1 Million+ Step-by-step solutions

Addition with 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.

Theorems on errors prove Theorem 1(a) for addition.

Show that in Example 1 the absolute value of the error of x2 = 2.000/39.95 = 0.05006 is less than 0.00001.

Nested form evaluate f(x) = x3 – 7.5x2 +11.2x + 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 nested form, is usually preferable since it minimizes the number of operations and thus the effect of rounding.

Estimate the error in Prob. 1 by (5)

Error bounds derive error bounds for p2 (9.2) in Example 2 from (5)

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

Interpolation and extrapolation Calculate p2© in Example 2 computer 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.

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

Compute f(6.5) from by cubic interpolation, using(10)

Sub-tabulation compute the Bessel function J1(x) for x = 0.1, 0.3, ∙∙∙, 0.9 from J1(0) = 0, J1 (0.2) = 0.09950, J1(0.4) = 0.19603, J1(0.6) = 0.28670, J1(0.8) = 0.36884, J1(1.0) = 0.44005. Use (14) with n = 5.

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

(Comparison) compares the sp line g in Example 1 with the quadratic interpolation polynomial over the whole interval find the maximum deviations of g and p2 from f, comment.

Find the cobic sp line g(x) for the given data with k0 and kn as given f (– 2) = f (– 1) = f (1) = f (2) = 0 f (0) = 1, k0, k4 = 0.

If your CAS gives natural sp lines, find the natural sp lines when x is integer from – m to m, and y(0) = 1 and all other y equal to 0. Graph each such sp line along with the interpolation polynomial p2m, Do this for m = 2 to 10 (or more). What happens with increasing m?

Rounding for the following matrix A find det A, what happens if you round off the fiven entries to

(a) 5S,

(b) 4S,

(c) 3S,

(d) 2S,

(e) 1S?

What is the practical implication of yourwork?

(a) 5S,

(b) 4S,

(c) 3S,

(d) 2S,

(e) 1S?

What is the practical implication of yourwork?

In Prob.7, compute C

(a) If you solve the first equation for x1, the second for x2, the third for x3, proving convergence;

(b) If you non-sensically solve the third equation for x1, the first for x2 the second for x3, proving divergence.

(a) If you solve the first equation for x1, the second for x2, the third for x3, proving convergence;

(b) If you non-sensically solve the third equation for x1, the first for x2 the second for x3, proving divergence.

Cubic parabola derives the formula for the normal equations of a cubic least squares parabola.

By what integer factor can you at most reduce the Gerschgorin circl with center 3 in Prob. 6?

Extended Gerschgorin theorem Prove Theorem 2

Normal matrices show that Hermitian, skew-Hermitian, and unitary matrices (hence real symmetric, ske-symmetric, and orthogonal matrices) are normal, why is this of practical interest?

Eigen values on the circle Illustrate with a 2 x 2 matrix that an Eigen value may very well lie on a Gerschgorin circle (so that Gerchgorin disks can generally not be replaced with smaller disks without losing the inclusion property).

CAS Experiment

(a) Write a program for the iteration in Example 4 (with any A and x0) that at each step prints the mid point (why?) the endpoints, and the length of the inclusion interval.

(b) Apply the program to symmetric matrices of your choice. Explore how convergence depends on the choice of initial vectors. Can you construct cases in which the lengths of the inclusion intervals are not monotone decreasing? Can you explain the reason? Can you experiment on the effect of rounding?

(a) Write a program for the iteration in Example 4 (with any A and x0) that at each step prints the mid point (why?) the endpoints, and the length of the inclusion interval.

(b) Apply the program to symmetric matrices of your choice. Explore how convergence depends on the choice of initial vectors. Can you construct cases in which the lengths of the inclusion intervals are not monotone decreasing? Can you explain the reason? Can you experiment on the effect of rounding?

Optimality of δ in Prob. 2 Choose x0 = [3 – 1]t and show that q = 0 and δ = 1 for all steps and that the Eigen values are ±1, so that the interval [q – δ, q + δ] cannot be shortened in general! Experiment with other x0.

