 4.9.1: Use Simpsons Composite rule and the given values of n to approximat...
 4.9.2: Use the Composite Simpsons rule and the given values of n to approx...
 4.9.3: Use the transformation t = x1 and then the Composite Simpsons rule ...
 4.9.4: The improper integral 0 f (x) dx cannot be converted into an integr...
 4.9.5: Suppose a body of mass m is traveling vertically upward starting at...
 4.9.6: The Laguerre polynomials {L0(x), L1(x)...} form an orthogonal set o...
 4.9.7: The Laguerre polynomials L0(x) = 1, L1(x) = 1 x, L2(x) = x2 4x + 2,...
 4.9.8: Use the quadrature formulas derived in Exercise 7 to approximate th...
 4.9.9: Use the quadrature formulas derived in Exercise 7 to approximate th...
Solutions for Chapter 4.9: Improper Integrals
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 4.9: Improper Integrals
Get Full SolutionsSince 9 problems in chapter 4.9: Improper Integrals have been answered, more than 12546 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Chapter 4.9: Improper Integrals includes 9 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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 .

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

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

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(D» O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).