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Q1: (6) The sequence {an} is defined by a1 = 1 and a_{n+1}=\left(2 a_{n}+1\right)^{\frac{1}{2}}-\frac{1}{4}\left(2 a_{n}+1\right)^{\frac{1}{4}}, \quad n=1,2, \ldots Show that the sequence is convergent.See Answer
Q2:3. Let f(x)=x²-6. With Po=3 and p₁=2, find p3. a. Use the Secant method.See Answer
Q3:8. Use the Secant method. b. Use the method of False Position. 5. Use Newton's method to find solutions accurate to within 10 for the following problems. a. b. C. d. ¹-2x²-5-0, [1,4] x-cos x = 0, [0, π/2] x+3x²-1=0. [-3,-2] x 0.8-0.2 sin x = 0, [0, π/2]See Answer
Q4:= 1. Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. a. x f(x) f'(x) 0.5 0.4794 0.6 0.5646 0.7 0.6442 0.0 0.2 0.4 f(x) f'(x) 0.00000 0.74140 1.3718See Answer
Q5:a. f(x) = 2 cos 2x -x b. f(x)=x² Inx+1 5. Use the most accurate three-point formula to determine each missing entry in the following tables. a. x f(x) f'(x) b. x f(x) f'(x) 1.1 9.025013 1.2 11.02318 1.3 13.46374 1.4 16.44465 8.1 8.3 8.5 8.7 16.94410 17.56492 18.19056 18.82091See Answer
Q6:1. If an integer n equals 2.k and k is an integer, then n is even. O equals 2.0 and O is an integer.See Answer
Q7:3. Consider the following algorithm segment: if x.y > 0 then do y := 3. x x := x + 1 end do z :=x.y Find the value of z if prior to execution x and y have the values given below. a. x = 2, y =3 b.x = 1, y = 1See Answer
Q8:1. Solve the following system of linear equations by hand, employing Cramer's rule. Verify your solution employing x = inv(A)*d and x=A\d. x12x2 + 3x3 = 0 -4x1 + x2 - 2x3 = -1 -3x16x2 + 2x3 = 2See Answer
Q9:2. Given the equations 10x1 + 2x2x3 = 27 -3x16x2 + 2x3 = -123/2 x1 + x2 + 5x3 = -43/2 Solve, by hand, using Gauss elimination. Show all steps of the computation. Verify using the function gauss.See Answer
Q10:3. Given the equations 2x2 + 3x3 = 8 4x1 + 6x2 + 7x3 = -3 2x13x2 + 6x3 = 5 Solve, by hand, using Gauss elimination with partial pivoting. Show all steps. Verify using the function gauss_pivot.See Answer
Q11:4. Find, by hand, the exact solution for the system 0.0003x1 + 3x2 = 2.0001 x1 + x2 = 1 Then, solve it using gauss and gauss_pivot. Set format to long. Which function is more accurate to use? Why?See Answer
Q12:6. Apply the Jacobi and guass_seidel functions to solve the system of equations in Problem 2. Assume the initial guess is [111]¹.See Answer
Q13:1. Bessel functions often arise in engineering analysis such as the study of electric filters. The following table provides 4 uniformly spaced samples for the zero-order Bessel function of the first kind. X J1(x) 1.8 0.5815 2.0 0.5767 2.2 0.5560 2.4 0.5202 generate the third-order Newton's interpolating polynomial. Use the newtintr function to interpolate at x = 2.1. Validate your answer using Matlab's polyfit and polyval functions.See Answer
Q14:2. Ohm's law states that the voltage, resistance and current in a resistor are related by the linear equation V=RI. In practice, however, real resistors may not always obey this law. Suppose that you performed a precise experiment in the lab and obtained the following data for the resistor. I V -2 -637 -1 -96.5 -0.5 -20.5 0.5 20.5 1 96.5 2 637 Use a fifth-order polynomial to fit the data and compute V for 1 = 0.1. Employ function newtintr. Next, use Matlab's polyfit to generate the interpolation polynomial. Plot the data points and the polynomial, on the same graph, for x-axis limits [-2.2 2.2].See Answer
