TEAM PROJECT. Romberg Integration (W. Romberg. Norske

(Floating point) Write 98.17, -100.987, 0.0057869, - 13600 in floating-point form, rounded to 4S (4 significant digits).

Write -0.0286403. 11.25845. - 3168\.55 in f1oatingpoint form rounded to 6S.

Small differences of large numbers may be particularly strongly affected by rounding errors. l\Iu~trate this by computing 0.36443/(17.862 - 17.798) as given with 5S. then rounding stepwise to 4S. 3S. and 2S. where "stepwise" means: round the rounded numbers. not the given ones.

Do the work in Prob. 3 with numbers of your choice that give even more drastically different results. How can you avoid such difficulties?

The quotient in Prob. 3 is of the form a/(b - c). Write it as alb + c:)/(b2 - c2 ). Compute it first with 5S, then rounding numerator 12.996 and denominator 2.28 stepwise as in Prob. 3. Compare and comment

(Quadratic equation) Solve.\"2 - 20x + I = 0 by (6) and by (7). using 6S in the computation. Compare and comment.

Do the computations in Prob. 6 with 4S and 2S.

Solve.\"2 + 100x + 2 = 0 by (6) and (7) with 5S and compare.

Calculate lIe = 0.367879 (6S) from the partial sums of 5 to 10 terms of the Maclaurin series (a) of e-" with x = 1, (b) of eX with x = 1 and then taking the reciprocal. Which is more accurate?

Addition with a fixed number of significant digits depends on the order in which you add the numbers. Illustrate this with an example. Find an empirical rule for the best order.

Approximations of 7T = 3.141 592 653 589 79 ... are 22/7 and 355/1 13. Determine the corresponding errors and relative errors to 3 significant digits.

Compute 7T by Machin's approximation 16 arctan ( 115) - 4 arctan (1/239) to lOS (which are correct). (In 19X6, D. H. Bailey computed almost 30 million decimals of 7T on a CRA Y -2 in less than 30 hours. The race for more and more decimals is continuing. )

(Rounding and adding) Let al' ... , an be numbers with aj correctly rounded to Dj decimals. In calculating the sum (II + ... + (In' retmmng D = min DJ decimals, is it essential that we first add and then round the result or that we first round each number to D decimals and then add?

(Theorems on errors) Prove Theorem I(a) for addition.

Prove Theorem I(b) for division.

Show that in Example 1 the absolute value of the error of X2 = 2.000/39.95 = 0.05006 is less than 0.00001.

Overflow and underflow can sometimes be avoided by simple changes in a formula. Explain this in terms of V.~ + y2 = xV I + (ylx)2 with x 2 ~ y2 and x so large that x 2 would cause ovelt10w. Invent examples of your own.

(Nested form) Evaluate I(x) = x 3 - 7.5x2 + 1l.2x + 2.8 = x - 7.5)x + 11.2)x + 2.8 at x = 3.94 using 3S arithmetic and rounding. in both of the given forms. The latter, called the nestedfamz, is usually preferable since it minimizes the number of operations and thus the effect of rounding.

CAS EXPERIMENT. Chopping and Rounding. (a) Let x = 4/7 andy = 113. Find the error~ Echop, Eround and the relative errors Er.ch, Er.rd of x + y. x - y. xy. x I)" in chopping and rounding to 5S. Experiment with other fractions of your choice. (b) Graph Echop and Eround (for 5S) of k121 as a function of k = I, 2, . , . , 21 on common axes. What average value can you read from the graph for Echop? For Eround? Experiment with other integers that give similar graphs. Different types of graphs. Can you characteri7e the ditTerent types in terms of prime factors? (c) How does the situation in (b) change if you take 4S instead of 5S? (d) Write programs for the work in (a)-(c).

WRITI.'JG PROJECT. Numerics. In your own words write about the overall role of numeric methods in applied mathematics, wby they are impol1ant, where and when they must be used or can be used, and how they are influenced by the use of the computer in engineering and other wor

Apply fixed-point iteration and answer related questions where indicated. Show details of your work.x = 1.4 sin x, Xo = 1.4

Apply fixed-point iteration and answer related questions where indicated. Show details of your work.00 the iterations indicated at the end of Example 2. Sketch a figure similar to Fig. 424.

