 11.5.1: Use the Piecewise Linear Algorithm to approximate the solution to t...
 11.5.2: Use the Piecewise Linear Algorithm to approximate the solution to t...
 11.5.3: Use the Piecewise Linear Algorithm to approximate the solutions to ...
 11.5.4: Use the Cubic Spline Algorithm with n 3 to approximate the solution...
 11.5.5: Use the Cubic Spline Algorithm with n = 3 to approximate the soluti...
 11.5.6: Repeat Exercise 3 using the Cubic Spline Algorithm
 11.5.7: The lead example ofthis chapter concerned the boundary value proble...
 11.5.8: In Exercise 8 of Section 11.3 the deflection of a uniformly loaded ...
 11.5.9: Show that the boundaryvalue problem ^{pix )y') + q(x )y = fix), ...
 11.5.10: Use Exercise 10 and the Piecewise Linear Algorithm with n = 9 to ap...
 11.5.11: Repeal Exercise 9 using the Cubic Spline Algorithm.
 11.5.12: Show that the boundaryvalue problem ^(P(x)y') + qiX)y = f(x), a
 11.5.13: Show that the piecewiselinear basis functions {,)"=] are linearly ...
 11.5.14: Show that the cubic spline basis functions {
 11.5.15: Show that the matrix given by the piecewise linear basis functions ...
 11.5.16: Show that the matrix given by the cubic spline basis functions is p...
Solutions for Chapter 11.5: The RayleighRitz Method
Full solutions for Numerical Analysis  10th Edition
ISBN: 9781305253667
Solutions for Chapter 11.5: The RayleighRitz Method
Get Full SolutionsThis textbook survival guide was created for the textbook: Numerical Analysis, edition: 10. This expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9781305253667. Chapter 11.5: The RayleighRitz Method includes 16 full stepbystep solutions. Since 16 problems in chapter 11.5: The RayleighRitz Method have been answered, more than 14968 students have viewed full stepbystep solutions from this chapter.

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

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).

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].

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

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

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!