- 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 fixed-point iteration to find solutions to the following nonlin...
- 10.1.7: Use the Gauss-Seidel method to approximate the fixed points in Exer...
- 10.1.8: Repeat Exercise 6 using the Gauss-Seidel 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
Remove row i and column j; multiply the determinant by (-I)i + j •
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. Block-diagonal: 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.
Gram-Schmidt 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.
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 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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.
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.
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+ (Moore-Penrose 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 ].