 7.6.1: The linear system 1 X + X2 1 1 A* + 3 12 has solution (xj, xa)' =...
 7.6.2: The linear system O.Ixi + 0.2x2 = 0.3, 0.2xi + 113x2 = 113.2, has s...
 7.6.3: The linear system 4. 5. 6. 7. 8. 9. Xi 4 1 2'2 + 1 3 X3 = 5 6' 1 2...
 7.6.4: Repeat Exercise 3 using singleprecision arithmetic on a computer
 7.6.5: Perform only two steps ofthe conjugate gradient method with C = C _...
 7.6.6: Repeat Exercise 5 using C 1 = D i/2 .
 7.6.7: Repeat Exercise 5 with TOL 103 in the norm. Compare the results in...
 7.6.8: Repeat Exercise 7 using C 1 = D _l/' 2 .
 7.6.9: Approximate solutions to the following linear systems Ax = b to wit...
 7.6.10: Solve the linear system in Exercise 14(b) of Exercise Set 7.3 using...
 7.6.11: Let A i 4 1 0 0 1 4 1 0 0 1 4 1 0 0 1 4 0 0 0 0 " 0 0 0 0 0 0...
 7.6.12: A coaxial cable is made up of a 0.1 inchsquare inner conductor an...
 7.6.13: Suppose that an object can be at any one ofn+1 equally spaced point...
 7.6.14: Use the transpose properties given in Theorem 6.14 on page 394 to p...
 7.6.15: a. Show that an Aorthogonal set of nonzero vectors associated with...
 7.6.16: Prove Theorem 7.33 using mathematical induction as follows: a. Show...
 7.6.17: In Example 3, the eigenvalues were found for the matrix A and the c...
Solutions for Chapter 7.6: The Conjugate Gradient Method
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 7.6: The Conjugate Gradient Method
Get Full SolutionsSince 17 problems in chapter 7.6: The Conjugate Gradient Method have been answered, more than 15125 students have viewed full stepbystep solutions from this chapter. Chapter 7.6: The Conjugate Gradient Method includes 17 full stepbystep solutions. 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. Numerical Analysis was written by and is associated to the ISBN: 9781305253667.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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.)

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.