 8.4.1: Determine all degree 2 Pad approximations for f (x) = e2x . Compare...
 8.4.2: Determine all degree 3 Pad approximations for f (x) = x ln(x +1). C...
 8.4.3: Determine the Pad approximation of degree 5 with n = 2 and m = 3 fo...
 8.4.4: Repeat Exercise 3 using instead the Pad approximation of degree 5 w...
 8.4.5: Determine the Pad approximation of degree 6 with n = m = 3 for f (x...
 8.4.6: Determine the Pad approximations of degree 6 with (a) n = 2,m = 4 a...
 8.4.7: Table 8.10 lists results of the Pad approximation of degree 5 with ...
 8.4.8: Express the following rational functions in continuedfraction form...
 8.4.9: Find all the Chebyshev rational approximations of degree 2 for f (x...
 8.4.10: Find all the Chebyshev rational approximations of degree 3 for f (x...
 8.4.11: Find the Chebyshev rational approximation of degree 4 with n = m = ...
 8.4.12: Find all Chebyshev rational approximations of degree 5 for f (x) = ...
 8.4.13: To accurately approximate f (x) = ex for inclusion in a mathematica...
 8.4.14: To accurately approximate sin x and cos x for inclusion in a mathem...
Solutions for Chapter 8.4: Rational Function Approximation
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 8.4: Rational Function Approximation
Get Full SolutionsSince 14 problems in chapter 8.4: Rational Function Approximation have been answered, more than 15971 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Chapter 8.4: Rational Function Approximation includes 14 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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.

Multiplier eij.
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).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pascal matrix
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).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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·