Rayleigh quotient why does q generally approximate the Eigen value of greatest absolute value? When will q be a good approximation?

Power method do 4 steps of the power method for the matrix in Prob. 24, starting from [1 1 1]T and computing the Rayleigh quotients and error bounds.

Kutta’s third-order method is define by yn + 1 = yn + 1/6 (k1 + 4k2 + k3*) with k1 and k2 as in RK Table 21.4 and k3* = hf (xn+1, yn – k1 + 2k2). Apply this method to (4) in example 1. Choose h = 0.2 and d0 5 steps. Compare with Table 21.6

How much can you reduce the error in Prob. 13 by halving h (20 steps, h = 0.05)? First guess then compute.

The ODE in Prob. 6 by what factor did the error decrease?

Bessel Function J0, xy'' + y' + xy = 0, y(1) = 0.765198 , y' (1) = – 0.440051, h = 0.5, 5 steps, (this igives the standard solution J0(x) in Fig. 107 in Sec. 5.5)

The system in Prob. 2, how much smaller is the error?

Verify the calculations in Example 1. Find out experimentally how many steps are needed to obtain the solution of the linear system with an accuracy of 3S.

3 x 3 grid solve example 1, choosing h = 3 and starting values 100, 100,∙∙∙.

Using the answer to Prob 11, try to sketch some isotherms.

Influences of starting values do Prob. 5 by Gauss-Seidel, starting from 0. Compare and comment.

ADI apply the ADI method to the Dirichlet problem in Prob 5, using the grid in Fig. 455, as before and starting values zero.

Solve the mixed boundary value problem for the Laplace equation ∆2u = 0 in the rectangle in Fig. 457a (using the grid in Fig. 457b) and the boundary conditions ux = 0 on the left edge, ux = 3 on the right edge, u = x2 on the lower edge, and u = x2 – 1 on the upper edge.

Solve the mixed boundary value problem for the Poisson equation ∆2u = 2(x2 + y2) in the region and for the boundary conditions shown in Fig. 461, using the indicated grid.

Solve ∆2u = – π2y sin 1/3πx for the grid in Fig. 461 and uy (1, 3) = uy (2, 3) = 1/2√243, u = 0 on the other three sides of the square.

If in Prob. 11 the axes are grounded (u = 0), what constant potential must the other portion of the boundary have in order to produce 100 volts at P11?

Solve the Poisson equation ∆2u = 2 in the region and for the boundary values shown in Fig. 463, using the grid also shown in the figure.

Show that from d' Alembert’s solution (13) in Sec, 12.4 with c = 1 it follows that (6) in the present section gives the exact value uij+1 = u (ih, (j + 1) h).

Solve Prob. 5 by the explicit method with h = 0.3 and k = 0.01. Do 8 steps, compare the last values with the Crank-Nicolson 3S- values 0.107, 0.175 and the exact 3S-values 0.108, 0.175.

If the left end of a laterally insulated bar extending from x = 0 to x = 1 is insulated, the boundary condition at x = 0, is un (0, t) = ux (0, t) = 0. Show that in the application of the explicit method given by (5), we can compute u0.j+1 by the formula

Solve Prob. 9 for f(x) = x if 0 < x < 0.5, f(x) =1 – x if 0.5 < x < 1, all the other data being as before. Can you expect the solution to satisfy u(x, t) = u (1 – x, t) for all t?

Compute approximate values in Prob. 5, using a finger grid (h = 0.1, k = 0.1) and notice the increase in accuracy.

Solve y' = 2xy, y(0) = 1, by the Euler method with h = 0.1, 10 steps, Compute the error.

Solve y' = (x + y – 4)2, y(0) = 4, by RK with h = 0.2, 7 steps.

Solve y' = (x + y)2 , y(0) = 0 by RK with h = 0.2, 5 steps.

Apply the multistep method in Prob. 23 to the initial value problem y' = x + y, y(0) = 0, choosing h = 0.2 and doing 5 step. Compare with the exact values.

