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Questions and Answers of Numerical Analysis

The data for Exercise 6 were generated using the following functions. Use the error formula to find a bound for the error, and compare the bound to the actual error for the cases n= 1 and n = 2. a. f
Use Neville's method to obtain the approximations for Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f (8.4) if f (8.1) = 16.94410, f (8.3)
Neville's Algorithm is used to approximate f (0) using f (−2), f (−1), f (1), and f (2). Suppose f (−1) was overstated by 2 and f (1) was understated by 3. Determine the error in the original
Construct a sequence of interpolating values yn to f (1 + √10), where f (x) = (1 + x2)−1 for −5 ≤ x ≤ 5, as follows: For each n = 1, 2, . . . , 10, let h = 10/n and yn = Pn(1+√10), where
Construct an algorithm that can be used for inverse interpolation.
Use Neville's method to obtain the approximations for Lagrange interpolating polynomials of degrees one, two, and three to approximate each of the following: a. f (0.43) if f (0) = 1, f (0.25) =
Use Neville's method to approximate√3 with the following functions and values. a. f (x) = 3x and the values x0 = −2, x1 = −1, x2 = 0, x3 = 1, and x4 = 2. b. f (x) =√x and the values x0 = 0,
Let P3(x) be the interpolating polynomial for the data (0, 0), (0.5, y), (1, 3), and (2, 2). Use Neville's method to find y if P3(1.5) = 0.
Neville's Algorithm is used to approximate f (0) using f (−2), f (−1), f (1), and f (2). Suppose f (−1) was understated by 2 and f (1) was overstated by 3. Determine the error in the original
Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f (8.4)
a. Show that the cubic polynomialsP(x) = 3 ˆ’ 2(x + 1) + 0(x + 1)(x) + (x + 1)(x)(x ˆ’ 1)AndQ(x) = ˆ’1 + 4(x + 2) ˆ’ 3(x + 2)(x + 1) + (x + 2)(x + 1)(x)Both interpolate the datab. Why
Given Pn(x) = f [x0] + f [x0, x1](x − x0) + a2(x − x0)(x − x1) + a3(x − x0)(x − x1)(x − x2)+· · · + an(x − x0)(x − x1) · · · (x − xn−1), use Pn(x2) to show that a2 = f [x0,
Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials. a. f (0.43)
Show thatFor some Î¾(x). [From Eq. (3.3),Considering the interpolation polynomial of degree n + 1 on x0, x1, . . . , xn, x, we havef (x) = Pn+1(x) = Pn(x) + f [x0, x1, . . . , xn, x](x ˆ’ x0) ·
Let i0, i1, . . . , in be a rearrangement of the integers 0, 1, . . . , n. Show that f [xi0 , xi1 , . . ., xin] = f [x0, x1, . . ., xn]. [Consider the leading coefficient of the nth Lagrange
Use Newton the forward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the
Use the Newton forward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the
Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the
Use the Newton backward-difference formula to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the
a. Use Algorithm 3.2 to construct the interpolating polynomial of degree three for the unequally spaced points given in the following table: x f (x) −0.1 5.30000 0.0 2.00000 0.2 3.19000 0.3
a. Use Algorithm 3.2 to construct the interpolating polynomial of degree four for the unequally spaced points given in the following table: x f (x) 0.0 −6.00000 0.1 −5.89483 0.3
a. Approximate f (0.05) using the following data and the Newton forward-difference formula:b. Use the Newton backward-difference formula to approximate f (0.65).c. Use Stirling's formula to
Use Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data.a.b.c.d.
A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the
a. Show that H2n+1(x) is the unique polynomial of least degree agreeing with f and f'at x0. . . xn. Assume that P(x) is another such polynomial and consider D = H2n+1 ˆ’ P and D' at x0, x1. . .
Let z0 = x0, z1 = x0, z2 = x1, and z3 = x1. Form the following divided-difference table.Show that the cubic Hermite polynomial H3(x) can also be written as f [z0] + f [z0, z1](x ˆ’ x0) + f
Use Theorem 3.9 or Algorithm 3.3 to construct an approximating polynomial for the following data.a.b.c.d.
The data in Exercise 1 were generated using the following functions. Use the polynomials constructed in Exercise 1 for the given value of x to approximate f (x), and calculate the absolute error. a.
