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- 1. For the function a). f (x) = x³ + 2a2 - rewrite it as x = g(x) and write a program that uses the Fixed point method to iterate to the solution starting at xo = 1.5. Use the following for g(x) (c) g(x) = x+ f/3 (d) g(a) = (x³ – f)/x? (e) g(x) = (2x? - f)/(2x) %3D %3D -PYTHON: Write a program (without using built-in functions from scipy, etc) to find the roots of the equation: x - 1.5 - 0.5sin(x) = 0 by the Newton-Raphson method. Solve for roots within the interval 0 < x < 2pi. Plot the error as a function of iteration number to see how quickly it converges.Assignment No. 1 (contd.) You will use Newton-Raphson method to find the eigenvalues. So we rewrite the equations as: fever (E) = VE + Vo sin (a VE+ Vo) V-E cos (a VE + Vo ) = o fodia( E) = VE + Vo cos (a VE+ Vo) + v-E sin (a VE + Vo) = 0 COS To find the eigenvalues of the even states, you must find the zeros of fever (E) and for the odd states you must find the zeros of fodd(E). To use the Newton-Raphson method, you will also need to find the deriva- tives of these functions. Write a MATLAB code to implement the Newton-Raphson Method. Now comes the need for physical insight. Recall three conclusions from your Phy 301 course: (1) For symmetric systems, the lowest eigenstate is always an even state, and (2) The eigenstates alternate between even and odd states, and (3) the bound state eigenvalues lie between 0 and -Vo. So first find the lowest eigenvalue Eo by solving ferer(E) = 0 and your starting guess should be close to -Vo. To find the next eigenvalue Ej solve fodad(E) = 0 and your…
- Write a pseudocode for an algorithm for finding real roots of equation ax2 + bx + c = 0 for arbitrary real coefficients a, b, and c. (You may assume the availability of the square root function sqrt(x).)?2. There is a more efficient algorithm (in terms of the number of multiplications and additions used) for evaluating polynomials than the conventional algorithm described in the previous exercise. It is called Horner's method. This pseudocode shows how to use this method to find the value of anx" + an-1xn ++ a₁x + ao = 0 at x = c. procedure Horner(c, ao, a₁, a2,..., an: real numbers) y := an for i:=1 to n y := y*c+an-i return y(y = anc" +an-1c"1+...+ a₁c + ao} 2.a Evaluate 3x² + x + 1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. 2.b Exactly how many multiplications and additions are used by this algorithm to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)Using Matlab. Monte Carlo methods are a class of computational methods that rely on repeated random sampling to compute their results. In this problem, you will use the Monte Carlo method to estimate the temperature at an arbitrary point in a uniform solid. Specifically, we will consider the solid object shown below. The temperatures of all surfaces are 500 des F with the exception of the two shaded surfaces, which are at 0 deg F. Inputs to the program should be: 1. The number of random walkers. 2. The (x,,2) coordinates of the point at which the temperature is to be estimated. 3. Plot interval (eg., you may want 1,000,000 random walkers in the simulation but may want to plot after every 10,000). Output from the program should be: 1. A graph of the estimated temperature, plotted according to the plot interval 2. Numerical value of the estimated temperature. USING MATLAB
- b) You are given the task of analyzing how joyful a person is. If you are given a list of numbers that represent the emotional value of an individual on each day, design a divide and conquer algorithm to find the most joyous interval of the person. The measure of joy is given as sum of the values in interval multiplied by the smallest integer in the interval.You are working on a problem where the size of each test case is between 1 and 100 integers and where the timeout is 4 seconds.You have developed a cubic time algorithm for the problem that gives the correct answeron all test cases.If you submit that solution, will you pass all test cases in time?Please answer YES or NO and then briefly explain your answer.Write a Python (or R) program that asks the user to enter an integer (X), then: Determines if X is prime or not • If X is not prime, compute and print the factors of that integer X • Evaluate and print the equation Y=8X²+ 1, for X values from -5 to 5 using the range function and for loop
- The velocity v and the falling distance d as a function of time of a skydiver that experience the air resistance can be approximated by: and d() = " in cosh :) kg v(t) = mg tanh where k = 0.24 Kg is a constant, m is the skydiver mass, g = 9.81 " is the acceleration due to gravity, and t is the time in seconds since the skydiver starts falling. Write m MATLAB code to determine the velocity v and the falling distance d at t = 8s for a 95kg skydiver. • Note: Velocity v and distance d are each worth 50% of the total points for this problem. Script e C Reset I MATLAB Docume 1 % Define variable(s) here. 3 v = % write the equation for v here, do not change the variable name. 4 d = % write the equation for d here, do not change the variable name.Start with a pile of n stones and successively split a pile into two smaller piles until each pile has only one Each time a split happens, multiply the number of stones in each of the two smaller piles. (For example, if a pile has 15 stones and you split it into a pile of 7 and another pile of 8 stones, multiply 7 and 8.) The goal of this problem is to show that no matter how the pile of n stones are split, the sum of the products computed at each split is equal to n(n - 1)/2. Using strong mathematical induction, prove that no matter how the pile of n stones are split, the sum of the products computed at each split is equal to n(n - 1)/2.ما 2. Write a program that implements the first order (linear) interpolations.. 3. Use the program to apply first order (linear) interpolation to the function f(x)=sin(x²) initially calculated in 10 uniform points in the interval [0.0, 5.0]. (1) 4. Write a program that implements n-point Lagrange interpolation. Treat n as an input parameter. 5. Use third and fifth order polynomial interpolation to interpolate the function f(x) Equation (1) above.