Solve y'' + y = 0, y (0) = 0, y'(0) = 1 by RKN with h = 0.2, 5 steps. Find the error.

Solve y'1 = – 5y1 + 3y2, y'2 = 3y1 – 5y2, y1 (0) = 2, y2 (0) = 2, by RK for systems, h = 0.1, 5 steps.

A laterally insulated homogeneous bar with ends at x = 0 and x = 1 has initial temperature 0. Its left and is kept at 0, whereas the temperature at the right end varies sinusoidally according to u(t, 1) = g(t) = sin 25/3πt. Find the temperature u(x, t) in the bar by the explicit method with h = 0.2 and r = 0.5(one period, that is, o < t < 0.24).

Find the solution of the vibrating string problem uu = uxx, u(x, 0) = x(1 – x), ut = 0, u(0, t) = u(1, t) = 0 by the method in Sec. 21.7 with h = 0.1 and k = 0.1 for t = 0.3

Water accounts for roughly 60% of total body weight. Assuming it can be categorized into six regions, the percentages go as follows. Plasma claims 4.5% of the body weight and is 7.5% of the total body water. Dense connective tissue and cartilage occupies 4.5% of the total body weight and 7.5% of the total body water. Interstitial lymph is 12% of the body weight, which is 20% of the total body water. Inaccessible bone water is roughly 7.5% of the total body water and 4.5% total body weight. If intracellular water is 33% of the total body weight and transcellular water is 2.5% of the total body water. What percent of total body weight must the transcellular water be and what percent of total body water must the intracellular water be?

A group of 30 students attend a class in a room that measures 10 m by 8 m by 3 m. Each student takes up about 0.075 m^{3} and gives out about 80 W of heat (l W = l J/s). Calculate the air temperature rise during the first 15 minutes of the class if the room is completely sealed and insulted. Assume the heat capacity, C_{υ}. for air is 0.718 kJ/(kg K). Assume air is an ideal gas at 20^{o}C and 101.325kPa. Note that the heat absorbed by the air Q is related to the mass of the air m, the heat capacity, and the change in temperature by the following relationship:

The mass of air can be obtained from the ideal gas law: PV = m/Mwt RT. Where P is the gas pressure, V is the volume of the gas, Mwt is the molecular weight of the gas (for air, 28.97 kg/kmol), and R is the ideal gas constant [8.314 kPa m^{3}/(kmol K)].

The following information is available for a bank account:

Use the conservation of cash to compute the balance on 6/1, 7/1, 8/1 and 9/1. Show each step in the computation. Is this a steady-state or a transient computation?

The volume flow rate through a pipe is given by Q = υA, where υ is the average velocity and A is the cross-sectional area, Use volume-country to solve for the required area in pipe 3.

Figure depicts the various ways in which an average man gains and loess water in one day. One liter is ingested as food, and

The body metabolically produces 0.3 L. In breathing air, the exchange is 0.05 L while inhaling and 0.4 L while exchange over a one-day period. The body will also 0.2, 1.4, 0.2, and 0.35 L through sweat, urine, faces, and through the skin, respectively. In order to maintain steady-state condition, how much water must be drunk perday?

For the free-falling parachutist with liner drag, assume a first jumper is 70 kg and has a drag coefficient of 12 kg/s. If a jumper has a drag coefficient of 15 kg/s and a mass of 75 kg, how long will it take him to reach the same velocity the first jumper reached in 10 s?

Use calculus to solve Eq. (1.9) for the case where the initial velocity, υ (0) is nonzero.

Repeat Example 1.2, Compute the velocity to t = 10 s, with a step size of (a) 1 and (b) 0.5 s. Can you make any statement regarding the errors of the calculation based on the results?

Rather than the linear relationship of Eq. (1.7), you might choose to model the upward force on the parachutist as a second-order relationship, F_{U} = – c’*v*^{2 }where c’ = second-order drag coefficient (kg/m).

(a) Using calculus, obtain the closed-form solution for the case where the jumper is initially at rest (υ = 0 at t = 0).