The data in Exercise 2 were generated using the following functions. Use the polynomials constructed in Exercise 2 for the given value of x to approximate f (x), and calculate the absolute error. a.
a. Use the following values and five-digit rounding arithmetic to construct the Hermite interpolating polynomial to approximate sin 0.34.b. Determine an error bound for the approximation in part (a),
Let f (x) = 3xex − e2x. a. Approximate f (1.03) by the Hermite interpolating polynomial of degree at most three using x0 = 1 and x1 = 1.05. Compare the actual error to the error bound. b. Repeat
Repeat Exercise 6 using the clamped cubic splines constructed in Exercise 8. In Exercise 6 a. f (x) = e2x ; approximate f (0.43) and f' (0.43). b. f (x) = x4 − x3 + x2 − x + 1; approximate f (0)
Construct a natural cubic spline to approximate f (x) = cos Ï€x by using the values given by f (x) at x = 0, 0.25, 0.5, 0.75, and 1.0. Integrate the spline over [0, 1], and compare the result to0.
Construct a natural cubic spline to approximate f (x) = eˆ’x by using the values given by f (x) at x = 0, 0.25, 0.75, and 1.0. Integrate the spline over [0, 1], and compare the result toUse the
Repeat Exercise 15, constructing instead the clamped cubic spline with f'(0) = f' (1) = 0.In Exercise 15Construct a natural cubic spline to approximate f (x) = cos Ï€x by using the values given by
Repeat Exercise 16, constructing instead the clamped cubic spline with f'(0) = ˆ’1, f'(1) = ˆ’eˆ’1.In Exercise 16Construct a natural cubic spline to approximate f (x) = eˆ’x by using the
Suppose that f (x) is a polynomial of degree 3. Show that f (x) is its own clamped cubic spline, but that it cannot be its own natural cubic spline.
Suppose the data {xi , f (xi))}ni=1 lie on a straight line. What can be said about the natural and clamped cubic splines for the function f?
Given the partition x0 = 0, x1 = 0.05, and x2 = 0.1 of [0, 0.1], find the piecewise linear interpolating function F for f (x) = e2x. ApproximateF(x) dx, and compare the results to the actual value.
Let f ˆˆ C2 [a, b], and let the nodes a = x0 In Exercise 21Given the partition x0 = 0, x1 = 0.05, and x2 = 0.1 of [0, 0.1], find the piecewise linear interpolating function F for f (x) = e2x.
Extend Algorithms 3.4 and 3.5 to include as output the first and second derivatives of the spline at the nodes.
Extend Algorithms 3.4 and 3.5 to include as output the integral of the spline over the interval [x0, xn].
Given the partition x0 = 0, x1 = 0.05, x2 = 0.1 of [0, 0.1] and f (x) = e2x:a. Find the cubic spline s with clamped boundary conditions that interpolates f.b. Find an approximation forc. Use Theorem
Let f be defined on [a, b], and let the nodes a = x0 < x1 < x2 = b be given. A quadratic spline interpolating function S consists of the quadratic polynomial S0(x) = a0 + b0(x − x0) + c0(x − x0)2
Determine a quadratic spline s that interpolates the data f (0) = 0, f (1) = 1, f (2) = 2 and satisfies S'(0) = 2.
