 19.3.19.1.46: (Linear interpolation) Calculate PI (x) in Example I.Compute from i...
 19.3.19.1.47: Estimate the enor in Prob. 1 by (5).
 19.3.19.1.48: (Quadratic interpolation) Calculate the Lagrange polynomial 1'2(X) ...
 19.3.19.1.49: (Error bounds) Derive enor bounds for P2(9.2) in Example 2 from (5).
 19.3.19.1.50: (Error function) Calculate the Lagrange polynomial P2(X) for the 50...
 19.3.19.1.51: Derive an error bound in Prob. 5 from (5).
 19.3.19.1.52: (Sine integral) Calculate the Lagrange polynomial P2(X) for the 40...
 19.3.19.1.53: (Linear and quadratic interpolation) Find eO. 25 and eO . 75 by l...
 19.3.19.1.54: (Cubic Lagrange interpolation) Calculate and sketch or graph Lo, L1...
 19.3.19.1.55: (Interpolation and extrapolation) Calculate P2(X) in Example 2. Com...
 19.3.19.1.56: (Extrapolation) Does a sketch or graph of the product of the (x  X...
 19.3.19.1.57: (Lower degree) Find the degree of the interpolation polynomial for ...
 19.3.19.1.58: (Newton's forward difference formula) Set up (14) for the data in P...
 19.3.19.1.59: Set up Newton's forward difference formula for the data in Prob. 3 ...
 19.3.19.1.60: (Newton's divided difference formula) Compute f(0.8) and f(0.9) fro...
 19.3.19.1.61: Compute f(6.5) from f(6.0) = O. J 506 f(7.0) = 0.3001 f(7.5) = 0.26...
 19.3.19.1.62: (Central differences) Write the difference in the table in Example ...
 19.3.19.1.63: (Subtabulation) Compute the Bessel function 11(X) for X = 0.1. 0.3,...
 19.3.19.1.64: (Notations) Compute a difference table of f(x) = x3 for X = O. 1, 2...
 19.3.19.1.65: CAS EXPERIMENT. Adding Terms in Newton Formulas. Write a program fo...
 19.3.19.1.66: WRITING PROJECT. Interpolation: Comparison of Methods. Make a list ...
 19.3.19.1.67: TEAM PROJECT. Interpolation and Extrapolation. (a) Lagrange practic...
Solutions for Chapter 19.3: [nterpolation
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter 19.3: [nterpolation
Get Full SolutionsSince 22 problems in chapter 19.3: [nterpolation have been answered, more than 44356 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter 19.3: [nterpolation includes 22 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their 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!

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

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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

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).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Iterative method.
A sequence of steps intended to approach the desired solution.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.