 Chapter 19.19.1: (Floating point) Write 98.17, 100.987, 0.0057869,  13600 in float...
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: (Linear interpolation) Calculate PI (x) in Example I.Compute from i...
 Chapter 19.19.4: WRITING PROJECT. Splines. In your own words, and using as few formu...
 Chapter 19.19.5: (Rectangular rule) Evaluate the integral in Example I by the rectan...
 Chapter 19.1: What is a numeric method? How has the computer influenced numeric m...
 Chapter 19.19.1: Write 0.0286403. 11.25845.  3168\.55 in f1oatingpoint form rounde...
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: Estimate the enor in Prob. 1 by (5).
 Chapter 19.19.4: (Individual polynomial qj) Show that qj(x) in (6) satisfies the int...
 Chapter 19.19.5: Derive a formula for lower and upper bounds for the rectangular rul...
 Chapter 19.2: What is floatingpoint representation of nwnbers? Overflow and unde...
 Chapter 19.19.1: Small differences of large numbers may be particularly strongly aff...
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: (Quadratic interpolation) Calculate the Lagrange polynomial 1'2(X) ...
 Chapter 19.19.4: Verify the differentiations that give (7) and (8) from (6).
 Chapter 19.19.5: Evaluate the integrals numelically as indicated and determine the e...
 Chapter 19.3: How do error and relative enor behave under addition? Under multipl...
 Chapter 19.19.1: Do the work in Prob. 3 with numbers of your choice that give even m...
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: (Error bounds) Derive enor bounds for P2(9.2) in Example 2 from (5).
 Chapter 19.19.4: (System for derivatives) Derive the basic linear system (9) for k1 ...
 Chapter 19.19.5: Evaluate the integrals numelically as indicated and determine the e...
 Chapter 19.4: Why are roundoff errors important? State the rounding rules.
 Chapter 19.19.1: The quotient in Prob. 3 is of the form a/(b  c). Write it as alb +...
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: (Error function) Calculate the Lagrange polynomial P2(X) for the 50...
 Chapter 19.19.4: (Equidistant nodes) Derive (14) from (9).
 Chapter 19.19.5: Evaluate the integrals numelically as indicated and determine the e...
 Chapter 19.5: What is an algorithm"! Which of its properties are important in sof...
 Chapter 19.19.1: (Quadratic equation) Solve.\"2  20x + I = 0 by (6) and by (7). usi...
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: Derive an error bound in Prob. 5 from (5).
 Chapter 19.19.4: (Coefficients) Give the details of the derivation of aj2 and aj3 in...
 Chapter 19.19.5: Evaluate the integrals numelically as indicated and determine the e...
 Chapter 19.6: Why is the selection of a good method at least as important on a la...
 Chapter 19.19.1: Do the computations in Prob. 6 with 4S and 2S.
 Chapter 19.19.2: Apply fixedpoint iteration and answer related questions where indi...
 Chapter 19.19.3: (Sine integral) Calculate the Lagrange polynomial P2(X) for the 40...
 Chapter 19.19.4: Verify the computations in Example I.
 Chapter 19.19.5: Evaluate the integrals numelically as indicated and determine the e...
 Chapter 19.7: Explain methods for solving equations, in particular fixedpoint it...
 Chapter 19.19.1: Solve.\"2 + 100x + 2 = 0 by (6) and (7) with 5S and compare.
 Chapter 19.19.2: CAS PROJECT. FixedPoint Iteration. (a) Existence. Prove that if g ...
 Chapter 19.19.3: (Linear and quadratic interpolation) Find eO. 25 and eO . 75 by l...
 Chapter 19.19.4: (Comparison) Compare the spline g in Example I with the quadratic i...
 Chapter 19.19.5: Evaluate the integrals numelically as indicated and determine the e...
 Chapter 19.8: Can the Newton (Raphson) method diverge? Is it fast? Same question...
 Chapter 19.19.1: Calculate lIe = 0.367879 (6S) from the partial sums of 5 to 10 term...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Cubic Lagrange interpolation) Calculate and sketch or graph Lo, L1...