a. The introduction to this chapter included a table listing the population of the United States from 1950 to 2000. Use natural cubic spline interpolation to approximate the population in the years
A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the
Construct the natural cubic spine for the following
The 2009 Kentucky Derby was won by a horse named Mine That Bird (at more than 50:1 odds) in a time of 2:02.66 (2 minutes and 2.66 seconds) for the 1 1/4 -mile race. Times at the quarter-mile,
It is suspected that the high amounts of tannin in mature oak leave inhibit the growth of the winter moth (Operophtera bromata L., Geometridae) larvae that extensively damage these trees in certain
The upper portion of this noble beast is to be approximated using clamped cubic spline interpellants. The curve is drawn on a grid from which the table is constructed. Use Algorithm 3.5 to construct
Repeat Exercise 32, constructing three natural splines using Algorithm 3.4.In Exercise 32The upper portion of this noble beast is to be approximated using clamped cubic spline interpellants. The
Construct the natural cubic spline for the following
The data in Exercise 3 were generated using the following functions. Use the cubic splines constructed in Exercise 3 for the given value of x to approximate f (x) and f'(x), and calculate the actual
The data in Exercise 4 were generated using the following functions. Use the cubic splines constructed in Exercise 4 for the given value of x to approximate f (x) and f'(x), and calculate the actual
Construct the clamped cubic spline using the data of Exercise 3 and the fact that a. f'(8.3) = 3.116256 and f'(8.6) = 3.151762 b. f'(0.8) = 2.1691753 and f'(1.0) = 2.0466965 c. f'(−0.5) = 0.7510000
Construct the clamped cubic spline using the data of Exercise 4 and the fact that a. f'(0) = 2 and f'(0.5) = 5.43656 b. f'(−0.25) = 0.437500 and f'(0.25) = −0.625000 c. f'(0.1) = −2.8004996 and
Repeat Exercise 5 using the clamped cubic splines constructed in Exercise 7. In Exercise 5 a. f (x) = x ln x; approximate f (8.4) and f'(8.4). b. f (x) = sin(ex − 2); approximate f (0.9) and
Let (x0, y0) = (0, 0) and (x1, y1) = (5, 2) be the endpoints of a curve. Use the given guide points to construct parametric cubic Hermite approximations (x(t), y(t)) to the curve, and graph the
Repeat Exercise 1 using cubic Bézier polynomials. In Exercise 1 Let (x0, y0) = (0, 0) and (x1, y1) = (5, 2) be the endpoints of a curve. Use the given guidepoints to construct parametric cubic
Construct and graph the cubic Bézier polynomials given the following points and guide points. a. Point (1, 1) with guide point (1.5, 1.25) to point (6, 2) with guide point (7, 3) b. Point (1,
Use the data in the following table and Algorithm 3.6 to approximate the shape of the letter N.
Suppose a cubic Bézier polynomial is placed through (u0, v0) and (u3, v3) with guide points (u1, v1) and (u2, v2), respectively. a. Derive the parametric equations for u(t) and v(t) assuming
Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables.a.b.
Use the formulas given in this section to determine, as accurately as possible, approximations for each missing entry in the following tables.a.b.
The data in Exercise 9 were taken from the following functions. Compute the actual errors in Exercise 9, and find error bounds using the error formulas and Maple.a. f (x) = tan xb. f (x) = ex/3 +
The data in Exercise 10 were taken from the following functions. Compute the actual errors in Exercise 10, and find error bounds using the error formulas and Maple.a. f (x) = tan 2xb. f (x) =
Use the following data and the knowledge that the first five derivatives of f are bounded on [1, 5] by 2, 3, 6, 12 and 23, respectively, to approximate f'(3) as accurately as possible. Find a bound
Repeat Exercise 13, assuming instead that the third derivative of f is bounded on [1, 5] by 4.In Exercise 13
Repeat Exercise 1 using four-digit rounding arithmetic, and compare the errors to those in Exercise 3.In Exercise 1a.b.
Repeat Exercise 5 using four-digit chopping arithmetic, and compare the errors to those in Exercise 7.In Exercise 5a.b.c.d.
Repeat Exercise 9 using four-digit rounding arithmetic, and compare the errors to those in Exercise 11.In Exercise 11a.b.
Consider the following table of data:a. Use all the appropriate formulas given in this section to approximate f'(0.4) and f''(0.4).b. Use all the appropriate formulas given in this section to
Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables.a.b.
Consider the following table of data:a. Use Eq. (4.7) to approximate f'(0.2).b. Use Eq. (4.7) to approximate f'(1.0).c. Use Eq. (4.6) to approximate f'(0.6).
Derive an O(h4) five-point formula to approximate f''(x0) that uses f (x0 − h), f (x0), f (x0 + h), f (x0 + 2h), and f (x0 + 3h). [Consider the expression Af (x0 − h) + Bf (x0 + h) + Cf (x0 + 2h)
a. Analyze the round-off errors, as in Example 4, for the formulab. Find an optimal h > 0 for the function given in Example 2.
In Exercise 10 of Section 3.4 data were given describing a car traveling on a straight road. That problem asked to predict the position and speed of the car when t = 10 s. Use the following times and
In a circuit with impressed voltage E(t) and inductance L, Kirchhoff's first law gives the relationship E(t) = L di/dt + Ri, Where R is the resistance in the circuit and i is the current. Suppose we
Derive a method for approximating f''' (x0) whose error term is of order h2 by expanding the function f in a fourth Taylor polynomial about x0 and evaluating at x0 ± h and x0 ± 2h.
