(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation
For example, if
a. Using the Newton-Raphson method, find the two roots of the equation
b. Extend the
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C++ for Engineers and Scientists
- Computer Science f(x)= Sin(x) – x 3 + C Where C = Max [0.15, (69/100)] x20, (x is in radian). Apply the false position method three iterations to find the root of the function (Decide the starting values of Xl and Xu yourself)arrow_forwardA vertical plate is partially submerged in water and has the indicated shape. 4 m 12 m- Express the hydrostatic force (in N) Enter a number. He of the plate as an integral (let the positive direction be upwards) and evaluate it. (Round your answer to the nearest whole number. Use 9.8 m/s for the acceleration due to gravity. Recall that the weight density of water is 1,000 kg/m3.) pg dy = Narrow_forwardNumerical Methods Lecture: The system of nonlinear equations given below x0 =y0=1 Obtain 4 iteration Newton-Raphson solutions starting with the estimated initial values.Discuss the digit precision in the accuracy of the final solutions. x = y+x2-0.5 y = x2-5xyarrow_forward
- (log n) = o(n) n° = o(n*+!) " = o((c+1)") (n/2)(m/2) = o(n!) n! = o(n"). %3D %3D (a) Rank the following functions based on their asymptotic value in the increasing order, i.e., list them as functions f1, S2, Sa, ..., S9 such that fi = 0(f2), S2 = 0(3),..., Js = 0(f9). Remember to write down your proof for each equation Si = 0(Si+1) in the sequence above. log n nlog n 100n 2 n! log narrow_forwardA) Given a real number x and a positive integer k, determine the number of multiplications used to find x2k starting with x and successively squaring (to find x2, x4, and so on). Is this a more efficient way to find x2k than by multiplying x by itself the appropriate number of times?arrow_forward(a) The Gregory-Leibniz formula for 7 dates from the 1670s (but was apparently known to Indian mathematician Madhava of Sangamagrama around 1400), and states that ㅠ 4 ㅠ 2 π=3+ = 3 + - 1 2 2 X 1 - 1 Write a function pi_gl(n) which takes as an input an integer n and returns an approximation of , by evaluating the Gregory-Leibniz formula with n terms. (For n=5 your function should return 3.3396825396.... Don't forget to multiply the sum of the series by 4.) (b) Another series expression for π, less well-known and discovered by Nikalantha (around 1500), is X + 3 5 4 2 × 3 × 4 1 3 4(−1)²+1 2i (2i + 1)(2i + 2) X 1 1 + 7 9 4 4× 5 × 6 4 5 Write a function pi_nik(n) which takes as an input an integer n and returns an approximation of , by evaluating this formula with n terms. (For n=3 your function should return 3.1333333....) (c) A third expression is due to Wallis (1656): X + 6 5 + -1) ². + ... 1 2i+1 X 4 4 6 × 7 ×8 8 × 9 × 10 +... X... X 2i 2i X 2i 1 2i + 1 X... Write a function pi_wallis…arrow_forward
- (please type answer fast.)arrow_forwardSolve part e and attach the output of the codearrow_forward(b) Let x = 1e16 (a very large number). How big is the "gap" nextfloat(x)-x? (c) The "gap" at 1.0 is like 1e-16, a small number. How small is the gap at 0.0. Answer with just the exponent (which would be -16 were we talking about 1.0).arrow_forward
- V:38) Given the following equations f(x) = x4 - x - 10 = 0. Determine the initial approximations for finding the smallest positive roots. Hence , use the secant method to find the root correct to three decimal places.arrow_forward(c) A third expression is due to Wallis (1656): ㅠ 2 2 4 4 6 6 ² = ( ² × ²3 ) × ( ² × ¦ ) × ( ÷ × ;) 2 1 3 5 5 7 X... X 2i 2i X 2i 1 2i + 1 :) X... Write a function pi_wallis (n) which takes an integer n as input and returns an approximation of by evaluating the Wallis formlua with n brackets in the formula. (For example, pi_wallis (3) should return 2.9257142857....)arrow_forwardA circle in the XY-coordinate system is specified by the center coordinates (x, y) and radius (r). Read the values for 2 circles- x1, y1, r1 for C1 and x2, y2, r2 for C2. (i) Determine whether the 2 circles intersect. To solve the problem it suffices to check if the distance between the 2 centers is lesser than the sum of radii of the 2 circles. (ii) Find the smallest circle that encloses the two circles and return its center coordinates and radius. programming language - carrow_forward
- C++ for Engineers and ScientistsComputer ScienceISBN:9781133187844Author:Bronson, Gary J.Publisher:Course Technology Ptr