 Chapter 19.19.4: (Natural spline condition) Using coefficients, verify that the spli...
 Chapter 19.19.5: Estimate the error by halving. In Prob. 5
 Chapter 19.9: What is the advamage of Newton's interpolation formulas over Lagran...
 Chapter 19.19.1: Addition with a fixed number of significant digits depends on the o...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Interpolation and extrapolation) Calculate P2(X) in Example 2. Com...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: Estimate the error by halving.In Prob. 6
 Chapter 19.10: What do you remember about errors in polynomial imerpolation?
 Chapter 19.19.1: Approximations of 7T = 3.141 592 653 589 79 ... are 22/7 and 355/1 ...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Extrapolation) Does a sketch or graph of the product of the (x  X...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: Estimate the error by halving.In Prob. 7
 Chapter 19.11: What is spline interpolation? Tts advantage over polynomial interpo...
 Chapter 19.19.1: Compute 7T by Machin's approximation 16 arctan ( 115)  4 arctan (1...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Lower degree) Find the degree of the interpolation polynomial for ...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: Estimate the error by halving.In Prob. 8
 Chapter 19.12: List and compare numeric integration methods. When would you appl} ...
 Chapter 19.19.1: (Rounding and adding) Let al' ... , an be numbers with aj correctly...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Newton's forward difference formula) Set up (14) for the data in P...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.13: In what sense is Gau~s integration optimal? Explain details.
 Chapter 19.19.1: (Theorems on errors) Prove Theorem I(a) for addition.
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: Set up Newton's forward difference formula for the data in Prob. 3 ...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.14: What does adaptive imegration mean? Why is it useful?
 Chapter 19.19.1: Prove Theorem I(b) for division.
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Newton's divided difference formula) Compute f(0.8) and f(0.9) fro...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.15: Why is numeric differentiation generally more delicate than numeric...
 Chapter 19.19.1: Show that in Example 1 the absolute value of the error of X2 = 2.00...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: Compute f(6.5) from f(6.0) = O. J 506 f(7.0) = 0.3001 f(7.5) = 0.26...
 Chapter 19.19.4: Find the cubic spline g(x) for the given data with ko and kn as giv...
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.16: Write 0.35287. 1274.799, 0.00614. 14.9482. 113, 8517 in floating...
 Chapter 19.19.1: Overflow and underflow can sometimes be avoided by simple changes i...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Central differences) Write the difference in the table in Example ...
 Chapter 19.19.4: (Natural conditions) Explain the remark after (II).
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.17: Compute (5.346  3.644)/(3.454  3.055) as given and then rounded s...
 Chapter 19.19.1: (Nested form) Evaluate I(x) = x 3  7.5x2 + 1l.2x + 2.8 = x  7.5)x...
 Chapter 19.19.2: Apply Newton's method (60 accuracy). First sketch the function(s) t...
 Chapter 19.19.3: (Subtabulation) Compute the Bessel function 11(X) for X = 0.1. 0.3,...
 Chapter 19.19.4: CAS EXPERIMENT. Spline versus Polynomial. [f your CAS gives natural...
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.18: Compute 0.29731/(4.1132  4.0872) with the numbers as given and the...
 Chapter 19.19.1: CAS EXPERIMENT. Chopping and Rounding. (a) Let x = 4/7 andy = 113. ...
 Chapter 19.19.2: TEAM PROJECT. Bisection Method. This simple but slowly convergent m...
 Chapter 19.19.3: (Notations) Compute a difference table of f(x) = x3 for X = O. 1, 2...
 Chapter 19.19.4: If a cubic spline is three times continuously differentiable (that ...
 Chapter 19.19.5: The following integrals cannot be evaluated by the usual methods of...
 Chapter 19.19: . Solve x 2  50x + 1 = 0 by (6) and by (7) in Sec. 19.1, using 5S ...
 Chapter 19.19.1: WRITI.'JG PROJECT. Numerics. In your own words write about the over...
 Chapter 19.19.2: TEAM PROJECT. Method of False Position (Regula falsi). Figure 427 s...
