 6.2.1: Find the row interchanges that are required to solve the following ...
 6.2.2: Find the row interchanges that are required to solve the following ...
 6.2.3: Repeat Exercise 1 using Algorithm 6.2.
 6.2.4: Repeat Exercise 2 using Algorithm 6.2.
 6.2.5: Repeat Exercise 1 using Algorithm 6.3.
 6.2.6: Repeat Exercise 2 using Algorithm 6.3.
 6.2.7: Repeat Exercise 1 using complete pivoting.
 6.2.8: Repeat Exercise 2 using complete pivoting.
 6.2.9: Use Gaussian elimination and threedigit chopping arithmetic to sol...
 6.2.10: Use Gaussian elimination and threedigit chopping arithmetic to sol...
 6.2.11: Repeat Exercise 9 using threedigit rounding arithmetic.
 6.2.12: Repeat Exercise 10 using threedigit rounding arithmetic.
 6.2.13: Repeat Exercise 9 using Gaussian elimination with partial pivoting.
 6.2.14: Repeat Exercise 10 using Gaussian elimination with partial pivoting.
 6.2.15: Repeat Exercise 9 using Gaussian elimination with partial pivoting ...
 6.2.16: Repeat Exercise 10 using Gaussian elimination with partial pivoting...
 6.2.17: Repeat Exercise 9 using Gaussian elimination with scaled partial pi...
 6.2.18: Repeat Exercise 10 using Gaussian elimination with scaled partial p...
 6.2.19: Repeat Exercise 9 using Gaussian elimination with scaled partial pi...
 6.2.20: Repeat Exercise 10 using Gaussian elimination with scaled partial p...
 6.2.21: Repeat Exercise 9 using Algorithm 6.1 in Maple with Digits:= 10.
 6.2.22: Repeat Exercise 10 using Algorithm 6.1 in Maple with Digits:= 10.
 6.2.23: Repeat Exercise 9 using Algorithm 6.2 in Maple with Digits:= 10.
 6.2.24: Repeat Exercise 10 using Algorithm 6.2 in Maple with Digits:= 10.
 6.2.25: Repeat Exercise 9 using Algorithm 6.3 in Maple with Digits:= 10.
 6.2.26: Repeat Exercise 10 using Algorithm 6.3 in Maple with Digits:= 10.
 6.2.27: Repeat Exercise 9 using Gaussian elimination with complete pivoting.
 6.2.28: Repeat Exercise 10 using Gaussian elimination with complete pivoting.
 6.2.29: Repeat Exercise 9 using Gaussian elimination with complete pivoting...
 6.2.30: Repeat Exercise 10 using Gaussian elimination with complete pivotin...
 6.2.31: Suppose that 2x1 + x2 + 3x3 = 1, 4x1 + 6x2 + 8x3 = 5, 6x1 + x2 + 10...
 6.2.32: Construct an algorithm for the complete pivoting procedure discusse...
 6.2.33: Use the complete pivoting algorithm to repeat Exercise 9 Maple with...
 6.2.34: Use the complete pivoting algorithm to repeat Exercise 10 Maple wit...
Solutions for Chapter 6.2: Pivoting Strategies
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 6.2: Pivoting Strategies
Get Full SolutionsSince 34 problems in chapter 6.2: Pivoting Strategies have been answered, more than 15776 students have viewed full stepbystep solutions from this chapter. Chapter 6.2: Pivoting Strategies includes 34 full stepbystep solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9780538733519.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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.

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.

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.

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

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Solvable system Ax = b.
The right side b is in the column space of A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

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. AI is also symmetric.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).