- 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
Column space C (A) =
space of all combinations of the columns of A.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
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.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Every v in V is orthogonal to every w in W.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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.