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# Solutions for Chapter 4.5: Romberg Integration

## Full solutions for Numerical Analysis | 10th Edition

ISBN: 9781305253667

Solutions for Chapter 4.5: Romberg Integration

Solutions for Chapter 4.5
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##### ISBN: 9781305253667

Since 19 problems in chapter 4.5: Romberg Integration have been answered, more than 13813 students have viewed full step-by-step solutions from this chapter. Chapter 4.5: Romberg Integration includes 19 full step-by-step 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.

Key Math Terms and definitions covered in this textbook
• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• Cayley-Hamilton 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 n-l - An).

• Condition number

cond(A) = c(A) = IIAIlIIA-III = 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. Block-diagonal: 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}.

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