Apply fixed-point iteration and answer related questions where indicated. Show details of your work.Why do we obtain a monotone sequence in Example I, but not in Example 2?

Apply fixed-point iteration and answer related questions where indicated. Show details of your work. = X4 - X + 0.2 = 0, the root near I, Xo =

Apply fixed-point iteration and answer related questions where indicated. Show details of your work. f as in Prob. 4, the root near 0, Xo = 0

Apply fixed-point iteration and answer related questions where indicated. Show details of your work. Find the smallest positive solution of sin x = e- 0 . 5X, Xo = I.

Apply fixed-point iteration and answer related questions where indicated. Show details of your work. (Bessel functions, drumhead) A partial sum of the Maclaurin series of JoCr) (Sec. 5.5) is f(x) = I - !x2 + i'4x4 - 23~4X6. Conclude from a sketch that f(x) = 0 near x = 2. Write f(x) = 0 as x = g(x) (by dividing f(x) by !x and taking the resulting x-term to the other side). Find the zero. (See Sec. 12.9 for the importance of these zeros.)

CAS PROJECT. Fixed-Point Iteration. (a) Existence. Prove that if g is continuous in a closed interval f and its range lies in f, then the equation x = g(x) has at least one solution in f. Illustrate that it may have more than one solution in f. (b) Convergence. Let f(x) = x 3 + 2r2 - 3x - 4 = o. Write this asx = g(x), for g choosing (1) (x 3 - /)113, (2) (r2 - !n1l2, (3) x + !f, (4) x(l + !f), (5) (x 3 - f)lx 2 , (6) (2x 2 - f)/2x, (7) x - flf' and in each case Xo = 1.5. Find out about convergence and divergence and the number of steps to reach exact 6S-values of a root.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

Apply Newton's method (60 accuracy). First sketch the function(s) to see what is going on.

TEAM PROJECT. Bisection Method. This simple but slowly convergent method for finding a solution of f(x) = 0 with continuous f is based on the intermediate value theorem, which states that if a continuous function f has opposite signs at some x = a and x = b (> a). that is, either f(a) < 0, feb) > 0 or f(a) > 0, feb) < 0, then f must be 0 somewhere on [a, b]. The solution is found by repeated bisection of the interval and in each iteration picking that half which also satisfies that sign condition. (a) Algorithm. Write an algorithm for the method. (b) Comparison. Solve x = cosx by Newton's method and by bisection. Compare. (e) Solve e-x = In x and eX + x4 + X = 2 by bisection.

TEAM PROJECT. Method of False Position (Regula falsi). Figure 427 shows the idea. We assume that f is continuous. We compute the x-intercept Co of the line through (ao, f(ao), (bo, f(bo. If f(co) = 0, we are done. If f(ao)f(c o) < 0 (as in Fig. 427), we set a 1 = ao, b 1 = Co and repeat to get Cl, etc. If f(ao)ffco) > 0, then f(co)f(b o) < 0 and we set al = Co, b 1 = bo, etc. (a) Algorithm. Show that Co = aof(bo) - bof(ao) f(bo) - f(ao) and write an algorithm for the method. (b) Comparison. Solve x 3 = 5x + 6 by Newton's method, the secant method, and the method of false position. Compare. (C) Solve X4 = 2, cos x = V~, and x + In x 2 by [he method of false position.

Solve. using Xo and Xl i1~ indicated. Prob. ll, Xo = 0.5, Xl = 2.0

Solve. using Xo and Xl i1~ indicated.e- x - tan X = o. Xo = I. Xl 0.7

Solve. using Xo and Xl i1~ indicated.Prob. 9, Xo = I. Xl = 0.5

Solve. using Xo and Xl i1~ indicated.Prob. 10, Xo = 0.5, Xl = I

WRITING PROJECT. Solution of Equations.Compare the methods in this section and problem set, discussing advantages and disadvantages using examples of your own.

(Linear interpolation) Calculate PI (x) in Example I.Compute from it In 9.4 = PI(9.4).

Estimate the enor in Prob. 1 by (5).

