- 3.2.1: Use Nevilles method to obtain the approximations for Lagrange inter...
- 3.2.2: Use Nevilles method to obtain the approximations for Lagrange inter...
- 3.2.3: Use Nevilles method to approximate 3 with the following functions a...
- 3.2.4: Let P3(x) be the interpolating polynomial for the data (0, 0), (0.5...
- 3.2.5: Nevilles method is used to approximate f (0.4), giving the followin...
- 3.2.6: Nevilles method is used to approximate f (0.5), giving the followin...
- 3.2.7: Suppose xj = j, for j = 0, 1, 2, 3 and it is known that P0,1(x) = 2...
- 3.2.8: Suppose xj = j, for j = 0, 1, 2, 3 and it is known that P0,1(x) = x...
- 3.2.9: Nevilles Algorithm is used to approximate f (0) using f (2), f (1),...
- 3.2.10: Nevilles Algorithm is used to approximate f (0) using f (2), f (1),...
- 3.2.11: Construct a sequence of interpolating values yn to f (1 + 10), wher...
- 3.2.12: Use iterated inverse interpolation to find an approximation to the ...
- 3.2.13: Construct an algorithm that can be used for inverse interpolation.
Solutions for Chapter 3.2: Data Approximation and Neville's Method
Full solutions for Numerical Analysis | 9th Edition
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.)
Remove row i and column j; multiply the determinant by (-I)i + j •
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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).
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.