- 4.7.1: Approximate the following integrals using Gaussian quadrature with ...
- 4.7.2: Approximate the following integrals using Gaussian quadrature with ...
- 4.7.3: Repeat Exercise 1 with n = 3.
- 4.7.4: Repeat Exercise 2 with n 3
- 4.7.5: Repeat Exercise 1 with n = 4.
- 4.7.6: Repeat Exercise 2 with n = 4.
- 4.7.7: Repeat Exercise 1 with n = 5.
- 4.7.8: Repeat Exercise 2 with n = 5.
- 4.7.9: Approximate the length of the graph of the ellipse 4x2 + 9y2 = 36 i...
- 4.7.10: Use Composite Gaussian Quadrature to approximate the integral <48 /...
- 4.7.11: Determine constants a, b, c, and d that will produce a quadrature f...
- 4.7.12: Determine constants a, b, c, and d that will produce a quadrature f...
- 4.7.13: Verify the entries for the values of = 2 and 3 in Table 4.12 on pag...
- 4.7.14: Show that the formula QiP) = Xw=i Q Pix,) cannot have degree of pre...
Solutions for Chapter 4.7: Gaussian Quadrature
Full solutions for Numerical Analysis | 10th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
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.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
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.
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
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·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.