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Solutions for Chapter 6.4: Finite Difference Numerical Methods for Partial Differential Equations
Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition
Solutions for Chapter 6.4: Finite Difference Numerical Methods for Partial Differential EquationsGet Full Solutions
Upper triangular systems are solved in reverse order Xn to Xl.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
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].
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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.
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.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.