- 8.5.1: Find the continuous least squares trigonometric polynomial S2(x) fo...
- 8.5.2: Find the continuous least squares trigonometric polynomial Sn(x) fo...
- 8.5.3: Find the continuous least squares trigonometric polynomial S3(x) fo...
- 8.5.4: Find the general continuous least squares trigonometric polynomial ...
- 8.5.5: Find the general continuous least squares trigonometric polynomial ...
- 8.5.6: Find the general continuous least squares trigonometric polynomial ...
- 8.5.7: Determine the discrete least squares trigonometric polynomial Sn(x)...
- 8.5.8: Compute the error E(Sn) for each of the functions in Exercise 7.
- 8.5.9: Determine the discrete least squares trigonometric polynomial S3(x)...
- 8.5.10: Repeat Exercise 9 using m = 8. Compare the values of the approximat...
- 8.5.11: Let f (x) = 2 tan x sec 2x, for 2 x 4. Determine the discrete least...
- 8.5.12: a. Determine the discrete least squares trigonometric polynomial S4...
- 8.5.13: Show that for any continuous odd function f defined on the interval...
- 8.5.14: Show that for any continuous even function f defined on the interva...
- 8.5.15: Show that the functions 0(x) = 1/2, 1(x) = cos x, ... , n(x) = cos ...
- 8.5.16: In Example 1 the Fourier series was determined for f (x) = |x|. Use...
- 8.5.17: Show that the form of the constants ak for k = 0, ... , n in Theore...
Solutions for Chapter 8.5: Trigonometric Polynomial Approximation
Full solutions for Numerical Analysis | 9th Edition
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
= Xl (column 1) + ... + xn(column n) = combination of columns.
A directed graph that has constants Cl, ... , Cm associated with the edges.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
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).
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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).