 8.5.1: Find the continuous least squares trigonometric polynomial Szix) fo...
 8.5.2: Find the continuous least squares trigonometric polynomial Sn(x) fo...
 8.5.3: Find the continuous least squares trigonometric polynomial Sjlx) 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 5(x) ...
 8.5.8: Compute the error EiS) for each ofthe functions in Exercise 7.
 8.5.9: Determine the discrete least squares trigonometric polynomial ^(x),...
 8.5.10: Repeat Exercise 9, using m = 8. Compare the values of the approxima...
 8.5.11: Let fix) 2 tan x sec2x, for 2 < x < 4. Determine the discrete least...
 8.5.12: a. Determine the discrete least squares trigonometric polynomial 54...
 8.5.13: The table lists the closing Dow Jones Industrial Averages (DJIA) at...
 8.5.14: The temperature u(x,t) in a bar of silver of length L = 10cm, densi...
 8.5.15: Show that for any continuous odd function / defined on the interval...
 8.5.16: Show that for any continuous even function / defined on the interva...
 8.5.17: Show that the functions 0oU) l/2,0i(x) cosx,... ,0(x) = cosnx, 0+i(...
 8.5.18: In Example 1, the Fourier series was determined for /(x) = x. Use...
 8.5.19: Show that the form ofthe constants a* for = (),... , n in Theorem 8...
Solutions for Chapter 8.5: Trigonometric Polynomial Approximation
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 8.5: Trigonometric Polynomial Approximation
Get Full SolutionsNumerical Analysis was written by and is associated to the ISBN: 9781305253667. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. Since 19 problems in chapter 8.5: Trigonometric Polynomial Approximation have been answered, more than 13804 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 8.5: Trigonometric Polynomial Approximation includes 19 full stepbystep solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (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 IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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).

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 Fn1c can be computed with ne/2 multiplications. Revolutionary.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.