[Solved] MATH 151A Assignment 1

1. Consider the following non-linear equation:

f(x) := x² 0.7x = 0 on [0.5,1]

• Show that f(x) has exactly one root on [0.5,1] without solving the equation.
• Consider the bisection algorithm starting with the interval [0.5,1], i.e. consider [a₁,b₁] = [0.5,1] and p₁ = 0. Find the minimum number of iterations required to approximate the solution with an absolute error of less than 10⁵.

2. Suppose f(x) is continuous on [a,b], and f(x) [a,b] for any x [a,b]. Show that f has at least one fixed point on [a,b]. (Hint: use Intermediate Value Theorem.) 3. Given the following sequence:

(

p p₀ is given.

• Calculate p₁ and p₂ with p₀ = 3.
• Find all the possible limits of sequence (for all possible p₀).
• Show by definition that the given sequence is actually a sequence generated by Newtons method to find a solution of the equation x² 3 = 0.

4. Consider the following non-linear equation:

f(x) := x² 3 = 0 on [0,4]

• Write the formula for the next term in the sequence in terms of the two previousterms for the secant method p_n to solve the above equation, with the starting point at and p₁ = 3, and calculate p₂ and p₃.
• Calculate p₂ and p₃ in the sequence of the method of false position p_n to solve the above equation, with the starting point at and p₁ = 3.

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5. (Programming problem)

Solve the following equation by the following methods:

sin(x) x = 0 on [0,

• Write a complete program to implement the secant method to solve the equationwithin an accuracy of at least 10⁵ (using |p_np_n1|) with initial values p₀ and p₁ as and Plot the graph of the sequence {p_n} generated by the secant

method.

• Write a complete programme to implement the Newtons Method to solve the equation within an accuracy of at least 10⁵ (using |p_n p_n1|) with initial value p₀ as . Plot the graph of the sequence {p_n} generated by Newtons method.

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