- 3.1.1: For the given functions /(a), let Aq = 0, A| = 0.6, and X2 = 0.9. C...
- 3.1.2: For the given functions/(a), let aq = 1,A| 1.25, and A2 = 1.6. Cons...
- 3.1.3: Use Theorem 3.3 to find an error bound for the approximations in Ex...
- 3.1.4: Use Theorem 3.3 to find an error bound for the approximations in Ex...
- 3.1.5: Use appropriate Lagrange interpolating polynomials of degrees one, ...
- 3.1.6: Use appropriate Lagrange interpolating polynomials of degrees one, ...
- 3.1.7: The data for Exercise 5 were generated using the following function...
- 3.1.8: The data for Exercise 6 were generated using the following function...
- 3.1.9: Let Pjix) be the interpolating polynomial forthe data (0, 0), (0.5,...
- 3.1.10: Let fix) = -fx x 2 and //(x) be the interpolation polynomial on xq ...
- 3.1.11: Use the following values and four-digit rounding arithmetic to cons...
- 3.1.12: Use the Lagrange interpolating polynomial of degree three or less a...
- 3.1.13: Construct the Lagrange interpolating polynomials for the following ...
- 3.1.14: Construct the Lagrange interpolating polynomials for the following ...
- 3.1.15: Let /(x) = e x , for 0 < x < 2. a. Approximate /(0.25) using linear...
- 3.1.16: Let f(x) e~x cosx, for 0 < x < I. a. Approximate /(0.25) using line...
- 3.1.17: Suppose you need to construct eight-decimal-place tables for the co...
- 3.1.18: In Exercise 24 of Section 1.1, a Maclaurin series was integrated to...
- 3.1.19: a. The introduction to this chapter included a table listing the po...
- 3.1.20: It is suspected that the high amounts of tannin in mature oak leave...
- 3.1.21: Show that max |g(x)| =/z2 /4, where g(x) = (x -/7z)(x - (y + 1)A).
- 3.1.22: Prove Taylor's Theorem 1.14 by following the procedure in the proof...
- 3.1.23: The Bernstein polynomial of degree n for / C[0, 1] is given by wher...
Solutions for Chapter 3.1: Interpolation and the Lagrange Polynomial
Full solutions for Numerical Analysis | 10th Edition
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.
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).
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
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.
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).
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).
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
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.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).