(Quadratic interpolation) Calculate the Lagrange polynomial 1'2(X) for the 4D-values of the Gamma function [(24), App. 3.1J r(l.oO) = 1.0000. r(l.02) = 0.9888, r(l.04) = 0.9784, and from it approximations of r{x) for x = 1.005, 1.010. 1.015. 1.025. 1.030. 1.035.

(Error bounds) Derive enor bounds for P2(9.2) in Example 2 from (5).

(Error function) Calculate the Lagrange polynomial P2(X) for the 50-values of the enor functionj(x) = erf x = (2/\'";) J~ e-w2 dw, namely. 1(0.25) = 0.27633 .. 0.5) = 0.52050. f( I) = 0.84270. and fromp2 an approximation of j(O.75) (= 0.71116. 50).

Derive an error bound in Prob. 5 from (5).

(Sine integral) Calculate the Lagrange polynomial P2(X) for the 40-values of the sine integral Si(x) [(40) in App. 3.1], namely, Si(O) = O. Si(l) = 0.9461. Si(2) = 1.6054, and from P2 approximations of Si(O.5) (= 0.4931. 40) and Si(1.5) (= 1.3247,401.

(Linear and quadratic interpolation) Find e-O. 25 and e-O . 75 by linear interpolation with xo = 0, Xl = 0.5 and Xo = 0.5, Xl = I. respectively. Then find pix)interpolating e-x with Xo = O. Xl = 0.5. X2 = 1 and from it e-O. 25 and e-O. 75 Compare the errors of these linear and quadratic interpolations. Use 4D-values of e-x .

(Cubic Lagrange interpolation) Calculate and sketch or graph Lo, L1 . L2 , L3 for x = O. 1.2.3 on common axes. Find P3(X) fOT the data (0. I) (1,0.765198) (2, 0.223891 ) (3. -0.260052) [values of the Bessel function 10(x)]. Find P3 for X = 0.5, 1.5. 2.5 by interpolation.

(Interpolation and extrapolation) Calculate P2(X) in Example 2. Compute from it approximations of In 9.4, In 10, In 10.5. In 11.5. In 12, compute the errors by using exact 4D-values. and comment.

(Extrapolation) Does a sketch or graph of the product of the (x - Xj) in (5) for the data in Prob. 10 indicate that extrapolation is likely to involve larger errors than interpolation does?

(Lower degree) Find the degree of the interpolation polynomial for the data (-2.33) (0.5) (2.9) (4,45) (6, 113).

(Newton's forward difference formula) Set up (14) for the data in Prob. 7 and derive P2(x) from (14).

Set up Newton's forward difference formula for the data in Prob. 3 and compute [(1.01), [(1.03), [(1.05).

(Newton's divided difference formula) Compute f(0.8) and f(0.9) from f(0.5) = 0.479 f(I.O) = 0.841 f(2.0) = 0.909 by quadratic interpolation.

Compute f(6.5) from f(6.0) = O. J 506 f(7.0) = 0.3001 f(7.5) = 0.2663 fO.7) = 0.2346 by cubic interpolation, using (10).

(Central differences) Write the difference in the table in Example 5 in central difference notation.

(Subtabulation) Compute the Bessel function 11(X) for X = 0.1. 0.3, .... 0.9 from 11(0) = 0,11(0.2) = 0.09950. h(O.4) = 0.19603. 11(0.6} = 0.28670.11(0.8) = 0.36884, 11(1.0) = 0.44005. Use (14) with II = 5.

(Notations) Compute a difference table of f(x) = x3 for X = O. 1, 2. 3.4. 5. Choose Xo = 2 and write all occurring numbers in tenTIS of the notations (a) for central differences, (b) for forward differences, (c) for backward differences

CAS EXPERIMENT. Adding Terms in Newton Formulas. Write a program for the forward formula ( 14). Experiment on the increase of accuracy by successively adding terms. As data use values of some function of your choice for which your CAS gives the values needed in determining errors.

WRITING PROJECT. Interpolation: Comparison of Methods. Make a list of 5-6 ideas that you feel are most basic in this section. Arrange them in the best logical order. Discuss them in a 2-3 page report.

