Reference no: EM132463556

SEHS4612 - Numerical Methods for Engineers - The Hong Kong Polytechnic University

Problem 1

(a) Convert (137.65625)₁₀ to base-2.

(b) Convert (10111.0110101)₂ to base-10.

(c) A hypothetical computer stores real numbers in floating point format in 9-bit words. The first bit is used for the sign of the number, the second bit for the sign of the exponent, the next two bits for the magnitude of the exponent, and the next five bits for the magnitude of the mantissa. Find the 9-bit format of the number 0.3624.

(d) The force of attraction F between two particles with masses m₁ and m₂, being distance r apart, is given by F = Gm₁m₂/r² + d

where G is a known constant and d is the experimental error. Suppose now the maximum errors are 2% in the measurements of each mass and 1% in the measurement of the distance between the particles. Also, in obtaining the value of d, it is expected that a maximum error of 4% will be occurred. Find the maximum propagation percentage error in evaluating the force F.

Problem 2

(a) Apply the Naive Gaussian elimination method to solve the system and use 4 decimal places for all numerical values.

(b) Apply the Gaussian elimination method with partial pivoting to solve the system and use 4 decimal places for all numerical values.

Problem 3

(a) Let s be the root of x = 2 cos x.

(i) Verify that s lies in the interval (Π/4, Π/3) .

(ii) Based on the fact that x ≈ Π/4, explain why the fixed-point iteration formula

x_i+1 = 2 cos x_i

is not suitable for finding the root s.

(iii) Suggest ONE suitable fixed-point iteration formula for finding s. Please give reasons.

(b) If r is a root of the equation f (x) = 0 and suppose that f (x) is di?erentiable infinitely many times. Show that if f′(r) ≠ 0 and if the Newton-Raphson formula

x_i+1 = x_i - f(x_i)/f′(x_i)  (n = 0, 1, 2, . . .)

converges to r, then the order of convergence of the iteration must be at least 2.

(c) Apply the Newton-Raphson method to find the root s in part (a) using the starting value x₀ = π/4.

Problem 4

(a) Suppose that the bisection method is used to find a zero of f (x) in the interval [14, 19]. How many times must this interval be bisected to guarantee that the approximated root has at least an accuracy of 1/2 × 10^-8?

(b) Figure 1(a) shows a uniform beam subject to a linearly increasing distributed load. The equation for the resulting elastic curve is (see Figure 1(b))

y = w₀/120EIL(- x⁵ + 2L²x³ - L⁴x)  (1)

where L = 600 cm, E = 50, 000 kN/cm², I = 30, 000 cm⁴, and w₀ = 2.5 kN/cm.

(i) Determine the point of maximum deflection (i.e., the value of x where dy/dx = 0) by using four iterations of the bisection method with two initial guesses of x_l = 200 and x_u = 400. Hence, find the relative approximate error for the second to fourth iterations.

(ii) Use the result of (b)(i) to determine the value of the maximum deflection.

Figure 1

(c) Employ the secant method to determine the value of the maximum deflection, using two initial guesses of x_-1 = 220 and x₀ = 240. Perform the computation until the relative approximate error is less than 0.2%. Please use 4 decimal places for all numerical values.

Problem 5

(a) Find the first four nonzero terms of the Maclaurin Series for

f (x) = (1 + x)^-1/2

(b) Use the result of (a) to find the first four nonzero terms of the Maclaurin Series for

sinh^-1 x = ₀∫^x dt/√(1+t²)

(c) Use the first three terms of the series in (b) to estimate sinh^-1 (0.45). Please correct your answer in four decimal places.

(d) Use the remainder formula to find an upper bound for the magnitude of the estimation error. Please correct your answer in four decimal places.