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mathematics
numerical analysis

Questions and Answers of Numerical Analysis

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
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) =
(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
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?
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
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
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
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
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 =
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
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
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,
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
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 m3 and gives out about 80 W of heat (l W = l J/s). Calculate the air temperature
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
The volume flow rate through a pipe is given by Q = ?A, where ? 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
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
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, FU = – c’v2 where c’ = second-order drag
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
In our example of the free-falling parachutist, we assumed that the acceleration due to gravity was a constant value of 9.8 m/s2. Although this is a decent approximation when we are examining falling
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
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
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
Suppose that a spherical droplet of liquid evaporates at a race that is proportional to its surface area.dV/dt = – kAWhere V = volume (mm3), t = time (h), k = the evaporation rater (mm/hr), and A =
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
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
A fluid is pumped into the network shown in Figure. 1f Q2 = 0.6, Q3 = 0.4, Q7 = 0.2, and Q8 = 0.3 m3/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 indentationDOi = i + 1IF z > 50 EXITx = x + 5IF x > 5 THENy = xELSEy = 0ENDIFz = x + yNDDO
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
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
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
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
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
The divide and average? method, an old-time method for approximating the square root of any positive number ? can be formulated as x = x + a/x / 2. (a) Write well-structured pseudosode to implement
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
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
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 =
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
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
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
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 coordinatesThe
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.
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
Convert the following base-2 numbers to base-10:(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
The infinite series converges on a value of ? (n) = ?4?/ 90 as n approaches infinity. Write a program in single precision to calculate ? (n) for n = 10,000 by computing the sum from i = 1 to
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 – 3x2)2 is given by 6x / (1 – 3x2)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 = x3 – 7x2 + 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
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

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