TEAM PROJECT. Interpolation and Extrapolation. (a) Lagrange practical error estimate (after Theorem I). Apply this to PI (9.2) and P2(9.2) for the data Xo = 9.0. Xl = 9.5. X2 = 11.0. fo = In Xo. fl = In XI> f2 = In X2 (6S-values). (b) Extrapolation. Given (.~i' f(x) = (0.2.0.9980). (0.4. 0.9686). (0.6. 0.8443), (0.8, 0.5358), (1.0. 0). Find f(0.7) from the quadratic interpolation polynomials based on (a) 0.6, 0.8. 1.0, ({3) 0.4. 0.6. 0.8. (y) 0.2, 0.4, 0.6. Compare the errors and comment. [Exact f(x) = cos (!7TX 2), f(O.7) = 0.7181 (45).1 (c) Graph the product of factors (x - xi> in the error formula (5) for 11 = 2.' . '. 10 separately. What do these graphs show regarding accuracy of interpolation and extrapolation? (d) Central differences. Show that 82fm = fm+l - 2fm + fm-I> and. furthermore 83 fm+I12 = fm+2 - 3fYII f 1 + 3fm - f m - 1 8n fm = t!.."fm-n/2 = vnfm+n/2' (e) Everett's interpolation formula f(x) = (I - r)fo + rf1 (20) (2 - r)O - r)( -r) + 8 2 fo 3! is an example of a formula involving only even-order differences. Use it (0 compute the Bessel function Jo(x) for x = 1.72 from 10(l.60} = 0.4554022 and 10(1.7). 10(1.8),10(1.9) III Example 6.

WRITING PROJECT. Splines. In your own words, and using as few formulas as possible, write a short report on spline interpolation, its motivation, a comparison with polynomial interpolation, and its applications.

(Individual polynomial qj) Show that qj(x) in (6) satisfies the interpolation condition (4) as well as the derivative condition (5).

Verify the differentiations that give (7) and (8) from (6).

(System for derivatives) Derive the basic linear system (9) for k1 , ... , kn - 1 as indicated in the text.

(Equidistant nodes) Derive (14) from (9).

(Coefficients) Give the details of the derivation of aj2 and aj3 in (13)

Verify the computations in Example I.

(Comparison) Compare the spline g in Example I with the quadratic interpolation polynomial over the whole interval. Find the maximum deviations of g and P2 from f. Comment.

(Natural spline condition) Using coefficients, verify that the spline in satisfies g"(x) = 0 at the ends.

Find the cubic spline g(x) for the given data with ko and kn as given.f( -2) = .f( -1) = f(1) = f(2) = O. f(O) = I. ko = k4 = 0

Find the cubic spline g(x) for the given data with ko and kn as given.. If we started from the piecewise linear function in Fig. 435. we would obtain g(x) in Prob. 10 as the spline satisfying g' (-2) = f' (-2) = 0, g' (2) = f' (2) = O. Find and sketch or graph the corresponding interpolation polynomial of 4th degree and compare it with the spline. Comment. I ~2_-&-----=--2 ------/ 0 ---- Fig. 435. Spline and interpolation polynomial in Problems 10 and 11

Find the cubic spline g(x) for the given data with ko and kn as given. fo = f(O) h = f(6) 1, fl = f(2) = 9, f2 = f(4) = 41, 41, ko = 0, k3 = -12

Find the cubic spline g(x) for the given data with ko and kn as given.fo = f(-I) = O. fl = f(O) = 4. f2 = f(1) = O. ko = 0, k2 = O. Is g(x) even? (Give reason.)

Find the cubic spline g(x) for the given data with ko and kn as given.fo = f(O) = o . .ft = fO) = L f2 = f(Z) = 6. f3 = f(3) = 10. ko = 0, k3 = 0

Find the cubic spline g(x) for the given data with ko and kn as given.fo = f(O) = \. fl = fO) = O. f2 = feZ) = -\. f3 = f(3) = 0.1..0 = O. k3 = -6

Find the cubic spline g(x) for the given data with ko and kn as given.It can happen that a spline is given by the same polynomial in two adjacent subintervals. To illustrate this, find the cubic spline g(x) for f(x) = sin x corresponding to the partition Xo = -7TI1. Xl = 0, X2 = 7T/2 of the interval -7T/Z ~ x ~ 7T/Z and satisfying g'(-7T/Z) = f'(-7T/2) and g'(7T/Z) = f'(7T/2).