(b) Repeat the numerical calculation in Example 1.2 with the same initial condition and parameter values. Use a value of 0.225 kg/m for c’.

Compute the velocity of a free-falling parachutist using Euler’s method for the case where m = 80 kg and c = 10 kg/s. Per-form the calculation from t = 0 to 20 s with a step size of 1 s. Use an initial condition that the parachutist has an upward velocity of 20 m/s at t = 0. At t = 10 s, assume that the chute is instantaneously deployed so that the drag coefficient jumps to 50 kg/s.

In our example of the free-falling parachutist, we assumed that the acceleration due to gravity was a constant value of 9.8 m/s^{2}. Although this is a decent approximation when we are examining falling objects near the surface of the earth, the gravitational force decreases as we move above sea level. A more general representation based on Newton’s inverse square law of gravitational attraction can be written as g(x) = g(0) R^{2}/(r + x)^{2}

Where g (x) = gravitational attraction at altitude x (in m) measured upwards from the earth’s surface (m/s^{2}), g(0) = gravitational attraction at the earth’s surface (≡ 9.8 m/s^{2}), and R = the earth’s radius (≡ 6.37 x 10^{6 }m).

(a) In a fashion similar to the derivation of Eq. (1.9) use a force balance to derive a differential equation for velocity as a function of time that utilizes this more complete representation of gravitation. However, for this derivation, assume that upward velocity is positive.

(b) For the case where drag is negligible, use the chain rule to express the differential equation as a function of altitude rather than time. Recall that the chain rule is dv/dt = dv/dx dx/dt

(c) Use calculus to obtain from solution where υ = υ_{0} at x = 0.

(d) Use Euler’s method to obtain a numerical solution from x = 0 to 100,000 m using a step of 10,000 m where the initial velocity is 1400 m/s upwards. Compute your result with the analytical solution.

The amount of a uniformly distributed radioactive contaminant contained in a close reactor is measured by its concentration c (Becquerel/liter or Bq/L). The contaminant decreases at a decay rate proportional to its concentration-that is Decay rate = -kc where k is a constant with units of day^{–1}. Therefore, according to Eq. (1.13), a mass balance for the reactor can be written as

(a) Use Euler’s method to solve this equation from t = 0 to 1 d with k = 0.2d^{–1}. Employ a step size of Δt = 0.1. The concentration at t = 0 is 10 Bq/L.

(b) Plot the solution on a semilog graph (i.e., in c versus t) and determine the slope. Interpret your results.

A storage tank contains a liquid at depth y where y = 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Q sin^{2 }(t).

Equation (1.13) can be written for this system as

Or, since the surface area A is constant

dy/dx = 3 Q/A sin^{2} (t) – Q/A

Use Euler’s method to solve for the depth y from t = 0 to 10 d with a step size of 0.5 d. The parameter values are A = 1200 m^{2} and Q = 500 m^{3}/d. Assume that the initial condition is y = 0.

For the same storage tank described in Prob. 1.13, suppose that the outflow is not constant but rather depends on the depth. For this case, the differential equation for depth can be written as

Use Euler’s method to solve for the depth y from t = 0 to 10 d with a step size of 0.5d. The parameter values are A = 1200 m^{2}, Q = 500 m^{3}/d, and α = 300. Assume that the initial condition is y = 0.

Suppose that a spherical droplet of liquid evaporates at a race that is proportional to its surface area.

dV/dt = – kA

Where V = volume (mm^{3}), t = time (h), k = the evaporation rater (mm/hr), and A = surface area (mm^{2}). Use Euler’s method to compute the volume of the droplet from t = 0 to 10 min using a step size of 0.25 min. Assume that k = 0.1 mm/min and that the droplet initially has a radius of 3 mm. Assess the validity of your results by determining the radius of your final computed volume and verifying that it is consistent with the evaporation rate.

