 10.1.1: The nonlinear system + 1) + 2x2 = 18, (x,  I)2 + (xz  6)2 = 25 ha...
 10.1.2: The nonlinear system x 2 x 2 + Ax, 2 = 0, x 2 + 3x 4 = 0 has two ...
 10.1.3: The nonlinear system x 2 10xi + XT +8 = 0, XIXT+X] 10x2 + 8 = 0 can...
 10.1.4: The nonlinear system 5x2 xj 0, X2 0.25(sinxi + COSX2) = 0 has a sol...
 10.1.5: Use Theorem 10.6 to show that G ; D C E 3 E 3 has a unique fixed po...
 10.1.6: Use fixedpoint iteration to find solutions to the following nonlin...
 10.1.7: Use the GaussSeidel method to approximate the fixed points in Exer...
 10.1.8: Repeat Exercise 6 using the GaussSeidel method.
 10.1.9: In Exercise 6 of Section 5.9, we considered the problem of predicti...
 10.1.10: The population dynamics ofthree competing species can be described ...
 10.1.11: Show that the function F : M 3 R 3 defined by F(xi, X2, X3) = (X +...
 10.1.12: Give an example of a function F ; R 2 > R 2 that is continuous at e...
 10.1.13: Show thatthe first partial derivatives in Example 2 are continuous ...
 10.1.14: Show that a function F mapping D C R" into R" is continuous at Xq 6...
 10.1.15: Let A be an n x n matrix and Fbe the function from R" to R" defined...
Solutions for Chapter 10.1: Fixed Points for Functions of Several Variables
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 10.1: Fixed Points for Functions of Several Variables
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9781305253667. Chapter 10.1: Fixed Points for Functions of Several Variables includes 15 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. Since 15 problems in chapter 10.1: Fixed Points for Functions of Several Variables have been answered, more than 15166 students have viewed full stepbystep solutions from this chapter.

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

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

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.

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

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].