm 4.6 with n=m p = 5 to approximate Jxyz dV, where5'ist...
- 4.8.16: Use Algorithm 4.6 with n m p 4to approximate xy sin(yz) dV, s where...
- 4.8.17: A plane lamina is a thin sheet of continuously distributed mass. If...
- 4.8.18: Repeat Exercise 17 using Algorithm 4.5 with n = m = 5
- 4.8.19: The area ofthe surface described by z = /(x, y) for (x, y) in R is ...
- 4.8.20: Repeat Exercise 19 using Algorithm 4.5 with n = m = 4
Solutions for Chapter 4.8: Multiple Integrals
Full solutions for Numerical Analysis | 10th Edition
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
peA) = det(A - AI) has peA) = zero matrix.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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).
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.
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