Newton’s law of cooling says that the temperature of a body changes at a rate proportional to the difference between its temperature and that of the surrounding medium (the ambient temperature),

dT/dt = – k(T – T*a*)

Where T = the temperature of the body (^{o}C), t = time (min), k = the proportionality constant (per minute), and T_{a} = the ambient temperature (^{o}C). Suppose that a cup of coffee originally has a temperature of 68^{o}C. Use Euler’s method to compute the temperature from t = 0 to 10 min using a step size of 1 min if T_{a} = 21^{o}C and k = 0.017/min.

Cancer cells grow exponentially with a doubling time of 20 h when they have an unlimited nutrient supply. However, as the cells start to from a solid spherical tumor without a blood supply, growth at the center of the tumor becomes limited, and eventually cells start to die.

(a) Exponential growth of cell ∙ number N can be expressed as shown, where µ is the growth rate of the cells. For cancer cells, find the value of µ.

dN/dt = µN

(b) Write an equation that will describe the rate of change of tumor volume during exponential growth given that the diameter of an individual cell is 20 microns.

(c) After a particular type of tumor exceeds 500 microns in diameter, the cells at the center of the tumor die (but continue to take up space in the tumor). Determine how long it will take for the tumor to exceed this critical size.

(a) Exponential growth of cell ∙ number N can be expressed as shown, where µ is the growth rate of the cells. For cancer cells, find the value of µ.

dN/dt = µN

(b) Write an equation that will describe the rate of change of tumor volume during exponential growth given that the diameter of an individual cell is 20 microns.

(c) After a particular type of tumor exceeds 500 microns in diameter, the cells at the center of the tumor die (but continue to take up space in the tumor). Determine how long it will take for the tumor to exceed this critical size.

A fluid is pumped into the network shown in Figure. 1f Q_{2} = 0.6, Q_{3} = 0.4, Q_{7} = 0.2, and Q_{8} = 0.3 m^{3}/s, determine the other flows.

Write pseudocode to implement the flowchart depicted in Figure. Make sure that proper indentation is included to make the structureclear.

Rewrite the following pseudocode using proper indentation

DO

i = i + 1

IF z > 50 EXIT

x = x + 5

IF x > 5 THEN

y = x

ELSE

y = 0

ENDIF

z = x + y

NDDO

A value for the concentration of a pollutant in a lake is recorded on each card in a set of index cards. A card marked “end of data” is placed at the end of set. Write an algorithm to determine the sum, the average, and the maximum of these values.

Write a structured flowchart for prob.2.3

Develop, debug, and document a program to determine the roots of a quadratic equation, ax2 + bx + c, in either a high-level language or a macro language of your choice. Use a subroutine procedure to compute the roots (either real or complex). Perform test runs for the cases

(a) a = 1, b = 6, c = 2;

(b) a = 0, b = -4, c = 1.6;

(c) a = 3, b = 2.5, c = 7.

The cosine function can be evaluated by the following infinite series:

Write an algorithm to implement this formula so that it computes and prints out the values of cos x as each term in the series is added. In other words, compute and print in sequence the values for

cos x = 1

cos x = 1 – x^{2}/2!

cos x = 1 – x^{2}/2! + x^{4}/x!

Up to the order term n of your choosing. For each of the preceding, compute and display the percent relative error as % error = true – series approximation/true x 100%

Write the algorithm for prob. 2.6 as

(a) a structured flowchart and

(b) Pseudocode.

Develop, debug, and document a program for Prob. 2.6 in either a high-level language or a macro language of your choice. Employ the library function for the cosine in your computer to determine the true value. Have the program print out the series approximation and the error at each step. As a test case, employ the program to compute cos (1.25) for up to and including the term x^{10}_{ }/ 10!. Interpret your results.

The following algorithm is designed to determine a grade for a course that consists of quizzes, homework, and a final exam:

Step 1: Input course number and name.

Step 2: Input weighting factors for quizzes (WQ), homework (WH), and the final exam (WF).

Step 3: Input quiz grades and determine an average quiz grade (AQ).

Step 4: Input homework grades and determine an average homework grade (AH).

Step 5: If this course has a final grade, continue to step 6. If not, go to step 9.

