 6.6.6.6.1: (a) Show that Q < 1 if < 1 2 in SOR. (b) Determine the optimal ...
 6.6.6.6.2: (a) If = 1 2 (1 /N), show that Q 1 /2N (for large N) in SOR. (b...
 6.6.6.6.3: Describe a numerical scheme to solve Poissons equation 2u = f(x, y)...
 6.6.6.6.4: Describe a numerical scheme (based on Jacobi iteration) to solve La...
 6.6.6.6.5: Modify Exercise 6.6.4 for GaussSeidel iteration.
 6.6.6.6.6: Show that Jacobi iteration corresponds to the twodimensional diffu...
 6.6.6.6.7: What partial differential equation does SOR correspond to? (Hint:...
 6.6.6.6.8: Consider Laplaces equation on a square 0 x 1, 0 y 1 with u = 0 on t...
Solutions for Chapter 6.6: Finite Difference Numerical Methods for Partial Differential Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems  5th Edition
ISBN: 9780321797056
Solutions for Chapter 6.6: Finite Difference Numerical Methods for Partial Differential Equations
Get Full SolutionsApplied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.6: Finite Difference Numerical Methods for Partial Differential Equations includes 8 full stepbystep solutions. Since 8 problems in chapter 6.6: Finite Difference Numerical Methods for Partial Differential Equations have been answered, more than 8066 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5.

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

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

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

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

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.