Q15:X J1(x) 1.8 0.5815 -2 -637 2.0 0.5767 generate the third-order Newton's interpolating polynomial. Use the newtintr function to interpolate at x = 2.1. Validate your answer using Matlab's polyfit and polyval functions. -1 -96.5 2.2 0.5560 2. Ohm's law states that the voltage, resistance and current in a resistor are related by the linear equation V=RI. In practice, however, real resistors may not always obey this law. Suppose that you performed a precise experiment in the lab and obtained the following data for the resistor. -0.5 -20.5 2.4 0.5202 0.5 20.5 1 96.5 2 637 Use a fifth-order polynomial to fit the data and compute V for 1 = 0.1. Employ function newtintr. Next, use Matlab's polyfit to generate the interpolation polynomial. Plot the data points and the polynomial, on the same graph, for x-axis limits [-2.2 2.2]. 1 3. Consider the function f(x) = 1+25x2 Generate five equidistantly spaced points (xi, f (xi)) over the interval: [-1 1]. Fit this data with (a) a fourth-order polynomial, (b) a linear spline and (c) a cubic spline. Assume spline derivatives are zero at the end points. Compare your results graphically (use the x-axis range [-1 1]).See Answer
Q16: EGN 3433 NAME The concentration of a drug in a person's stomach can be described with the following linear ODE. Where ● dc dt = - KA, (c-cs) c is the concentration of the drug in the stomach k is the mass transfer transfer coefficient (4.1) A, is the ratio of the surface area of the stomach lining to its volume Sep 14, 2023 B. What is the forcing function? c, is the concentration of the drug in the blood stream, which can be a function of time. A. What are the independent and the dependent variables? dy C. Rewrite the ODE in the form: t + y = f(t) where y and x are the dependent and dx independent variables, respectively, Given that, k = : 0.9 m/s, A, = .01 m¯¹, what is the value T? Use correct units! 1 EGN 3433 Sep 14, 2023 D. If at time zero, c = 12 mg/ml, what is the amount of time needed for the concentration of the drug in the stomach to be 3 mg/mL? Use separation of variables. Here you can assume, that Cs = .01 mg/mL. In the problem, k 0.9 m/s, A. = .01 m¯¹. = E. Let's say that c¸ is not a constant, but slowly increases with time. For instance, mg cs(t) = 0.001- t +0.002. ml. s mg ml Can you use separation of variables to solve? Why or why not? Suggest an alternative technique that you can use to solve. 2See Answer
Q17:Problems Problem 1. Consider the matrix 1/√2 -1/√2] 5/2 7/2 L-1/√2 7/2 5/2 5 A = 1/√2 a) Find a Householder transformation H such that B = HAHT is a tridiagonal matrix. This involves some bookkeeping, so it may help to use the aid of a computer for calculating the linear algebra in steps that you describe. b) Use Gershgorin's second theorem in combination with a similarity transformation to estimate the spectrum of A. Show that all eigenvalues are distinct and describe the location of the eigenvalues as accurately as you can. Hint: The matrix [100] 0 a 0 0 0 1 combined with B for a suitable a > 0 may be helpful. Ta = c) Computer exercise: Approximate the largest eigenvalue of A using power iteration and approximate the other two eigenvalues with inverse iteration (a shift is needed for the middle eigenvalue). Draw a random vector 2) as start vector (for instance, with the components being independent, standard normal distributed). Include your implementation code and the output of 10 iterations for each approximation.See Answer
Q18:The problem: a) Consider the function f(x,y: 3) = (x - ₁)² + (y - 3₂)² - Bwith arguments (x, y) = R² and parameters BER³. Given 3, the level set of all points (x, y) such that f = 0 describes the circle centered at (81, 82) and with radius [33]. We are given the following list of 20 noisy measurements of points on a circle (and the measurements be downloaded as a mat file from this link) y -2.3073 -3.5569 -1.6627 -4.5479 4.4413 -0.1823 -2.1021 -0.7938 -2.0460 -0.3176 5.1864 -3.9465 -2.6359 -4.3716 -1.0931 1.9392 4.7061 -3.2146 4.9005 -2.3397 0.5035 0.9895 0.5666 -5.9482 3.3504 -4.7778 1.4214 0.7869 1.7892 1.2574 -1.1229 -4.4437 0.3167 -6.1661 0.6024 4.7246 4.9007 0.7662 -1.2589 -3.6789 and we seek to find the circle 3 such that f(MB) = 0 (5) holds in least squares sense. Describe the function R(3) for the nonlinear least squares problem for determining this circle and compute its Jacobian JR(3) € R20x3 i = 1,..., 20 Extracting data points from file: For part b) and c) of this exercise. If you prefer to download the "circle-measurements.mat" from the aforementioned link rather than copying the above measurements, then, assuming you store the down- loaded file in the same folder as you run Matlab or Pyton from, you can open it and extract its information by the command load('circle-measurements.mat') in Matlab, and by the following commands in Python: import scipy.10 measurement Data = scipy.io. loadnat ('circle-measurements. mat ³) x = measurement Data ['x'] y measurement Data [¹y¹]See Answer
