 4.2.1: Apply the extrapolation process described in Example 1 to determine...
 4.2.2: Add another line to the extrapolation table in Exercise 1 to obtain...
 4.2.3: Repeat Exercise 1 using fourdigit rounding arithmetic.
 4.2.4: Repeat Exercise 2 using fourdigit rounding arithmetic.
 4.2.5: The following data give approximations to the integral M= sinx dx. ...
 4.2.6: The following data can be used to approximate the integral M cosx d...
 4.2.7: Show that the fivepoint formula in Eq. (4.6) applied to /(x) = xex...
 4.2.8: The forwarddifference formula can be expressed as /'(*o) = pf/Uo +...
 4.2.9: Suppose that N(h) is an approximation to M for every h > 0 and that...
 4.2.10: Suppose that N(h) is an approximation to M for every h > 0 and that...
 4.2.11: In calculus, we learn that e = lim/,^o(l + h)^h . a. Determine appr...
 4.2.12: a. Show that f 2 + h \ l//' lim  = e. h*0 \2 h J b. Compute app...
 4.2.13: Suppose the following extrapolation table has been constructed to a...
 4.2.14: Suppose that N\ (A) is a formula that produces O(h) approximations ...
 4.2.15: The semiperimeters of regular polygons with k sides that inscribe a...
Solutions for Chapter 4.2: Richardson's Extrapolation
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 4.2: Richardson's Extrapolation
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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 space C (A) =
space of all combinations of the columns of 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 II· Condition numbers measure the sensitivity of the output to change in the input.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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 ().

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

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