Consider the function e(h) = ε/h+ h2/6 M, Where M is a bound for the third derivative of a function Show that e(h) has a minimum at 3√(3ε/M).
The data in Exercise 1 were taken from the following functions. Compute the actual errors and find error bounds using the error formulas.a. f (x) = sin xb. f (x) = ex ˆ’ 2x2 + 3x ˆ’ 1In
The data in Exercise 2 were taken from the following functions. Compute the actual errors and find error bounds using the error formulas.a. f (x) = 2 cos 2x - xb. f (x) = x2 ln x + 1In Exercise 2a.b.
Use the most accurate three-point formula to determine each missing entry in the following tables.a.b. c. d.
Use the most accurate three-point formula to determine each missing entry in the following tables.a.b. c. d.
The data in Exercise 5 were taken from the following functions. Compute the actual errors in Exercise 5, and find error bounds using the error formulas.a. f (x) = e2xb. f (x) = x ln xc. f (x) = x cos
The data in Exercise 6 were taken from the following functions. Compute the actual errors in Exercise 6, and find error bounds using the error formulas.a. f (x) = e2x ˆ’ cos 2xb. f (x) =
Use the formulas given in this section to determine, as accurately as possible, approximations for each missing entry in the following tables.a.b.
Apply the extrapolation process described in Example 1 to determine N3(h), an approximation to f' (x0), for the following functions and step sizes. a. f (x) = ln x, x0 = 1.0, h = 0.4 b. f (x) = x +
Suppose that N (h) is an approximation to M for every h > 0 and that M = N (h) + K1h2 + K2h4 + K3h6 +· · · , For some constants K1, K2, K3 . . . Use the values N (h), N(h/3), and N(h/9) to produce
In calculus, we learn that e = limh→0(1 + h)1/h.a. Determine approximations to e corresponding to h = 0.04, 0.02, and 0.01.b. Use extrapolation on the approximations, assuming that constants K1, K2
a. Show thatlimh→0 ((2 + h)/(2 - h))1/h = e.b. Compute approximations to e using the formula N(h) = ((2+h)/(2−h))1/h, for h = 0.04, 0.02, and 0.01.c. Assume that e = N(h)+K1h+K2h2 +K3h3 +· · ·
Suppose the following extrapolation table has been constructed to approximate the number M with M = N1(h) + K1h2 + K2h4 + K3h6:a. Show that the linear interpolating polynomial P0,1(h) through (h2,
Suppose that N1(h) is a formula that produces O(h) approximations to a number M and that M = N1(h) + K1h + K2h2 +· · · ,For a collection of positive constants K1, K2 . . . Then N1(h), N1(h/2),
The semi perimeters of regular polygons with k sides that inscribe and circumscribe the unit circle were used by Archimedes before 200 b.c.e. to approximate π, the circumference of a semicircle.
Add another line to the extrapolation table in Exercise 1 to obtain the approximation N4(h).In Exercise 1a. f (x) = ln x, x0 = 1.0, h = 0.4b. f (x) = x + ex , x0 = 0.0, h = 0.4c. f (x) = 2x sin x, x0
Repeat Exercise 1 using four-digit rounding arithmetic.In Exercise 1a. f (x) = ln x, x0 = 1.0, h = 0.4b. f (x) = x + ex , x0 = 0.0, h = 0.4c. f (x) = 2x sin x, x0 = 1.05, h = 0.4d. f (x) = x3 cos x,
Repeat Exercise 2 using four-digit rounding arithmetic.In Exercise 2a. f (x) = ln x, x0 = 1.0, h = 0.4b. f (x) = x + ex , x0 = 0.0, h = 0.4c. f (x) = 2x sin x, x0 = 1.05, h = 0.4d. f (x) = x3 cos x,
Show that the five-point formula in Eq. (4.6) applied to f (x) = xex at x0 = 2.0 gives N2(0.2) in Table 4.6 when h = 0.1 and N2(0.1) when h = 0.05.
The forward-difference formula can be expressed as f(x0) = 1/h[f (x0 + h) − f (x0)] - h/2f''(x0) - h2/6 f'''(x0) + O(h3). Use extrapolation to derive an O (h3) formula for f'(x0).

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