Q19:Problem 3. Interpolation and numerical integration in 2D. a) Let Pn,n = {Σ-o-j¹|aj R 0≤ij≤n} denote the set of 2-variate polynomials in z and y that are of degree ≤n in z and of degree ≤n in y. For a given 2D square [0, 1]2 with a square mesh (r.. y) = (ih, jh) for 0 ≤i, j≤n and h = 1/n, and f = C([0, 1]2), consider the interpolation problem: find p € Pn,n such that (6) P(Z₁yj) = f(x,y) for all 0≤i,j≤n. We recall from the lectures that this problem has a solution 12 12 p(z,y)=f(x,ye)Lx (T) Le(y), k-0 1-0 where Le (2) is a univariate polynomial in z of degree n and Le(y) is a univariate polynomial of degree n in y. Describe L₁ (2) and (y) and show that p(x, y) in (7) with the functions L and Le that you define indeed is a solution of (6). b) Leaning on the fact that any univariate polynomial p of degree ≤n that has n+1 or more zeros is equal to the 0 function, meaning p = 0, show that the solution to the 2 dimensional interpolation problem (6) is unique. Hint: Let q € Pn, be another solution and consider the polynomial r = p-q- Then r € Pn,n, so it can be written 12 n r(z,y) = Σcjz¹y 1-0 j-0/nfor some c₁ € R. For any fixed ye, for 0 ≤ ≤n, the function r(x, y) is a univariate polynomial in z of degree ≤n, namely, n 12 r(x,y) = Σ(Σ)n - Σαμπ = (v². j-0 1-0 -(Me) How many zeros does r(x, y) have, what does this imply about the coefficients a(ye) for all 0 ≤ i ≤n, and how can this be used to show that c = 0 for all 0 ≤i, j≤n? c) The composite trapezoidal rule in 2D on the square mesh presented above can be described as follows: 12 [.. f(x, y)dx dy ≈ [Ë"*" ["“_ p., (x,y) dx dy =: T(n), 1-1-1-1 where pij € P₁,1 denotes the unique polynomial that goes through f in the four points (-1,3-1), (Fi, Yj-1), (Fi-1, y₁) and (zi, yj). Determine pij(x,y) and verify that the 2D square-mesh trapezoidal rule is given by 12 T(n)=ΐΣΣ(f(-1,31 si−1, Yj−1) + ƒ(zi, Yj−1) + f(x−1, 9j) +; i-1 j-1 + f(24.3))). d) For na = 2¹+*, 8 = 1,2,..., 7, compute T(n.) to estimate the integral I 1 = exp(-(x - sin(y²))³3) dr dy. Since we do not know the alue of I, we approximate the error E(8) = |I-T(n,), using the pseudo-reference solution I := T(210). Estimate numerically the order of convergence r in E(n)= en+O(n-(r+1)) for example, through estimating the slope of the curve (log(n,), log(E(n.)) in a plot, or by studying the ratio log(E(n¸)) – log(E(n,-1)) log(n.) -log(n-1) which typically becomes more accurate for larger values of s. (See this link for more on using loglog plots for convergence estimates.)See Answer
Q20:EXERCISE SET 4.1 1. Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. a. f(x) f'(x) 0.5 0.4794 0.6 0.5646 0.7 0.6442 b. x f(x) | f'(x) 0.0 0.00000 0.2 0.74140 0.4 1.3718 2. Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. a. x f(x) | f'(x) -0.3 1.9507 -0.1 2.0421 -0.1 2.0601 b. xf(x) f'(x) 1.0 1.0000 1.2 1.2625 1.4 1.6595 3. The data in Exercise I were taken from the following functions. Compute the actual errors in Exercise 1, and find error bounds using the error formulas. a. f(x) = sinx b. f(x)=-2x²+3x-1 4. The data in Exercise 2 were taken from the following functions. Compute the actual errors in Exercise 2 and find error bounds using the error formulas. a. f(x)=2 cos 2x-x b. f(x) = x²lnx+1 5. Use the most accurate three-point formula to determine each missing entry in the following tables. a. x f(x) f'(x) b. x f(x) f'(x) 1.1 9.025013 8.1 16.94410 1.2 11.02318 8.3 17.56492 1.3 13.46374 8.5 18.19056 1.4 16.44465 8.7 18.82091 c. f(x) f'(x) d. f(x) f'(x) 2.9 -4.827866 2.0 3.6887983 3.0 -4.240058 2.1 3.6905701 3.1 -3.496909 3.2 -2.596792 2.2 3.6688192 2.3 3.6245909See Answer

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