 Chapter 19.19.3: CAS EXPERIMENT. Adding Terms in Newton Formulas. Write a program fo...
 Chapter 19.19.4: TEAM PROJECT. Hermite Interpolation and Bezier Curves. In Hermite i...
 Chapter 19.19.5: (Stability) Prove that the trapezoidal rule is stable with respect ...
 Chapter 19.20: Solve x 2  100x + 4 = 0 by (6) and by (7) in Sec. 19.1, using 5S i...
 Chapter 19.19.2: Solve. using Xo and Xl i1~ indicated. Prob. ll, Xo = 0.5, Xl = 2.0
 Chapter 19.19.3: WRITING PROJECT. Interpolation: Comparison of Methods. Make a list ...
 Chapter 19.19.5: Integrate by (11) with II = 5:IIx from I to 3
 Chapter 19.21: Let 4.81 and 12.752 be correctly rounded to the number of digits sh...
 Chapter 19.19.2: Solve. using Xo and Xl i1~ indicated.e x  tan X = o. Xo = I. Xl 0.7
 Chapter 19.19.3: TEAM PROJECT. Interpolation and Extrapolation. (a) Lagrange practic...
 Chapter 19.19.5: Integrate by (11) with II = 5:co~ x from 0 to!7T
 Chapter 19.22: Answer the question in Prob. 21 for the difference 4.81  11.752.
 Chapter 19.19.2: Solve. using Xo and Xl i1~ indicated.Prob. 9, Xo = I. Xl = 0.5
 Chapter 19.19.5: Integrate by (11) with II = 5: ex" from 0 to I
 Chapter 19.23: What is the relative error of l1a in terms of that of a?
 Chapter 19.19.2: Solve. using Xo and Xl i1~ indicated.Prob. 10, Xo = 0.5, Xl = I
 Chapter 19.19.5: Integrate by (11) with II = 5: sin <x2 ) from 0 to 1.25
 Chapter 19.24: Show that the relative error of a2 is about twice that of a.
 Chapter 19.19.2: WRITING PROJECT. Solution of Equations.Compare the methods in this ...
 Chapter 19.19.5: (Given TOL) Find the smallest 11 in computing the integral of 1Ix f...
 Chapter 19.25: Compute the solution of x 5 = x + 0.2 near J. = 0 by transforming t...
 Chapter 19.19.5: TEAM PROJECT. Romberg Integration (W. Romberg. Norske Videllskab. T...
 Chapter 19.26: Solve cos x = x by iteration (6S, Xo = I), writing it as x = (O.74x...
 Chapter 19.19.5: Consider f(x) = X4 for Xo = 0, Xl = 0.2, X2 = 0.4, X3 = 0.6, X4 = 0...
 Chapter 19.27: Solve X4  x 3  2x  34 = 0 by Newton's method with Xo = 3 and 6S ...
 Chapter 19.19.5: A "fourpoint formula" for the derivative is Apply it to f(x) = X4 ...
 Chapter 19.28: Solve cos x  x = 0 by the method of false position.
 Chapter 19.19.5: The derivative f' (x) can also be approximated in terms of firstor...
 Chapter 19.29: Compute f(1.28) from .fO.Q) = 3.00000 f(l.2) = 1.98007 f(1.4) = 2.9...
 Chapter 19.19.5: Derive the formula in Prob. 29 from (14) in Sec. 19.3
 Chapter 19.30: Find the cubic spline for the data f(I) = 3 f(1) = I f(3) = 23 f(5...
 Chapter 19.31: Compute the integral of X3 from 0 to I by the trapezoidal rule with...
 Chapter 19.32: Compute the integral of cos (X2) from 0 to I by Simpson's rule with...
 Chapter 19.33: Compute the integral of cos x from 0 to !rr by the threeeights ru...
Solutions for Chapter Chapter 19: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 19
Get Full SolutionsSince 150 problems in chapter Chapter 19 have been answered, more than 44460 students have viewed full stepbystep solutions from this chapter. Advanced Engineering Mathematics was written by and is associated to the ISBN: 9780471488859. Chapter Chapter 19 includes 150 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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

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