intro to numerical analysisThe programming question use Force 2.0(in Fortran 95).A copy of the source code you have written anda copy of the output produced by your program(screenshot is ok). all in one word doc.doc file
CSCI/MATH 2308: Assignment #3 Nov. 23rd, 2020For computer problems, make sure to submit a copy of the source code you have written anda copy of the output produced by your program. All computer programs should be writtenin Fortran95. Except for questions (1), (2) and (3), use the library software discussed inclass for the numerical solution of non-linear equations.
(a) Show that if for some iteration you get xk1 = x or xk = x, when you use the Secantmethod to solve a nonlinear equation having a root x, then xk+1 must also equal x.(b) Show that if for some iteration you get xk = x, when you use Newton’s method tosolve a nonlinear equation having a root x, then xk+1 must also equal x:
For what starting values, if any, will Newton’s method fail for the nonlinear systemx1 1 = 0; x1x2 1 = 0?
(a) Show that the iterative methodxi+1 =xi1f(xi) xif(xi1)f(xi) f(xi1)is mathematically equivalent to the secant method (discussed in class), for solving ascalar nonlinear equation f(x) = 0.(b) When implemented in nite precisionoating point arithmetic, what advantagesor disadvantages does the formula given in part(a) have compared with the formula forthe secant method discussed in class?Programming Questions
In neutron transport theory, the critical length of a fuel rod is determined by the rootsof the equationcot(x) =x2 12x:Find the smallest positive root of this equation. Plot the graph of the function on theinterval, [0, 5], to see where the curve cuts the x-axis, then use this information to obtainsuitable initial guesses.
The natural frequencies of vibrations of a uniform beam of unit length clamped on oneend and free on the other, satisfy the equationtan(x) tanh(x) = 1:Find the smallest positive root of this equation. Plot the graph of the function on theinterval, [0, 10], to see where the curve cuts the x-axis, then use this information to obtainsuitable initial guesses.v3.1
Bioremediation involves the use of bacteria to consume toxic wastes. At steady state, thebacterial density, x, and nutrient concentration, y, satisfy the equations
xy x(1 + y) = 0 xy + ( y)(1 + y) = 0;whereand are parameters that depend on various physical features of the system;typical values are= 5 and = 1. Find one solution to this problem with a non-zerobacterial density and one solution to this problem for which the bacterial population hasdied out. (More information on how to choose the initial guesses for the two cases will beprovided later).
The American mathematician, Edward Lorenz, derived a simple system of ordinary dier-ential equations describing buoyant convection in auid, as a crude model for atmosphericcirculation. At steady state, the convective velocity, x, temperature gradient, y, and theheatow, z, satisfy the equations(y x) = 0; rx y xz = 0; xy bz = 0;where the Prandtl number, r, the Rayleigh number, and b, are positive constants thatdepend on properties of theuid, the applied temperature gradient, and the geometry ofthe problem. Typical values Lorenz used are = 10, r = 28, and b = 8=3. Find threedierent solutions to this problem. (More information on how to choose the initial guessesfor the two cases will be provided later).v3.1
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