 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: RepeatExercise 9 using Gaussian elimination with partial pivoting a...
 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 Gaussian elimination with complete pivoting.
 6.2.22: Repeat Exercise 10 using Gaussian elimination with complete pivoting.
 6.2.23: Repeat Exercise 9 using Gaussian elimination with complete pivoting...
 6.2.24: Repeat Exercise 10 using Gaussian elimination with complete pivotin...
 6.2.25: The following circuit has four resistors and two voltage sources. T...
 6.2.26: Suppose that 2x\ + ^2 + 3x3 = '' 4xi + 6x2 + 8x3 = 5, 6X1 + 0!X2 + ...
Solutions for Chapter 6.2: Pivoting Strategies
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 6.2: Pivoting Strategies
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. Since 26 problems in chapter 6.2: Pivoting Strategies have been answered, more than 14072 students have viewed full stepbystep solutions from this chapter. Chapter 6.2: Pivoting Strategies includes 26 full stepbystep solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Outer product uv T
= column times row = rank one matrix.

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.

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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