- 5.8.1: Use the Extrapolation Algorithm with tolerance TOL I0-4 , hmax 0.25...
- 5.8.2: Use the Extrapolation Algorithm with TOL 10~4 to approximate the so...
- 5.8.3: Use the Extrapolation Algorithm with tolerance TOL 10-6 , hmax 0.5,...
- 5.8.4: Use the Extrapolation Algorithm with tolerance TOL = 10~6 , hmax = ...
- 5.8.5: Suppose a wolf is chasing a rabbit. The path ofthe wolftoward the r...
- 5.8.6: The Gompertz population model was described in Exercise 26 ofSectio...
Solutions for Chapter 5.8: Extrapolation Methods
Full solutions for Numerical Analysis | 10th Edition
ISBN: 9781305253667
Solutions for Chapter 5.8
23
5
Numerical Analysis was written by and is associated to the ISBN: 9781305253667. This expansive textbook survival guide covers the following chapters and their solutions. Since 6 problems in chapter 5.8: Extrapolation Methods have been answered, more than 14107 students have viewed full step-by-step solutions from this chapter. Chapter 5.8: Extrapolation Methods includes 6 full step-by-step solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10.
Key Math Terms and definitions covered in this textbook
-
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.
-
Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
-
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
-
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
-
Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
-
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.
-
Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and - ).
-
Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
-
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
-
Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
-
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
-
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.
-
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
-
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
-
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
-
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
-
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
-
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
-
Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
-
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.