 4.5.1: Use Romberg integration to compute R3 3 for the following integrals...
 4.5.2: Use Romberg integration to compute R3.3 for the following integrals...
 4.5.3: Calculate R4.4 for the integrals in Exercise 1.
 4.5.4: Calculate ^4,4 for the integrals in Exercise 2.
 4.5.5: Use Romberg integration to approximate the integrals in Exercise I ...
 4.5.6: Use Romberg integration to approximate the integrals in Exercise 2 ...
 4.5.7: Use the following data to approximate J,5 /(x) dx as accurately as ...
 4.5.8: Use the following data to approximate J0 6 f{x) dx as accurately as...
 4.5.9: Romberg integration is used to approximate ^ f(x)dx. If /(2) = 0.51...
 4.5.10: Romberg integration is used to approximate /' x 2 / ; 7 (lx  Jo 1 ...
 4.5.11: Romberg integration for approximating /j' f(x) dx gives R\\ =8, R22...
 4.5.12: Romberg integration for approximating /(x) r/x gives Ru = 4 and R22...
 4.5.13: Use Romberg integration to compute the following approximations to ...
 4.5.14: In Exercise 24 of Section 1.1, a Maclaurin series was integrated to...
 4.5.15: Find an approximation to within 104 ofthe value ofthe integral con...
 4.5.16: In Section 4.4 Exercise 19, the Composite Simpson's method was used...
 4.5.17: Show thatthe approximation obtained from Rk.2 isthe same asthat giv...
 4.5.18: Show that, for any k, 2*"2 2*"2 1 = S + + Yl f (  a + ihk\)
 4.5.19: Use the result of Exercise 18 to verify Eq. (4.34); that is, show t...
Solutions for Chapter 4.5: Romberg Integration
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 4.5: Romberg Integration
Get Full SolutionsSince 19 problems in chapter 4.5: Romberg Integration have been answered, more than 13813 students have viewed full stepbystep solutions from this chapter. Chapter 4.5: Romberg Integration includes 19 full stepbystep solutions. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Companion matrix.
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 nl  An).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.