 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
ISBN: 9781305253667
Solutions for Chapter 4.8: Multiple Integrals
Get Full SolutionsChapter 4.8: Multiple Integrals includes 20 full stepbystep solutions. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. Since 20 problems in chapter 4.8: Multiple Integrals have been answered, more than 14801 students have viewed full stepbystep solutions from this chapter.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Fundamental Theorem.
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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Norm
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.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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. AI is also symmetric.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.