(Natural conditions) Explain the remark after (II).

CAS EXPERIMENT. Spline versus Polynomial. [f your CAS gives natural splines, find the natural splines when x is integer from -111 to Ill, and yeO) = I and all other y equal to O. Graph each such spline along with the interpolation polynomial P2m' Do this for 111 = Z to 10 (or more). What happens with increasing 111?

If a cubic spline is three times continuously differentiable (that is, it has continuous first, second. and third derivatives). show that it must be a single polynomial

TEAM PROJECT. Hermite Interpolation and Bezier Curves. In Hermite interpolation we are looking for a polynomial p(x) (of degree ZI1 + I or less) such that p(x) and its derivative p' (x) have given values at 11 + I nodes. (More generally, p(x). p' (x), p"(xJ, ... may be required to have given values at the nodes.) (a) Curves with given endpoints and tangents. Let C be a curve in the x),-plane parametrically represented by ret) = [x(t), y(t)], 0 ~ t ~ I (see Sec. 9.5). Show that for given initial and terminal points of a curve and given initial and terminal tangents. say, A: ro = [x(O). yeo)] [xo, Yo]. 8: r 1 [x(l). y(J)] [Xl' yd Vo = [x'(O), ),'(0)] [x~, y~]. VI [x'(I), y'(I)] [x~. y;] we can find a curve C, namely, reT) = ro + vot (15) + (3(r1 ro) - (ZVo + vtl)t2 + (Z(ro - r 1 ) + Vo + V 1 )t3 ; in components. \"(t) = Xo + X~f + (3(Xl - xo) - (2x~ -t- X;f2 + (2(xo - Xl) + X~ + X;)t 3 y(t) = Yo + Y~f + (3(Yl - l'O) - (2y~ + y;t 2 + (2(yo - YI) + y~ + y;)3. Note that this is a cubic Hennite interpolation polynomiaL and 11 = I because we have two nodes (the endpoints of C). (This has nothing to do with the Hermite polynomials in Sec. 5.S.) The two points = [xo + x~. Yo + Y~] and = [Xl - x;, YI - Y;] are called guidepoints because the segments AGA and BGB specify the tangents graphically. A, 8, GA , GB determine C. and C can be changed quickly by moving the points. A curve consisting of such Hemlite interpolation polynomials is called a Bezier curve. after the French engineer P. Bezier of the Renault Automobile Company. who introduced them in the early 1960s in designing car bodies. Bezier curves (and surfaces) are used in computer-aided design (CAD) and computer-aided manufacturing (CAM). (For more detaib, ~t!e Ref. [E21] in App. \.) (b) Find and graph the Bezier curve and its guidepoints if A: [0, 0]. 8: [I, 0], Vo = [i, n VI = [-i, -!vi3]. (c) Changing guidepoints changes C. Moving guide points farther away makes C "staying near the tangents for a longer time." Confirm this by changing Vo and VI in (b) to Zvo and 2Vl (see Fig. 436). (d) Make experiments of your own. What happens if you change VI in (b) to -VI' If you rotate the tangents? If you multiply Vo and VI by positive factors less than I? B x Fig. 436. Team Project 20(b) and (c): Bezier curves

(Rectangular rule) Evaluate the integral in Example I by the rectangular rule (1) with a subinterval of length 0.1.

Derive a formula for lower and upper bounds for the rectangular rule and apply it to Prob. I.

Evaluate the integrals numelically as indicated and determine the error by using an integration formula known from calculus. J x dx* IX dx* F(x) = - . G(x) 2 1 x* 0 cos x* ' H(x) = 1~\:*e-~* dx*F(2l by (2). Il = 10

Evaluate the integrals numelically as indicated and determine the error by using an integration formula known from calculus. J x dx* IX dx* F(x) = - . G(x) 2 1 x* 0 cos x* ' H(x) = 1~\:*e-~* dx*F(2) by (7), n = [0

