 3.4.1: Use Theorem 3.9 or Algorithm 3.3 to construct an approximating poly...
 3.4.2: Use Theorem 3.9 or Algorithm 3.3 to construct an approximating poly...
 3.4.3: The data in Exercise 1 were generated using the following functions...
 3.4.4: The data in Exercise 2 were generated using the following functions...
 3.4.5: a. Use the following values and fivedigit rounding arithmetic to c...
 3.4.6: Let f (x) = 3xex e2x . a. Approximate f (1.03) by the Hermite inter...
 3.4.7: Use the error formula and Maple to find a bound for the errors in t...
 3.4.8: Use the error formula and Maple to find a bound for the errors in t...
 3.4.9: The following table lists data for the function described by f (x) ...
 3.4.10: A car traveling along a straight road is clocked at a number of poi...
 3.4.11: a. Show that H2n+1(x) is the unique polynomial of least degree agre...
 3.4.12: Let z0 = x0, z1 = x0, z2 = x1, and z3 = x1. Form the following divi...
Solutions for Chapter 3.4: Hermite Interpolation
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 3.4: Hermite Interpolation
Get Full SolutionsThis textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Chapter 3.4: Hermite Interpolation includes 12 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 12 problems in chapter 3.4: Hermite Interpolation have been answered, more than 16069 students have viewed full stepbystep solutions from this chapter.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.