 4.3.1: Approximate the following integrals using the Trapezoidal rule. a. ...
 4.3.2: Approximate the following integrals using the Trapezoidal rule. a. ...
 4.3.3: Find a bound for the error in Exercise 1 using the error formula, a...
 4.3.4: Find a bound for the error in Exercise 2 using the error formula, a...
 4.3.5: Repeat Exercise 1 using Simpsons rule.
 4.3.6: Repeat Exercise 2 using Simpsons rule.
 4.3.7: Repeat Exercise 3 using Simpsons rule and the results of Exercise 5.
 4.3.8: Repeat Exercise 4 using Simpsons rule and the results of Exercise 6.
 4.3.9: Repeat Exercise 1 using the Midpoint rule.
 4.3.10: Repeat Exercise 2 using the Midpoint rule.
 4.3.11: Repeat Exercise 3 using the Midpoint rule and the results of Exerci...
 4.3.12: Repeat Exercise 4 using the Midpoint rule and the results of Exerci...
 4.3.13: The Trapezoidal rule applied to 2 0 f (x) dx gives the value 4, and...
 4.3.14: The Trapezoidal rule applied to 2 0 f (x) dx gives the value 5, and...
 4.3.15: Find the degree of precision of the quadrature formula 1 1 f (x) dx...
 4.3.16: Let h = (b a)/3, x0 = a, x1 = a + h, and x2 = b. Find the degree of...
 4.3.17: The quadrature formula 1 1 f (x) dx = c0f (1) + c1f (0) + c2f (1) i...
 4.3.18: The quadrature formula 2 0 f (x) dx = c0f (0) + c1f (1) + c2f (2) i...
 4.3.19: Find the constants c0, c1, and x1 so that the quadrature formula 1 ...
 4.3.20: Find the constants x0, x1, and c1 so that the quadrature formula 1 ...
 4.3.21: Approximate the following integrals using formulas (4.25) through (...
 4.3.22: Given the function f at the following values, x 1.8 2.0 2.2 2.4 2.6...
 4.3.23: Suppose that the data of Exercise 22 have roundoff errors given by...
 4.3.24: Derive Simpsons rule with error term by using x2 x0 f (x) dx = a0f ...
 4.3.25: Prove the statement following Definition 4.1; that is, show that a ...
 4.3.26: Derive Simpsons threeeighths rule (the closed rule with n = 3) wit...
 4.3.27: Derive the open rule with n = 1 with error term by using Theorem 4.3.
Solutions for Chapter 4.3: Elements of Numerical Integration
Full solutions for Numerical Analysis  9th Edition
ISBN: 9780538733519
Solutions for Chapter 4.3: Elements of Numerical Integration
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Numerical Analysis was written by and is associated to the ISBN: 9780538733519. Chapter 4.3: Elements of Numerical Integration includes 27 full stepbystep solutions. This textbook survival guide was created for the textbook: Numerical Analysis, edition: 9. Since 27 problems in chapter 4.3: Elements of Numerical Integration have been answered, more than 13756 students have viewed full stepbystep solutions from this chapter.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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.

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.

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].

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).