Evaluate the integrals numelically as indicated and determine the error by using an integration formula known from calculus. J x dx* IX dx* F(x) = - . G(x) 2 1 x* 0 cos x* ' H(x) = 1~\:*e-~* dx* Gn) by (2).11 = 10

Evaluate the integrals numelically as indicated and determine the error by using an integration formula known from calculus. J x dx* IX dx* F(x) = - . G(x) 2 1 x* 0 cos x* ' H(x) = 1~\:*e-~* dx*G(I) by (7),11 = 10

Evaluate the integrals numelically as indicated and determine the error by using an integration formula known from calculus. J x dx* IX dx* F(x) = - . G(x) 2 1 x* 0 cos x* ' H(x) = 1~\:*e-~* dx*H(4) by (2), Il 10

Evaluate the integrals numelically as indicated and determine the error by using an integration formula known from calculus. J x dx* IX dx* F(x) = - . G(x) 2 1 x* 0 cos x* ' H(x) = 1~\:*e-~* dx*H(4) by (7), Il = 10

Estimate the error by halving. In Prob. 5

Estimate the error by halving.In Prob. 6

Estimate the error by halving.In Prob. 7

Estimate the error by halving.In Prob. 8

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

The following integrals cannot be evaluated by the usual methods of calculus. Evaluate them as indicated. I x sinx* Si(x) = --. - dx~'. o x~ Sex) = {'Sin (X*2) dx*. C(x) = {"cos (X*2) dx* o 0 Si(x) is the sine integral. S(x) and C(x) are the Fresnel integrals. (See App. 3.1.)

(Stability) Prove that the trapezoidal rule is stable with respect to rounding.

Integrate by (11) with II = 5:IIx from I to 3

Integrate by (11) with II = 5:co~ x from 0 to!7T

Integrate by (11) with II = 5: e-x" from 0 to I

Integrate by (11) with II = 5: sin <x2 ) from 0 to 1.25

(Given TOL) Find the smallest 11 in computing the integral of 1Ix from I to 2 for which 50-accuracy is guaranteed (a) by (4) in the use of (2). (b) by (9) in the use of (7). Compare and comment.

TEAM PROJECT. Romberg Integration (W. Romberg. Norske Videllskab. Trolldheil11, FfJrh. 28, Nr. 7, 1955). This method uses the trapezoidal rule and gains precision stepwise by halving h and adding an error estimate. Do this for the integral of f(x) = e-X ti'om x = 0 to x = 2 with TOL = 10-3 , as follows. Step 1. Apply the trapezoidal rule (2) with h = 2 (hence n = I) to get an approximation In. Halve II and use (2) to gel 121 and an error estimate 1 E21 = 22 _ I (121 - 1u) If IE211 ~ TOL, stop. The result is 122 = 121 + E21' Step 2. Show that E21 = -0.066596, hence 1"211 > TOL and go on. Use (2) with hl4 to get 131 and add to it the error estimate 1"31 = i(131 - 1 21 ) to get the better 132 = 131 + 1"31- Calculate I I E32 = 24 _ I (132 - 1 22) = 15 (132 - 1 22),If IE3d ~ TOL, stop. The result is is3 = 132 + E32. (Why does 24 = 16 come in?) Show that we obtain 1"32 = -0.000266, so that we can stop. Arrange your 1- and E-values in a kind of "difference table." If IE3d were greater than TOL, you would have [0 go on and calculate in the next step 141 from (1) with h = ~: then I 142 = 141 + E41 with 1"41 = "3 (141 - 1 31) I 143 = 142 + 1"42 with q2 = 15 (142 - 1 32) I 144 = 143 + E43 with E43 = 63 (J43 - is3) where 63 = 26 - 1. (How does this come in?) Apply the Romberg method to the integral of f( 1:) = ~7TX4 cos !7TX from x = 0 to 2 with TOL = 10-4 .

Consider f(x) = X4 for Xo = 0, Xl = 0.2, X2 = 0.4, X3 = 0.6, X4 = 0.8. Calculate f~ from (14a), (14b), (14c), (15). Determine the errors. Compare and comment.

A "four-point formula" for the derivative is Apply it to f(x) = X4 with Xl' ... , X4 as in Prob. 27, determine the error. and compare it with that in the case of (15).

