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3 years ago

Assignment #4

Assignment #4

Assignment

AER336: SCIENTIFIC COMPUTINGAssignment #41. Using either the C, C++, or FORTRAN programming languages, write a computer programthat uses Newton’s method to solve the following nonlinear system of equations:3x 1 − cos(x 2 x 3 ) − 1 2 = 0 ,x 2 1 − 81(x 2 + 0.10) 2 + sin x 3 + 1.06 = 0 ,e −x 1x 2+ 20x 3 + 10π − 3 = 0 .3Start with an initial approximation (k = 0) of⎡x (0) ⎤ ⎡ ⎤x (0) 1= ⎢⎣ x (0)0.10⎥2 ⎦ = ⎢ ⎥⎣ 0.10 ⎦ ,x (0) −0.103and iterate until ||x (k) − x (k−1) || ∞ < 10 −6 . Use Cramer’s rule to solve the system of linearequations for each step or interation, k. Plot the values of x 1 , x 2 , and x 3 as a function of k.Confirm that the solution is given by2. The nonlinear system of equationsx ≈⎡⎢⎣0.500.00−0.52359877⎤⎥⎦ .3x 1 − cos(x 2 x 3 ) − 1 2 = 0 ,x 2 1 − 625x 2 2 − 1 4 = 0 ,e −x 1x 2+ 20x 3 + 10π − 3 = 0 .3has a singular Jacobian matrix at the solution. Using your computer program of problem 1,apply Newton’s method with⎡x (0) ⎤ ⎡ ⎤x (0) 1= ⎢⎣ x (0)1⎥2 ⎦ = ⎢ ⎥⎣ 1 ⎦ ,x (0) −13and show that convergence may be slow or not occur within a reasonable number of iterations.3. Finally, using either the C, C++, or FORTRAN programming languages, write a secondcomputer program that uses Newton’s method to find a local minimum, x = [x, y, z] T , of theodjective function, f(x, y, z), given byf(x, y, z) = x 2 + 2y 2 + z 2 − 2xy + 2x − 5 2 y − z + 2exyz .Assignment #4 — Page 1 of 2

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