 9.5.9.1.63: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.64: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.65: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.66: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.67: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.68: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.69: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.70: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.71: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.72: In 110 use the finite difference method and the indicated value of ...
 9.5.9.1.73: Rework Example 1 using n 8.
 9.5.9.1.74: The electrostatic potential u between two concentric spheres of rad...
 9.5.9.1.75: Consider the boundaryvalue problem y xy 0, y(0) 1, y(1) 1. (a) Fin...
 9.5.9.1.76: Consider the boundaryvalue problem y y sin (xy), y(0) 1, y(1) 1.5....
Solutions for Chapter 9.5: Numerical Solutions of Ordinary Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 9.5: Numerical Solutions of Ordinary Differential Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Since 14 problems in chapter 9.5: Numerical Solutions of Ordinary Differential Equations have been answered, more than 21243 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Chapter 9.5: Numerical Solutions of Ordinary Differential Equations includes 14 full stepbystep solutions.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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.

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

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.

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.

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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.

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.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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·