Step 6: Input final exam grade (FE).

Step 7: Determine average grade AG according to

Step 8: Go to step 10.

Step 9: Determine average grade AG according to

Step 10: Print out course number, name, and average grade.

Step 11: Tarminate computation.

(a) Write well-structured pseudocode to implement this algorithm.

(b) Write debug, and document a structured computer program based on this algorithm. Test it using the following data to calculate a grade without the final exam and a grade with the final exam: WQ = 35; WH = 30; WF = 35; quizzes = 98, 85, 90, 65, 99; homework = 95, 90, 87, 100, 92, 77; and final exam =92.

The divide and average” method, an old-time method for approximating the square root of any positive number α can be formulated as x = x + a/x / 2.

(a) Write well-structured pseudosode to implement this algorithm as depicted in Figure. Use proper indentation so that the structure is clear.

(b) Develop, debug, and document a program to implement this equation in either a high-level language or a macro language of your choice. Structure your code according to Figure.

An amount of money P is invested in an account where interest is compounded at the end of the period. The future worth F yielded at an interest rate i after n periods may be determined from the following formula: F = P (l + i) n

Write a program that will calculate the future worth of an investment for each year from l through n. The input to the function should include the initial investment P, the interest rate i (as a decimal), and the number of years n for which the future worth is to be calculated. The output should consist of a table with headings and columns for n and F. Run the program for P = $100,000, i = 0.06, and n = 5 years.

Write a program that will calculate the future worth of an investment for each year from l through n. The input to the function should include the initial investment P, the interest rate i (as a decimal), and the number of years n for which the future worth is to be calculated. The output should consist of a table with headings and columns for n and F. Run the program for P = $100,000, i = 0.06, and n = 5 years.

Economic formulas are available to compute annual payments for loans. Suppose that you borrow an amount of money P and agree to repay it in n annual payments at an interest rate of i. The formula to compute the annual payment A is

Write a program to compute A. Test it with P = $55,000 and an interest rate of 6.6 % ( i = 0.066). Compute results for n = 1, 2, 3, 4, and 5, display the results as a table with headings and columns for n andA.

The average daily temperature for an area can be approximated by the following function,

T = T _{mean} + (T _{peak} – T _{mean}) cos (ω (t – t _{peak}))

Where T _{mean} = the average annual temperature, T _{peak} = the peak temperature, ω = the frequency of the annual variation (= 2π/365), and t _{peak} = day of the peak temperature (≈ 205 d).Develop a program that computes the average temperature between two days of the year for a particular city. Test it for

(a) January – February (t = 0 to 59) in Miami, Florida (T mean = 22.1^{o} C; T _{peak} = 28.3^{o} C), and

(b) July–August (t = 180 to 242) in Boston, Massachusetts (T _{mean }= 10.7^{o} C; T _{peak }= 22.9^{o }C).

Develop, debug, and test a program in either a high-level language or a macro language of your choice to compute the velocity of the falling parachutist as outlined in Example 1.2 Design the program so that it allows the user to input values for the drag coefficient and mass. Test the program by duplicating the results from Example 1.2 Repeat the computation but employ step sizes of 1 and 0.5 s. Compare your results with the analytical solution obtained previously in Example 1.1 Does a smaller step size make the results better or worse? Explain your results.

The bubble sort is an inefficient, but easy-to-program, sorting technique. The idea behind the sort is to move down through an array comparing adjacent pairs and swapping the values if they are out of order. For this method to sort the array completely, it may need to pass through it many times. As the passes proceed for an ascending-order sort the smaller elements in the array to rise toward the top like bubbles. Eventually, there will be a pass through the array where no swaps are required. Then, the array is sorted. After the first pass, the largest value in the array drops directly to the bottom.Consequently, the second pass only has to proceed to the second-to-last value, and so on. Develop a program to set up an array of 20 random numbers and sort them in ascending order with the bubble sort(figure)

