- 8.3.1: Use the zeros of T3 to construct an interpolating polynomial of deg...
- 8.3.2: Use the zeros offtto construct an interpolating polynomial ofdegree...
- 8.3.3: Find a bound for the maximum error ofthe approximation in Exercise ...
- 8.3.4: Repeat Exercise 3 for the approximations computed in Exercise 3.
- 8.3.5: Use the zeros of Tj and transformations ofthe given interval to con...
- 8.3.6: Find the sixth Maciaurin polynomial for xex and use Chebyshev econo...
- 8.3.7: Find the sixth Maciaurin polynomial for sinx and use Chebyshev econ...
- 8.3.8: The Chebyshev polynomials 7(x) are solutionsto the differential equ...
- 8.3.9: An interesting fact is that Tn (x) equals the determinant of the tr...
- 8.3.10: Show that for any positive integers i and j with / > j, we have Ti(...
- 8.3.11: Show that for each Chebyshev polynomial Tn(x), we have /' [r(x)J2 =...
- 8.3.12: Show that for each n, the Chebyshev polynomial r(x) has n distinct ...
- 8.3.13: Show that for each n, the derivative of the Chebyshev polynomial r(...
Solutions for Chapter 8.3: Chebyshev Polynomials and Economization of Power Series
Full solutions for Numerical Analysis | 10th Edition
Solutions for Chapter 8.3: Chebyshev Polynomials and Economization of Power SeriesGet Full Solutions
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
A = CTC = (L.J]))(L.J]))T for positive definite A.
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].
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
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.
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.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).