The derivative f' (x) can also be approximated in terms of first-order and higher order differences (see Sec. 19.3): I 3 1 4 ) + "3 .1. fo - 4" ...\ fo + - . .. . Compute t' (0.4) in Prob. 27 from this fOlmula usino differences up to and including first order, ~econd order, third order, fourth order.

Derive the formula in Prob. 29 from (14) in Sec. 19.3

What is a numeric method? How has the computer influenced numeric methods?

What is floating-point representation of nwnbers? Overflow and underflow?

How do error and relative enor behave under addition? Under multiplication?

Why are roundoff errors important? State the rounding rules.

What is an algorithm"! Which of its properties are important in software implementation?

Why is the selection of a good method at least as important on a large computer as it is on a small one?

Explain methods for solving equations, in particular fixed-point iteration and its convergence.

Can the Newton (-Raphson) method diverge? Is it fast? Same questions for the bisection method.

What is the advamage of Newton's interpolation formulas over Lagrange's?

What do you remember about errors in polynomial imerpolation?

What is spline interpolation? Tts advantage over polynomial interpolation?

List and compare numeric integration methods. When would you appl} them'!

In what sense is Gau~s integration optimal? Explain details.

What does adaptive imegration mean? Why is it useful?

Why is numeric differentiation generally more delicate than numeric integration?

Write -0.35287. 1274.799, -0.00614. 14.9482. 113, 8517 in floating-point form with 5S (5 significant digits, properly rounded).

Compute (5.346 - 3.644)/(3.454 - 3.055) as given and then rounded stepwise to 3S. 2S. I S. ("Stepwise" means rounding the four rounded numbers. not the given ones.) Comment on your results.

Compute 0.29731/(4.1132 - 4.0872) with the numbers as given and then rounded stepwise (that is. rounding the rounded numbers) to 4S. 3S, 2S. Comment.

. Solve x 2 - 50x + 1 = 0 by (6) and by (7) in Sec. 19.1, using 5S in the computation. Compare and comment.

Solve x 2 - 100x + 4 = 0 by (6) and by (7) in Sec. 19.1, using 5S in the computation. Compare and comment.

Let 4.81 and 12.752 be correctly rounded to the number of digits shown. Determine the smallest interval in which the sum (using the true instead of the rounded values) must lie.

Answer the question in Prob. 21 for the difference 4.81 - 11.752.

What is the relative error of l1a in terms of that of a?

Show that the relative error of a2 is about twice that of a.

Compute the solution of x 5 = x + 0.2 near J. = 0 by transforming the equation into the f01Tl1 x = g(x) and starting from Xo = O. (Use 6S.)

Solve cos x = x by iteration (6S, Xo = I), writing it as x = (O.74x + cosx)/1.74. obtainingx4 = 0.739085 (exact to (is!). Why does this converge so rapidly?

Solve X4 - x 3 - 2x - 34 = 0 by Newton's method with Xo = 3 and 6S accuracy.

Solve cos x - x = 0 by the method of false position.

Compute f(1.28) from .fO.Q) = 3.00000 f(l.2) = 1.98007 f(1.4) = 2.92106 f( 1.6) = 1.111534 f( 1.8) = 2.69671 f(l.O) = 2.54030 by linear interpolation. By quadratic interpolation. using f( 1.2), fOA-), IO.6).

Find the cubic spline for the data f(-I) = 3 f(1) = I f(3) = 23 f(5) = -1-5 ko = k3 = 3.

Compute the integral of X3 from 0 to I by the trapezoidal rule with 11 = 5. What error bounds are obtained from (4) in Sec. 19.5? What is the actual error of the result? Why is this result larger than the exact value?

Compute the integral of cos (X2) from 0 to I by Simpson's rule with 2111 = 2 and 2171 = 4 and estimate the error by (10) in Sec. 19.5. (This is the Fresnel integral (38) in App. 3.1 with x = 1.)

Compute the integral of cos x from 0 to -!rr by the three-eights rule f b 3 f(x) dx = "8 hUo + 3.fl + 3f2 + f3) a 1 . - - (b - a)h4!'IV)(t) 80 and give error bounds; here a ~ f ~ band Xj = a + (b - a)jI3, j = 0, ... , 3.