 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
ISBN: 9780538733519
Solutions for Chapter 8.5: Trigonometric Polynomial Approximation
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9780538733519. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Since 17 problems in chapter 8.5: Trigonometric Polynomial Approximation have been answered, more than 16039 students have viewed full stepbystep solutions from this chapter. Chapter 8.5: Trigonometric Polynomial Approximation includes 17 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

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.

lAII = 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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Partial pivoting.
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(DÂ» 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+ (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).

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.

Skewsymmetric 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.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).