Figure shows a cylindrical tank with a conical base. If the liquid level is quite low in the conical part, the volume is simply the conical volume of liquid. If the liquid level is midrange in the cylindrical part, the total volume of liquid includes the filled conical part and the partially filled cylindrical part. Write a well-structured function procedure to compute the tank’s volume as a function of given values R and d. Use decisional control structure (like If/Then, Elself, Else, End If). Design the function so that it returns the volume for all cases where the depth is less than 3R. Return an error massage ("Overtop") if you overtop the tank that is d > 3R.Test it with following data:

Two distances are required to specify the location of a point relative to an origin in two-dimensional space (Figure):

- The horizontal and vertical distances ( x . y ) in Cartesian coordinates
- The radius and angle (r, θ) in radial coordinates.

It is relatively straightforward to compute Cartesian coordinates (x, y) on the basis of polar coordinates (r, θ). The reverse process is not so simple. The radius can be computed by the following formula: r = √ x^{2} + y^{2}

If the coordinates lie within the first and fourth coordinates (i.e., x > 0), then a simple formula can be used to compute θ

θ = tan – 1 (y/x)

The difficulty arises for the other cases. The following table summarizes the possibilities:

(a) Write a well-structured flowchart for a subroutine procedure to calculate r and θ as a function of x and y. Express the final results for θ in degrees.

(b) Write a well-structured function procedure based on your flowchart. Test your program by using it to fill out following table:

Develop a well-structured function procedure that is passed a numeric grade from 0 to 100 and returns a letter grade according toscheme:

Develop a well-structured function procedure to determine

(a) The factorial;

(b) The minimum value in a vector; and

(c) The average of the values in a vector.

(a) The factorial;

(b) The minimum value in a vector; and

(c) The average of the values in a vector.

Develop well-structured programs to

(a) Determine the square root of the sum of the square of the elements of a two-dimensional array (i.e., a matrix) and

(b) Normalize a matrix by dividing each row by the maximum absolute in the row so that the maximum elements in each row is 1.

(a) Determine the square root of the sum of the square of the elements of a two-dimensional array (i.e., a matrix) and

(b) Normalize a matrix by dividing each row by the maximum absolute in the row so that the maximum elements in each row is 1.

Convert the following base-2 numbers to base-10:

(a) 101101,

(b) 101.101,

(c) 0.01101.

(a) 101101,

(b) 101.101,

(c) 0.01101.

Compose your own program based on Figure and use it to determine your computer’s machine epsilon.

In a fashion similar to that in Figure, write a short program to determine the smallest number, xmin, used on the computer you will be employing along with this book. Note that your computer will be unable to reliably distinguish between zero and a quantity that is smaller than this number.

The infinite series converges on a value of ƒ (n) = π^{4}_{ }/ 90 as n approaches infinity.

Write a program in single precision to calculate ƒ (n) for n = 10,000 by computing the sum from i = 1 to 10,000. Then repeat the calculation but in reverse order-that is, from i = 10,000 to 1 using increments of – 1.In each case, compute the true percent relative error. Explain the results.

Evaluate e-5 using two approaches

And

And compare with the true value of 6.737947 x 10^{-3}.Use 20 terms to evaluate each series and compute true and approximate relative errors as terms are added.

The derivative of ƒ (x) = l/ (l – 3^{x2})^{2} is given by 6x / (1 – 3x^{2})^{2}. Do you expect to have difficulties evaluating this function at x = 0.577? Try it using 3-and 4-digit arithmetic with chopping.

(a) Evaluate the polynomial y = x^{3} – 7x^{2} + 8x – 0.35, at x = 1.37. Use 3-digit arithmetic with chopping. Evaluate the percent relative error.

(b) Repeat (a) but express y as y = ((x – 7) x +) x – 0.35. Evaluate the error and compare with part (a).

Calculate the random access memory (RAM) in megabytes necessary to store a multidimensional array that is 20 x 40 x 120.This array is double precision, and each value requires a 64-bit word. Recall that a 64-bit word = 8 bytes and 1 kilobyte = 2^{10} bytes. Assume that the index starts at 1.

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