Question

Numerical Analysis: Apply the BFGS Method to minimize the
function f(x) = x_{1}^{2} - 2x_{1}x_{2
+} 4x_{2}^{2} with the starting point
x_{0} = [-3,1]^{T}. Thanks!

Answer #1

If we want to minimize a function f(x) = e^(x^2)
over R, then it is equivalent to finding the root of f '(x).
Starting with x0 = 1, can you perform 4 iterations of Newton's
method to estimate the minimizer
of f(x)? (Correct to four decimal places at each iteration).

Suppose we want to apply Newton’s method to solving f(x) =
0where f is such that |f′′(x)| <10 and |f′(x)|> 2 for all x.
How close must x0 be to τ for the method to converge?

Suppose we want to apply Newton’s method to solving f(x) =
0where f is such that |f′′(x)| < 10 and |f′(x)| >2 for all x.
How close must x0 be to τ for the method to converge?

Suppose we want to apply Newton’s method to solving f(x) = 0
where f is such that |f′′(x)| ? 10 and |f′(x)| ? 2 for all x. How
close must x0 be to τ for the method to converge?

find the slop of the function f(x)=5x2-1/x using the
analytical as well as the numerical method. find the values at x=10
x=5 and x=0.5

Consider the Rosenbrock function, f(x) = 100(x2 -
x12)2 + (1
-x1)2. Let x*=(1,1), the minimum of the
function.
Let r(x) be the second order Taylor series for f(x) about the
base point x*, r(x) will be a quadratic, and therefore can be
written:
r(x) = A11(x1-x1*)2
+ 2A12(x1-x1*)(x2-x2*) +
A22(x2-x2*)2 +
b1(x1-x1*) +
b2(x2-x2*) + c
Find all the coefficients in the formula for r(x) - What is
A11, A12, A22, b1,
b2, and c?

Apply Newton's Method to f and initial guess
x0
to calculate
x1, x2, and x3.
(Round your answers to seven decimal places.)
f(x) = 1 − 2x sin(x), x0 = 7

Problem: Let y=f(x)be a differentiable function
and let P(x0,y0)be a point that is not on the graph of function.
Find a point Q on the graph of the function which is at a
minimum distance from P.
Complete the following steps. Let Q(x,y)be a point on the graph
of the function
Let D be the square of the distance PQ¯. Find an expression for
D, in terms of x.
Differentiate D with respect to x and show that
f′(x)=−x−x0f(x)−y0
The...

Use Newton's method to estimate the two zeros of the function
f(x)=x^4−x−12. Start with x0=−1 for the left-hand zero and with
x0=1 for the zero on the right. Then, in each case, find x2 .
for the zero on the right. Then, in each case, find
x 2x2.

Numerical analysis problem, please show all steps thank you.
A four times continuously differentiable function f is given by
the following data: f(1.1)=2, f(1.3)=1.5, f(1.5)=1.2, f(1.7)=1.6.
Assume that |f ''''(t)|=<100 for 1.1<t<1.7. Find the
estimate for f '' (1.3). Give an error bound.

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